Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 156786 8 pageshttpdxdoiorg1011552013156786

Research ArticleCayley Bipolar Fuzzy Graphs

Noura O Alshehri1 and Muhammad Akram2

1 Department of Mathematics Faculty of Sciences (Girls) King Abdulaziz University Jeddah Saudi Arabia2 Punjab University College of Information Technology University of the Punjab Old Campus Lahore Pakistan

Correspondence should be addressed to Muhammad Akram makrampucitedupk

Received 18 September 2013 Accepted 7 October 2013

Academic Editors M I Ali Y Cao and M Finger

Copyright copy 2013 N O Alshehri and M Akram This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We introduce the concept of Cayley bipolar fuzzy graphs and investigate some of their properties We present some interestingproperties of bipolar fuzzy graphs in terms of algebraic structures We also discuss connectedness in Cayley bipolar fuzzy graphs

1 Introduction

Graph theory is an extremely useful tool in solving thecombinatorial problems in different areas Point-to-pointinterconnection networks for parallel and distributed systemsare usually modeled by directed graphs (or digraphs) Adigraph is a graph whose edges have direction and are calledarcs (edges) Arrows on the arcs are used to encode thedirectional information an arc from vertex (node) 119909 to vertex119910 indicates that one may move from 119909 to 119910 but not from 119910

to 119909 The Cayley graph was first considered for finite groupsby Cayley in 1878 Max Dehn in his unpublished lectures ongroup theory from 1909 to 1910 reintroduced Cayley graphsunder the name Gruppenbild (group diagram) which ledto the geometric group theory of today His most importantapplication was the solution of the word problem for the fun-damental group of surfaces with genus which is equivalent tothe topological problem of deciding which closed curves onthe surface contract to a point [1]

The notion of fuzzy sets was introduced by Zadeh [2] asa method of representing uncertainty and vagueness Sincethen the theory of fuzzy sets has become a vigorous area ofresearch in different disciplines In 1994 Zhang [3] initiatedthe concept of bipolar fuzzy sets as a generalization of fuzzysets [4] A bipolar fuzzy set is an extension of fuzzy setswhose membership degree range is [minus1 1] In a bipolar fuzzyset the membership degree 0 of an element means thatthe element is irrelevant to the corresponding property themembership degree (0 1] of an element indicates that theelement somewhat satisfies the property and themembership

degree [minus1 0) of an element indicates that the elementsomewhat satisfies the implicit counterproperty

Kaufmannrsquos initial definition of a fuzzy graph [5] wasbased on Zadehrsquos fuzzy relations [2] Mordeson and Nair[6] introduced the fuzzy analogue of several basic graph-theoretic concepts Kauffman [5] defined the concept ofcomplement of fuzzy graph and studied some operations onfuzzy graphs Akram and Dudek [7ndash9] introducedmany newconcepts including bipolar fuzzy graphs complete bipolarfuzzy graphs regular bipolar fuzzy graphs and irregularbipolar fuzzy graphs Wu [4] discussed fuzzy digraphsShahzamanian et al [10] introduced the notion of rough-ness in Cayley graphs and investigated several propertiesNamboothiri et al [11] discussed Cayley fuzzy graphs Inthis paper we introduce the concept of Cayley bipolar fuzzygraphs and investigate some of their properties We presentsome interesting properties of bipolar fuzzy graphs in termsof algebraic structures We also discuss connectedness inCayley bipolar fuzzy graphs

2 Preliminaries

A digraph is a pair 119863lowast = (119881 119864) where 119881 is a finite set and119864 sube 119881 times 119881 Let 119863lowast

1= (1198811 1198641) and 119863

lowast

2= (1198812 1198642) be two

digraphsTheCartesian product of119863lowast1and119863

lowast

2gives a digraph

119863lowast

1times 119863lowast

2= (119881 119864) with 119881 = 119881

1times 1198812and

119864 = (119909 1199092) 997888rarr (119909 119910

2) | 119909 isin 119881

1 1199092997888rarr 1199102isin 1198642

cup (1199091 119911) 997888rarr (119910

1 119911) | 119909

1997888rarr 1199101isin 1198641 119911 isin 119881

2

(1)

2 The Scientific World Journal

In this paper we will write 119909119910 isin 119864 to mean 119909 rarr 119910 isin 119864 andif 119890 = 119909119910 isin 119864 we say 119909 and 119910 are adjacent such that 119909 is astarting node and 119910 is an ending node

The study of vertex transitive graphs has a long andrich history in discrete mathematics Prominent examplesof vertex transitive graphs are Cayley graphs which areimportant in both theory and applications for exampleCayley graphs are goodmodels for interconnection networks

Definition 1 Let 119866 be a finite group and let 119878 be a minimalgenerating set of 119866 A Cayley graph (119866 119878) has elements of 119866as its vertices the edge set is given by (119892 119892

119904) 119892 isin 119866 119904 isin 119878

Two vertices 1198921and 119892

2are adjacent if 119892

2= 1198921sdot 119904 where 119904 isin 119878

Note that a generating set 119878 is minimal if 119878 generates119866 but noproper subset of 119878 does

Theorem 2 All Cayley graphs are vertex transitive

Definition 3 Let (119881 lowast) be a group and let 119860 be any subset of119881 Then the Cayley graph induced by (119881 lowast 119860) is the graph119866 = (119881 119877) where 119877 = (119909 119910) 119909

minus1

119910 isin 119860

Definition 4 (see [2]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on119883times119883 By a fuzzy relationwemean a fuzzy binary relationgiven by 120583 119883 times 119883 rarr [0 1]

Definition 5 (see [11]) Let (119881 lowast) be a group and let 120583 be afuzzy subset of 119881 Then the fuzzy relation 119877 on 119881 defined by

119877 (119909 119910) = 120583 (119909minus1

lowast 119910) forall119909 119910 isin 119881 (2)

induces a fuzzy graph 119866 = (119881 119877) called the Cayley fuzzygraph induced by the (119881 lowast 120583)

Definition 6 (see [3]) Let 119883 be a nonempty set A bipolarfuzzy set 119861 in119883 is an object having the form

119861 = (119909 120583119875

119861(119909) 120583

119873

119861(119909)) | 119909 isin 119883 (3)

where 120583119875119861 119883 rarr [0 1] and 120583

119873

119861 119883 rarr [minus1 0] are mappings

We use the positive membership degree 120583119875

119861(119909) to denote

the satisfaction degree of an element 119909 to the propertycorresponding to a bipolar fuzzy set 119861 and the negativemem-bership degree 120583

119873

119861(119909) to denote the satisfaction degree of an

element 119909 to some implicit counterproperty correspondingto a bipolar fuzzy set 119861 If 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 it is the

situation that119909 is regarded as having only positive satisfactionfor 119861 If 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 it is the situation that 119909

does not satisfy the property of 119861 but somewhat satisfies thecounterproperty of119861 It is possible for an element119909 to be suchthat 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 when the membership function

of the property overlaps that of its counterpro perty over someportion of119883

For the sake of simplicity we shall use the symbol 119861 =

(120583119875

119861 120583119873

119861) for the bipolar fuzzy set

119861 = (119909 120583119875

119861(119909) 120583

119873

119861(119909)) | 119909 isin 119883 (4)

A nice application of bipolar fuzzy concept is a politicalacceptation (map to [0 1]) and nonacceptation (map to[minus1 0])

Definition 7 (see [3]) A bipolar fuzzy relation 119877 = (120583119875

119877(119909 119910)

120583119873

119877(119909 119910)) in a universe 119883 times 119884 (119877(119883 rarr 119884) for short) is a

bipolar fuzzy set of the form

119877 = ⟨(119909 119910) 120583119875

119877(119909 119910) 120583

119873

119877(119909 119910)⟩ | (119909 119910) isin 119883 times 119884 (5)

where 120583119875119877 119883 times 119884 rarr [0 1] and 120583

119873

119877 119883 times 119884 rarr [minus1 0]

Definition 8 Let 119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy equivalence relation on119883 if itsatisfies the following conditions

(a) 119877 is bipolar fuzzy reflexive that is119877(119909 119909) = (1 minus1) foreach 119909 isin 119883

(b) 119877 is bipolar fuzzy symmetric that is119877(119909 119910) = 119877(119910 119909)

for any 119909 119910 isin 119883(c) 119877 is bipolar fuzzy transitive that is 119877(119909 119911) ge

⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 9 Let 119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy partial order relation on 119883 ifit satisfies the following conditions

(a) 119877 is bipolar fuzzy reflexive that is 119877(119909 119909) = (1 minus1)

for each 119909 isin 119883(b) 119877 is bipolar fuzzy antisymmetric that is

119877(119909 119910) = 119877(119910 119909) for any 119909 119910 isin 119883(c) 119877 is bipolar fuzzy transitive that is 119877(119909 119911) ge

⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 10 Let119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy linear order relation on119883 if itsatisfies the following conditions

(a) 119877 is bipolar fuzzy partial relation(b) (120583119875

119877or 120583119875

119877minus1

)(119909 119910) gt 0 (120583119873119877

and 120583119873

119877minus1

)(119909 119910) lt 0 for all 119909119910 isin 119883

3 Cayley Bipolar Fuzzy Graphs

Definition 11 A bipolar fuzzy digraph of a digraph119863lowast is a pair

119863 = (119860 119861)where119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy set in119881 and

119861 = (120583119875

119861 120583119873

119861) is a bipolar fuzzy relation on 119864 such that

120583119875

119861(119909119910) le min (120583

119875

119860(119909) 120583

119875

119860(119910))

120583119873

119861(119909119910) ge max (120583119873

119860(119909) 120583

119873

119860(119910))

(6)

for all 119909119910 isin 119864 We note that 119861 need not to be symmetric

Definition 12 Let 119863 be a bipolar fuzzy digraph Theindegree of a vertex 119909 in 119863 is defined by ind(119909) =

(ind119875120583(119909) ind119873

120583(119909)) where ind119875

120583(119909) = sum

119910 = 119909120583119875

119860(119909119910) and

ind119873120583(119909) = sum

119910 = 119909120583119873

119860(119909119910) Similarly the outdegree

The Scientific World Journal 3

of a vertex 119909 in 119863 is defined by outd(119909)= (outd119875

120583(119909) outd119873

120583(119909)) where outd119875

120583(119909) = sum

119910 = 119909120583119875

119860(119909119910) and

outd119873120583(119909) = sum

119910 = 119909120583119873

119860(119909119910) A bipolar fuzzy digraph in which

each vertex has the same outdegree 119903 is called an outregulardigraph with index of outregularity 119903 In-regular digraphs aredefined similarly

Definition 13 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

the bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation119877 defined on 119881 by

119877 (119909 119910) = (120583119875

119877(119909minus1

119910) 120583119873

119877((119909minus1

119910))) forall119909 119910 isin 119881 (7)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 120583119875

119877 120583119873

119877)

We now introduce Cayley bipolar fuzzy graphs and provethat all Cayley bipolar fuzzy graphs are regular

Definition 14 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation 119877

on 119881 defined by

119877 (119909 119910) = 120583119875

119877(119909minus1

119910) 120583119873

119877(119909minus1

119910) forall119909 119910 isin 119881 (8)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 119860)

Example 15 Consider the group 1198853and take 119881 = 0 1 2

Define 120583119875

119860 119881 rarr [0 1] and 120583

119873

119860 119881 rarr [minus1 0] by 120583

119875

119860(0) =

120583119875

119860(1) = 120583

119875

119860(2) = 05 120583119873

119860(0) = 120583

119873

119860(1) = 120583

119873

119860(2) = minus04

Then the Cayley bipolar fuzzy graph 119866 = (119881 119877) induced by(1198853 + 119860) is given by Table 1 and Figure 1

We see that Cayley bipolar fuzzy graphs are actuallybipolar fuzzy digraphs Furthermore the relation 119877 in theabove definition describes the strength of each directed edgeLet 119866 denote a bipolar fuzzy graph 119866 = (119881 119877) induced by thetriple (119881 lowast 119860)

Theorem 16 The Cayley bipolar fuzzy graph 119866 is vertextransitive

Proof Let 119886 119887 isin 119881 Define 120595 119881 rarr 119881 by 120595(119909) = 119887119886minus1

119909 forall 119909 isin 119881 Clearly 120595 is a bijective map For each 119909 119910 isin 119881

119877 (120595 (119909) 120595 (119910)) = (119877120583119875 (120595 (119909) 120595 (119910))

119877120583119873 (120595 (119909) 120595 (119910)))

Now 119877120583119875 (120595 (119909) 120595 (119910)) = 119877

120583119875 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119875

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

119877120583119873 (120595 (119909) 120595 (119910)) = 119877

120583119873 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119873

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910) (9)

Therefore 119877(120595(119909) 120595(119910)) = 119877(119909 119910) Hence 120595 is an automor-phism on 119866 Also 120595(119886) = 119887 Hence 119866 is vertex transitive

Theorem 17 Every vertex transitive bipolar fuzzy graph isregular

Proof Let 119866 = (119881 119877) be any vertex transitive bipolar fuzzygraph Let 119906 V isin 119881 Then there is an automorphism 119891 on 119866

such that 119891(119906) = V Note that

ind (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119909 119906) 119877

120583119873 (119909 119906))

= sum

119909isin119881

(119877120583119875 (119891 (119909) 119891 (119906)) 119877

120583119873 (119891 (119909) 119891 (119906)))

= sum

119909isin119881

(119877120583119875 (119891 (119909) V) 119877

120583119873 (119891 (119909) V))

= sum

119909isin119881

(119877120583119875 (119910 V) 119877

120583119873 (119910 V))

= ind (V)

outd (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119906 119909) 119877

120583119873 (119906 119909))

= sum

119909isin119881

(119877120583119875 (119891 (119906) 119891 (119909)) 119877

120583119873 (119891 (119906) 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119891 (119909)) 119877

120583119873 (V 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119910) 119877

120583119873 (V 119910))

= outd (V) (10)

Hence 119866 is regular

Theorem 18 Cayley bipolar fuzzy graphs are regular

Proof Proof follows fromTheorems 16 and 17

Theorem 19 Let 119866 = (119881 119877) denote bipolar fuzzy graph Thenbipolar fuzzy relation 119877 is reflexive if and only if 120583119875

119860(1) = 1 and

120583119873

119860(1) = minus1

4 The Scientific World Journal

Table 1 119877(119886 119887) for Cayley bipolar fuzzy graph

119886 0 0 0 1 1 1 2 2 2119887 0 1 2 0 1 2 0 1 2(minus119886) + 119887 0 1 2 2 0 1 1 2 0119877 (119886 119887) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04)

(05 minus04)

(03 minus03)

(03 minus02)

(03 minus03)

(03 minus02)(03 minus02)

(05 minus04)

(03 minus02)(05 minus04)

0

1

2

G

Figure 1 Cayley bipolar fuzzy graph

Proof 119877 is reflexive if and only if 119877(119909 119909) = (1 minus1) for all 119909 isin

119881 Now

119877 (119909 119909) = (120583119875

119860(119909minus1

119909) 120583119873

119860(119909minus1

119909))

= (120583119875

119860(1) 120583

119873

119860(1)) forall119909 isin 119881

(11)

Hence 119877 is reflexive if and only if 120583119875119860(1) = 1 and 120583

119873

119860(1) =

minus1

Theorem 20 Let 119866 = (119881 119877) denote bipolar fuzzy graphThen bipolar fuzzy relation 119877 is symmetric if and only if(120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Proof Suppose that 119877 is symmetric Then for any 119909 isin 119881

(120583119875

119860(119909) 120583

119873

119860(119909)) = (120583

119875

119860(119909minus1

1199092

) 120583119873

119860(119909minus1

1199092

))

= 119877 (119909 1199092

) = 119877 (1199092

119909)

(since 119877 is symmetric)

= (120583119875

119860((1199092

)

minus1

119909) 120583119873

119860(1199092

)

minus1

119909)

= 120583119875

119860(119909minus2

119909) 120583119873

119860(119909minus2

119909)

= 120583119875

119860(119909minus1

) 120583119873

119860(119909minus1

)

(12)

Conversely suppose that (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

)

120583119873

119860(119909minus1

)) for all 119909 isin 119881 Then for all 119909 119910 isin 119881

119877 (119909 119910) = (120583119875

119860(119909minus1

119910) 120583119873

119860(119909minus1

119910))

= (120583119875

119860(119910minus1

119909) 120583119873

119860(119910minus1

119909))

= 119877 (119910 119909)

(13)

Hence 119877 is symmetric

Theorem21 Abipolar fuzzy relation119877 is antisymmetric if andonly if 119909 (120583

119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = (1 minus1)

Definition 22 Let (119878 lowast) be a semigroup Let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878Then119860 is said to be a bipolar fuzzysubsemigroup of 119878 if for all 119909 119910 isin 119878 120583119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 120583119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910)

Theorem23 Abipolar fuzzy relation119877 is transitive if and onlyif 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Proof Suppose that119877 is transitive and let 119909 119910 isin 119881Then1198772

le

119877 Now for any 119909 isin 119881 we have 119877(1 119909) = (120583119875

119860(119909) 120583119873

119860(119909))

This implies that 119877(1 119911) and 119877(119911 119909119910) 119911 isin 119881 = 1198772

(1 119909119910) le

119877(1 119909119910) That is or120583119875119860(119911) and 120583

119875

119860(119911minus1

119909119910) 119911 isin 119881 le 120583119875

119860(119909119910) and

and120583119873

119860(119911) or 120583

119873

119860(119911minus1

119909119910) 119911 isin 119881 ge 120583119873

119860(119909119910) Hence 120583

119875

119860(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119860(119909119910) le 120583

119875

119860(119909) or 120583

119873

119860(119910) Hence 119860 =

(120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Conversely suppose that 119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy

subsemigroup of (119881 lowast) That is for all 119909 119910 isin 119881120583119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Then for any

119909 119910 isin 119881

1198772

(119909 119910) = (1198772

120583119875

(119909 119910) 1198772

120583119873

(119909 119910))

1198772

120583119875

(119909 119910) = or 119877120583119875 (119909 119911) and 119877

120583119875 (119911 119910) 119911 isin 119881

= or 120583119875

119860(119909minus1

119911) and 120583119875

119860(119911minus1

119910) 119911 isin 119881

le 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

1198772

120583119873

(119909 119910) = and 119877120583119873 (119909 119911) or 119877

120583119873 (119911 119910) 119911 isin 119881

= and 120583119873

119860(119909minus1

119911) or 120583119873

119860(119911minus1

119910) 119911 isin 119881

ge 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910)

(14)

The Scientific World Journal 5

Hence 1198772

120583119875

(119909 119910) le 119877120583119875(119909 119910) and 119877

2

120583119873

(119909 119910) ge 119877120583119873(119909 119910)

Hence 119877 is transitive

We conclude that

Theorem 24 A bipolar fuzzy relation 119877 is a partial order ifand only if 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of

(119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1

Theorem 25 A bipolar fuzzy relation 119877 is a linear order ifand only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1(iii) 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 = 119881

Proof Suppose 119877 is a linear order Then by Theorem 24conditions (i) (ii) and (iii) are satisfied For any 119909 isin 119881(119877 or 119877

minus1

)(1 119909) gt 0 This implies that 119877(1 119909) or 119877(119909 1) gt 0Hence 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0Conversely suppose that conditions (i) (ii) and (iii) hold

Then by Theorem 24 119877 is partial order Now for any 119909 119910 isin

119881 we have (119909minus1

119910) (119910minus1119909) isin 119881 Then by condition (iv) 119909

120583119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 Therefore 119877 islinear order

Theorem 26 A bipolar fuzzy relation 119877 is a equivalencerelation if and only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup

of (119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) (120583119875119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Theorem 27 119866 is a Hasse diagram if and only if for anycollection 119909

1 1199092 1199093 119909

119899of vertices in 119881 with 119899 ge 2 and

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we have

120583119875

119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0

Proof Suppose 119866 is a Hasse diagram and let 1199091 1199092 119909

119899

be vertices in 119881 with 119899 ge 2 and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt

0 for 119894 = 1 2 3 119899 Then it is obvious that 119877(11990911199092sdot sdot sdot

119909119894minus1

11990911199092sdot sdot sdot 119909119894) = (120583

119875

119860(119909119894) 120583119873

119860(119909119894)) for 119894 = 1 2

119899 where 1199090

= 1 Therefore (1 1199091 11990911199092 119909

11199092sdot sdot sdot 119909119899)

is a path from 1 to 11990911199092sdot sdot sdot 119909119899 Since119866 is a Hasse diagram we

have 119877(1 11990911199092sdot sdot sdot 119909119899) = 0 This implies that 120583119875

119860(11990911199092sdot sdot sdot 119909119899) =

0 and 120583119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Conversely suppose that for

any collection 1199091 1199092 119909119899of vertices in 119881 with 119899 ge 2

and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we

have 120583119875119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Let

(1199090 1199091 1199092 119909119899) be a path in 119866 from 119909

0to 119909119899with 119899 ge

2 Then 119877(119909119894minus1

119909119894) gt 0 for 119894 = 1 2 119899 Therefore

120583119875

119860(119909minus1

119894minus1119909119894) gt 0 120583

119873

119860(119909minus1

119894minus1119909119894) lt 0 for 119894 = 1 2 119899

Now consider the elements 119909minus1

01199091 119909minus1

11199092 119909

minus1

119899minus1119909119899in 119881

Then by assumption 120583119875

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 and

120583119873

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 That is 120583119875

119860(119909minus1

0119909119899) = 0 and

120583119873

119860(119909minus1

0119909119899) = 0 Hence 119877(119909

0 119909119899) = 0 Thus 119866 is a Hasse

diagram

Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is connected (weakly connected semiconnected locallyconnected or quasi-connected) if and only if the inducefuzzy graph (119881 119877

+

0) is connected (weakly connected semi-

connected locally connected or quasi-connected)

Definition 28 Let (119878 lowast) be a semigroup and let 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then the subsemigroup gener-ated by119860 is the meeting of all bipolar fuzzy subsemigroups of119878 which contains 119860 It is denoted by ⟨119860⟩

Lemma 29 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then bipolar fuzzy subset ⟨119860⟩ isprecisely given by ⟨120583119875

119860⟩(119909) = or120583

119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

119909 = 11990911199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 ⟨120583119873

119860⟩(119909) =

and120583119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878

Proof Let1198601015840 = (119875

119860 119873

119860) be a bipolar fuzzy subset of 119878 defined

by 119875119860(119909) = or120583

119875

119860(1199091) and120583119875

119860(1199092) and sdot sdot sdot and 120583

119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 119873

119860(119909) = and120583

119873

119860(1199091) or

120583119873

119860(1199092)orsdot sdot sdotor120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119873

119860(119909119894) lt 0 for 119894 =

1 2 119899 for any 119909 isin 119878 Let 119909 119910 isin 119878 If 120583119875119860(119909) = 0 or 120583119875

119860(119910) =

0 then 120583119875

119860(119909) and 120583

119875

119860(119910) = 0 and 120583

119873

119860(119909) = 0 or 120583119873

119860(119910) = 0 and

then 120583119873

119860(119909) or 120583

119873

119860(119910) = 0 Therefore 119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Again if 120583119875

119860(119909) = 0 120583

119873

119860(119909) = 0

then by definition of 119875119860(119909) and

119873

119860(119909) we have

119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Hence (119875

119860 119873

119860)

is a bipolar fuzzy subsemigroup of 119878 containing (120583119875

119860 120583119873

119860)

Now let 119871 be any bipolar fuzzy subsemigroup of 119878 containing(120583119875

119860 120583119873

119860) Then for any 119909 isin 119878 with 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 we have 120583

119875

119871(119909119894) ge

120583119875

119871(1199091)and120583119875

119871(1199092)andsdot sdot sdotand120583

119875

119871(119909119899) ge 120583119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

and 120583119873

119871(119909119894) ge 120583

119873

119871(1199091) and 120583119873

119871(1199092) and sdot sdot sdot and 120583

119873

119871(119909119899) ge 120583

119873

119860(1199091) and

120583119873

119860(1199092)and sdot sdot sdotand120583

119873

119860(119909119899)Thus 120583119875

119871(119909) ge or120583

119875

119860(1199091)and120583119875

119860(1199092)and sdot sdot sdotand

120583119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899

120583119873

119871(119909) le and120583

119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878 Hence

120583119875

119871(119909) ge

119875

119860(119909) 120583119873

119871(119909) le

119873

119860(119909) for all 119909 isin 119878 Thus 119875

119860(119909) le

120583119875

119871(119909) 119873

119860(119909) ge 120583

119875

119860(119909) Thus 1198601015840 = (

119875

119860 119873

119860) is the meeting of

all bipolar fuzzy subsemigroups containing (120583119875

119860 120583119873

119860)

Theorem 30 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then for any 120572 isin [0 1](⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) and (⟨(120583

+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) =

(⟨120583119875

⟩

+

120572 ⟨120583119873

⟩

+

120572) where (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) denotes the subsemigroup

generated by (120583119875

120572 120583119873

120572) and ⟨(120583

119875

120583119873

)⟩ denotes bipolar fuzzysubsemigroup generated by (120583119875 120583119873)

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

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Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

In this paper we will write 119909119910 isin 119864 to mean 119909 rarr 119910 isin 119864 andif 119890 = 119909119910 isin 119864 we say 119909 and 119910 are adjacent such that 119909 is astarting node and 119910 is an ending node

The study of vertex transitive graphs has a long andrich history in discrete mathematics Prominent examplesof vertex transitive graphs are Cayley graphs which areimportant in both theory and applications for exampleCayley graphs are goodmodels for interconnection networks

Definition 1 Let 119866 be a finite group and let 119878 be a minimalgenerating set of 119866 A Cayley graph (119866 119878) has elements of 119866as its vertices the edge set is given by (119892 119892

119904) 119892 isin 119866 119904 isin 119878

Two vertices 1198921and 119892

2are adjacent if 119892

2= 1198921sdot 119904 where 119904 isin 119878

Note that a generating set 119878 is minimal if 119878 generates119866 but noproper subset of 119878 does

Theorem 2 All Cayley graphs are vertex transitive

Definition 3 Let (119881 lowast) be a group and let 119860 be any subset of119881 Then the Cayley graph induced by (119881 lowast 119860) is the graph119866 = (119881 119877) where 119877 = (119909 119910) 119909

minus1

119910 isin 119860

Definition 4 (see [2]) A fuzzy subset 120583 on a set 119883 is a map120583 119883 rarr [0 1] A fuzzy binary relation on119883 is a fuzzy subset120583 on119883times119883 By a fuzzy relationwemean a fuzzy binary relationgiven by 120583 119883 times 119883 rarr [0 1]

Definition 5 (see [11]) Let (119881 lowast) be a group and let 120583 be afuzzy subset of 119881 Then the fuzzy relation 119877 on 119881 defined by

119877 (119909 119910) = 120583 (119909minus1

lowast 119910) forall119909 119910 isin 119881 (2)

induces a fuzzy graph 119866 = (119881 119877) called the Cayley fuzzygraph induced by the (119881 lowast 120583)

Definition 6 (see [3]) Let 119883 be a nonempty set A bipolarfuzzy set 119861 in119883 is an object having the form

119861 = (119909 120583119875

119861(119909) 120583

119873

119861(119909)) | 119909 isin 119883 (3)

where 120583119875119861 119883 rarr [0 1] and 120583

119873

119861 119883 rarr [minus1 0] are mappings

We use the positive membership degree 120583119875

119861(119909) to denote

the satisfaction degree of an element 119909 to the propertycorresponding to a bipolar fuzzy set 119861 and the negativemem-bership degree 120583

119873

119861(119909) to denote the satisfaction degree of an

element 119909 to some implicit counterproperty correspondingto a bipolar fuzzy set 119861 If 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 it is the

situation that119909 is regarded as having only positive satisfactionfor 119861 If 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 it is the situation that 119909

does not satisfy the property of 119861 but somewhat satisfies thecounterproperty of119861 It is possible for an element119909 to be suchthat 120583119875

119861(119909) = 0 and 120583

119873

119861(119909) = 0 when the membership function

of the property overlaps that of its counterpro perty over someportion of119883

For the sake of simplicity we shall use the symbol 119861 =

(120583119875

119861 120583119873

119861) for the bipolar fuzzy set

119861 = (119909 120583119875

119861(119909) 120583

119873

119861(119909)) | 119909 isin 119883 (4)

A nice application of bipolar fuzzy concept is a politicalacceptation (map to [0 1]) and nonacceptation (map to[minus1 0])

Definition 7 (see [3]) A bipolar fuzzy relation 119877 = (120583119875

119877(119909 119910)

120583119873

119877(119909 119910)) in a universe 119883 times 119884 (119877(119883 rarr 119884) for short) is a

bipolar fuzzy set of the form

119877 = ⟨(119909 119910) 120583119875

119877(119909 119910) 120583

119873

119877(119909 119910)⟩ | (119909 119910) isin 119883 times 119884 (5)

where 120583119875119877 119883 times 119884 rarr [0 1] and 120583

119873

119877 119883 times 119884 rarr [minus1 0]

Definition 8 Let 119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy equivalence relation on119883 if itsatisfies the following conditions

(a) 119877 is bipolar fuzzy reflexive that is119877(119909 119909) = (1 minus1) foreach 119909 isin 119883

(b) 119877 is bipolar fuzzy symmetric that is119877(119909 119910) = 119877(119910 119909)

for any 119909 119910 isin 119883(c) 119877 is bipolar fuzzy transitive that is 119877(119909 119911) ge

⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 9 Let 119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy partial order relation on 119883 ifit satisfies the following conditions

(a) 119877 is bipolar fuzzy reflexive that is 119877(119909 119909) = (1 minus1)

for each 119909 isin 119883(b) 119877 is bipolar fuzzy antisymmetric that is

119877(119909 119910) = 119877(119910 119909) for any 119909 119910 isin 119883(c) 119877 is bipolar fuzzy transitive that is 119877(119909 119911) ge

⋁119910(119877(119909 119910) and 119877(119910 119911))

Definition 10 Let119877 be a bipolar fuzzy relation on universe119883Then 119877 is called a bipolar fuzzy linear order relation on119883 if itsatisfies the following conditions

(a) 119877 is bipolar fuzzy partial relation(b) (120583119875

119877or 120583119875

119877minus1

)(119909 119910) gt 0 (120583119873119877

and 120583119873

119877minus1

)(119909 119910) lt 0 for all 119909119910 isin 119883

3 Cayley Bipolar Fuzzy Graphs

Definition 11 A bipolar fuzzy digraph of a digraph119863lowast is a pair

119863 = (119860 119861)where119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy set in119881 and

119861 = (120583119875

119861 120583119873

119861) is a bipolar fuzzy relation on 119864 such that

120583119875

119861(119909119910) le min (120583

119875

119860(119909) 120583

119875

119860(119910))

120583119873

119861(119909119910) ge max (120583119873

119860(119909) 120583

119873

119860(119910))

(6)

for all 119909119910 isin 119864 We note that 119861 need not to be symmetric

Definition 12 Let 119863 be a bipolar fuzzy digraph Theindegree of a vertex 119909 in 119863 is defined by ind(119909) =

(ind119875120583(119909) ind119873

120583(119909)) where ind119875

120583(119909) = sum

119910 = 119909120583119875

119860(119909119910) and

ind119873120583(119909) = sum

119910 = 119909120583119873

119860(119909119910) Similarly the outdegree

The Scientific World Journal 3

of a vertex 119909 in 119863 is defined by outd(119909)= (outd119875

120583(119909) outd119873

120583(119909)) where outd119875

120583(119909) = sum

119910 = 119909120583119875

119860(119909119910) and

outd119873120583(119909) = sum

119910 = 119909120583119873

119860(119909119910) A bipolar fuzzy digraph in which

each vertex has the same outdegree 119903 is called an outregulardigraph with index of outregularity 119903 In-regular digraphs aredefined similarly

Definition 13 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

the bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation119877 defined on 119881 by

119877 (119909 119910) = (120583119875

119877(119909minus1

119910) 120583119873

119877((119909minus1

119910))) forall119909 119910 isin 119881 (7)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 120583119875

119877 120583119873

119877)

We now introduce Cayley bipolar fuzzy graphs and provethat all Cayley bipolar fuzzy graphs are regular

Definition 14 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation 119877

on 119881 defined by

119877 (119909 119910) = 120583119875

119877(119909minus1

119910) 120583119873

119877(119909minus1

119910) forall119909 119910 isin 119881 (8)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 119860)

Example 15 Consider the group 1198853and take 119881 = 0 1 2

Define 120583119875

119860 119881 rarr [0 1] and 120583

119873

119860 119881 rarr [minus1 0] by 120583

119875

119860(0) =

120583119875

119860(1) = 120583

119875

119860(2) = 05 120583119873

119860(0) = 120583

119873

119860(1) = 120583

119873

119860(2) = minus04

Then the Cayley bipolar fuzzy graph 119866 = (119881 119877) induced by(1198853 + 119860) is given by Table 1 and Figure 1

We see that Cayley bipolar fuzzy graphs are actuallybipolar fuzzy digraphs Furthermore the relation 119877 in theabove definition describes the strength of each directed edgeLet 119866 denote a bipolar fuzzy graph 119866 = (119881 119877) induced by thetriple (119881 lowast 119860)

Theorem 16 The Cayley bipolar fuzzy graph 119866 is vertextransitive

Proof Let 119886 119887 isin 119881 Define 120595 119881 rarr 119881 by 120595(119909) = 119887119886minus1

119909 forall 119909 isin 119881 Clearly 120595 is a bijective map For each 119909 119910 isin 119881

119877 (120595 (119909) 120595 (119910)) = (119877120583119875 (120595 (119909) 120595 (119910))

119877120583119873 (120595 (119909) 120595 (119910)))

Now 119877120583119875 (120595 (119909) 120595 (119910)) = 119877

120583119875 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119875

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

119877120583119873 (120595 (119909) 120595 (119910)) = 119877

120583119873 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119873

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910) (9)

Therefore 119877(120595(119909) 120595(119910)) = 119877(119909 119910) Hence 120595 is an automor-phism on 119866 Also 120595(119886) = 119887 Hence 119866 is vertex transitive

Theorem 17 Every vertex transitive bipolar fuzzy graph isregular

Proof Let 119866 = (119881 119877) be any vertex transitive bipolar fuzzygraph Let 119906 V isin 119881 Then there is an automorphism 119891 on 119866

such that 119891(119906) = V Note that

ind (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119909 119906) 119877

120583119873 (119909 119906))

= sum

119909isin119881

(119877120583119875 (119891 (119909) 119891 (119906)) 119877

120583119873 (119891 (119909) 119891 (119906)))

= sum

119909isin119881

(119877120583119875 (119891 (119909) V) 119877

120583119873 (119891 (119909) V))

= sum

119909isin119881

(119877120583119875 (119910 V) 119877

120583119873 (119910 V))

= ind (V)

outd (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119906 119909) 119877

120583119873 (119906 119909))

= sum

119909isin119881

(119877120583119875 (119891 (119906) 119891 (119909)) 119877

120583119873 (119891 (119906) 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119891 (119909)) 119877

120583119873 (V 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119910) 119877

120583119873 (V 119910))

= outd (V) (10)

Hence 119866 is regular

Theorem 18 Cayley bipolar fuzzy graphs are regular

Proof Proof follows fromTheorems 16 and 17

Theorem 19 Let 119866 = (119881 119877) denote bipolar fuzzy graph Thenbipolar fuzzy relation 119877 is reflexive if and only if 120583119875

119860(1) = 1 and

120583119873

119860(1) = minus1

4 The Scientific World Journal

Table 1 119877(119886 119887) for Cayley bipolar fuzzy graph

119886 0 0 0 1 1 1 2 2 2119887 0 1 2 0 1 2 0 1 2(minus119886) + 119887 0 1 2 2 0 1 1 2 0119877 (119886 119887) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04)

(05 minus04)

(03 minus03)

(03 minus02)

(03 minus03)

(03 minus02)(03 minus02)

(05 minus04)

(03 minus02)(05 minus04)

0

1

2

G

Figure 1 Cayley bipolar fuzzy graph

Proof 119877 is reflexive if and only if 119877(119909 119909) = (1 minus1) for all 119909 isin

119881 Now

119877 (119909 119909) = (120583119875

119860(119909minus1

119909) 120583119873

119860(119909minus1

119909))

= (120583119875

119860(1) 120583

119873

119860(1)) forall119909 isin 119881

(11)

Hence 119877 is reflexive if and only if 120583119875119860(1) = 1 and 120583

119873

119860(1) =

minus1

Theorem 20 Let 119866 = (119881 119877) denote bipolar fuzzy graphThen bipolar fuzzy relation 119877 is symmetric if and only if(120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Proof Suppose that 119877 is symmetric Then for any 119909 isin 119881

(120583119875

119860(119909) 120583

119873

119860(119909)) = (120583

119875

119860(119909minus1

1199092

) 120583119873

119860(119909minus1

1199092

))

= 119877 (119909 1199092

) = 119877 (1199092

119909)

(since 119877 is symmetric)

= (120583119875

119860((1199092

)

minus1

119909) 120583119873

119860(1199092

)

minus1

119909)

= 120583119875

119860(119909minus2

119909) 120583119873

119860(119909minus2

119909)

= 120583119875

119860(119909minus1

) 120583119873

119860(119909minus1

)

(12)

Conversely suppose that (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

)

120583119873

119860(119909minus1

)) for all 119909 isin 119881 Then for all 119909 119910 isin 119881

119877 (119909 119910) = (120583119875

119860(119909minus1

119910) 120583119873

119860(119909minus1

119910))

= (120583119875

119860(119910minus1

119909) 120583119873

119860(119910minus1

119909))

= 119877 (119910 119909)

(13)

Hence 119877 is symmetric

Theorem21 Abipolar fuzzy relation119877 is antisymmetric if andonly if 119909 (120583

119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = (1 minus1)

Definition 22 Let (119878 lowast) be a semigroup Let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878Then119860 is said to be a bipolar fuzzysubsemigroup of 119878 if for all 119909 119910 isin 119878 120583119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 120583119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910)

Theorem23 Abipolar fuzzy relation119877 is transitive if and onlyif 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Proof Suppose that119877 is transitive and let 119909 119910 isin 119881Then1198772

le

119877 Now for any 119909 isin 119881 we have 119877(1 119909) = (120583119875

119860(119909) 120583119873

119860(119909))

This implies that 119877(1 119911) and 119877(119911 119909119910) 119911 isin 119881 = 1198772

(1 119909119910) le

119877(1 119909119910) That is or120583119875119860(119911) and 120583

119875

119860(119911minus1

119909119910) 119911 isin 119881 le 120583119875

119860(119909119910) and

and120583119873

119860(119911) or 120583

119873

119860(119911minus1

119909119910) 119911 isin 119881 ge 120583119873

119860(119909119910) Hence 120583

119875

119860(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119860(119909119910) le 120583

119875

119860(119909) or 120583

119873

119860(119910) Hence 119860 =

(120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Conversely suppose that 119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy

subsemigroup of (119881 lowast) That is for all 119909 119910 isin 119881120583119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Then for any

119909 119910 isin 119881

1198772

(119909 119910) = (1198772

120583119875

(119909 119910) 1198772

120583119873

(119909 119910))

1198772

120583119875

(119909 119910) = or 119877120583119875 (119909 119911) and 119877

120583119875 (119911 119910) 119911 isin 119881

= or 120583119875

119860(119909minus1

119911) and 120583119875

119860(119911minus1

119910) 119911 isin 119881

le 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

1198772

120583119873

(119909 119910) = and 119877120583119873 (119909 119911) or 119877

120583119873 (119911 119910) 119911 isin 119881

= and 120583119873

119860(119909minus1

119911) or 120583119873

119860(119911minus1

119910) 119911 isin 119881

ge 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910)

(14)

The Scientific World Journal 5

Hence 1198772

120583119875

(119909 119910) le 119877120583119875(119909 119910) and 119877

2

120583119873

(119909 119910) ge 119877120583119873(119909 119910)

Hence 119877 is transitive

We conclude that

Theorem 24 A bipolar fuzzy relation 119877 is a partial order ifand only if 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of

(119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1

Theorem 25 A bipolar fuzzy relation 119877 is a linear order ifand only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1(iii) 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 = 119881

Proof Suppose 119877 is a linear order Then by Theorem 24conditions (i) (ii) and (iii) are satisfied For any 119909 isin 119881(119877 or 119877

minus1

)(1 119909) gt 0 This implies that 119877(1 119909) or 119877(119909 1) gt 0Hence 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0Conversely suppose that conditions (i) (ii) and (iii) hold

Then by Theorem 24 119877 is partial order Now for any 119909 119910 isin

119881 we have (119909minus1

119910) (119910minus1119909) isin 119881 Then by condition (iv) 119909

120583119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 Therefore 119877 islinear order

Theorem 26 A bipolar fuzzy relation 119877 is a equivalencerelation if and only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup

of (119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) (120583119875119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Theorem 27 119866 is a Hasse diagram if and only if for anycollection 119909

1 1199092 1199093 119909

119899of vertices in 119881 with 119899 ge 2 and

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we have

120583119875

119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0

Proof Suppose 119866 is a Hasse diagram and let 1199091 1199092 119909

119899

be vertices in 119881 with 119899 ge 2 and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt

0 for 119894 = 1 2 3 119899 Then it is obvious that 119877(11990911199092sdot sdot sdot

119909119894minus1

11990911199092sdot sdot sdot 119909119894) = (120583

119875

119860(119909119894) 120583119873

119860(119909119894)) for 119894 = 1 2

119899 where 1199090

= 1 Therefore (1 1199091 11990911199092 119909

11199092sdot sdot sdot 119909119899)

is a path from 1 to 11990911199092sdot sdot sdot 119909119899 Since119866 is a Hasse diagram we

have 119877(1 11990911199092sdot sdot sdot 119909119899) = 0 This implies that 120583119875

119860(11990911199092sdot sdot sdot 119909119899) =

0 and 120583119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Conversely suppose that for

any collection 1199091 1199092 119909119899of vertices in 119881 with 119899 ge 2

and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we

have 120583119875119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Let

(1199090 1199091 1199092 119909119899) be a path in 119866 from 119909

0to 119909119899with 119899 ge

2 Then 119877(119909119894minus1

119909119894) gt 0 for 119894 = 1 2 119899 Therefore

120583119875

119860(119909minus1

119894minus1119909119894) gt 0 120583

119873

119860(119909minus1

119894minus1119909119894) lt 0 for 119894 = 1 2 119899

Now consider the elements 119909minus1

01199091 119909minus1

11199092 119909

minus1

119899minus1119909119899in 119881

Then by assumption 120583119875

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 and

120583119873

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 That is 120583119875

119860(119909minus1

0119909119899) = 0 and

120583119873

119860(119909minus1

0119909119899) = 0 Hence 119877(119909

0 119909119899) = 0 Thus 119866 is a Hasse

diagram

Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is connected (weakly connected semiconnected locallyconnected or quasi-connected) if and only if the inducefuzzy graph (119881 119877

+

0) is connected (weakly connected semi-

connected locally connected or quasi-connected)

Definition 28 Let (119878 lowast) be a semigroup and let 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then the subsemigroup gener-ated by119860 is the meeting of all bipolar fuzzy subsemigroups of119878 which contains 119860 It is denoted by ⟨119860⟩

Lemma 29 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then bipolar fuzzy subset ⟨119860⟩ isprecisely given by ⟨120583119875

119860⟩(119909) = or120583

119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

119909 = 11990911199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 ⟨120583119873

119860⟩(119909) =

and120583119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878

Proof Let1198601015840 = (119875

119860 119873

119860) be a bipolar fuzzy subset of 119878 defined

by 119875119860(119909) = or120583

119875

119860(1199091) and120583119875

119860(1199092) and sdot sdot sdot and 120583

119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 119873

119860(119909) = and120583

119873

119860(1199091) or

120583119873

119860(1199092)orsdot sdot sdotor120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119873

119860(119909119894) lt 0 for 119894 =

1 2 119899 for any 119909 isin 119878 Let 119909 119910 isin 119878 If 120583119875119860(119909) = 0 or 120583119875

119860(119910) =

0 then 120583119875

119860(119909) and 120583

119875

119860(119910) = 0 and 120583

119873

119860(119909) = 0 or 120583119873

119860(119910) = 0 and

then 120583119873

119860(119909) or 120583

119873

119860(119910) = 0 Therefore 119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Again if 120583119875

119860(119909) = 0 120583

119873

119860(119909) = 0

then by definition of 119875119860(119909) and

119873

119860(119909) we have

119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Hence (119875

119860 119873

119860)

is a bipolar fuzzy subsemigroup of 119878 containing (120583119875

119860 120583119873

119860)

Now let 119871 be any bipolar fuzzy subsemigroup of 119878 containing(120583119875

119860 120583119873

119860) Then for any 119909 isin 119878 with 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 we have 120583

119875

119871(119909119894) ge

120583119875

119871(1199091)and120583119875

119871(1199092)andsdot sdot sdotand120583

119875

119871(119909119899) ge 120583119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

and 120583119873

119871(119909119894) ge 120583

119873

119871(1199091) and 120583119873

119871(1199092) and sdot sdot sdot and 120583

119873

119871(119909119899) ge 120583

119873

119860(1199091) and

120583119873

119860(1199092)and sdot sdot sdotand120583

119873

119860(119909119899)Thus 120583119875

119871(119909) ge or120583

119875

119860(1199091)and120583119875

119860(1199092)and sdot sdot sdotand

120583119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899

120583119873

119871(119909) le and120583

119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878 Hence

120583119875

119871(119909) ge

119875

119860(119909) 120583119873

119871(119909) le

119873

119860(119909) for all 119909 isin 119878 Thus 119875

119860(119909) le

120583119875

119871(119909) 119873

119860(119909) ge 120583

119875

119860(119909) Thus 1198601015840 = (

119875

119860 119873

119860) is the meeting of

all bipolar fuzzy subsemigroups containing (120583119875

119860 120583119873

119860)

Theorem 30 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then for any 120572 isin [0 1](⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) and (⟨(120583

+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) =

(⟨120583119875

⟩

+

120572 ⟨120583119873

⟩

+

120572) where (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) denotes the subsemigroup

generated by (120583119875

120572 120583119873

120572) and ⟨(120583

119875

120583119873

)⟩ denotes bipolar fuzzysubsemigroup generated by (120583119875 120583119873)

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

of a vertex 119909 in 119863 is defined by outd(119909)= (outd119875

120583(119909) outd119873

120583(119909)) where outd119875

120583(119909) = sum

119910 = 119909120583119875

119860(119909119910) and

outd119873120583(119909) = sum

119910 = 119909120583119873

119860(119909119910) A bipolar fuzzy digraph in which

each vertex has the same outdegree 119903 is called an outregulardigraph with index of outregularity 119903 In-regular digraphs aredefined similarly

Definition 13 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

the bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation119877 defined on 119881 by

119877 (119909 119910) = (120583119875

119877(119909minus1

119910) 120583119873

119877((119909minus1

119910))) forall119909 119910 isin 119881 (7)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 120583119875

119877 120583119873

119877)

We now introduce Cayley bipolar fuzzy graphs and provethat all Cayley bipolar fuzzy graphs are regular

Definition 14 Let (119881 lowast) be a group and let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119881 Then the bipolar fuzzy relation 119877

on 119881 defined by

119877 (119909 119910) = 120583119875

119877(119909minus1

119910) 120583119873

119877(119909minus1

119910) forall119909 119910 isin 119881 (8)

induces a bipolar fuzzy graph 119866 = (119881 119877) called the Cayleybipolar fuzzy graph induced by the (119881 lowast 119860)

Example 15 Consider the group 1198853and take 119881 = 0 1 2

Define 120583119875

119860 119881 rarr [0 1] and 120583

119873

119860 119881 rarr [minus1 0] by 120583

119875

119860(0) =

120583119875

119860(1) = 120583

119875

119860(2) = 05 120583119873

119860(0) = 120583

119873

119860(1) = 120583

119873

119860(2) = minus04

Then the Cayley bipolar fuzzy graph 119866 = (119881 119877) induced by(1198853 + 119860) is given by Table 1 and Figure 1

We see that Cayley bipolar fuzzy graphs are actuallybipolar fuzzy digraphs Furthermore the relation 119877 in theabove definition describes the strength of each directed edgeLet 119866 denote a bipolar fuzzy graph 119866 = (119881 119877) induced by thetriple (119881 lowast 119860)

Theorem 16 The Cayley bipolar fuzzy graph 119866 is vertextransitive

Proof Let 119886 119887 isin 119881 Define 120595 119881 rarr 119881 by 120595(119909) = 119887119886minus1

119909 forall 119909 isin 119881 Clearly 120595 is a bijective map For each 119909 119910 isin 119881

119877 (120595 (119909) 120595 (119910)) = (119877120583119875 (120595 (119909) 120595 (119910))

119877120583119873 (120595 (119909) 120595 (119910)))

Now 119877120583119875 (120595 (119909) 120595 (119910)) = 119877

120583119875 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119875

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

119877120583119873 (120595 (119909) 120595 (119910)) = 119877

120583119873 (119887119886minus1

119909 119887119886minus1

119910)

= 120583119873

119860((119887119886minus1

119909)

minus1

(119887119886minus1

119909))

= 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910) (9)

Therefore 119877(120595(119909) 120595(119910)) = 119877(119909 119910) Hence 120595 is an automor-phism on 119866 Also 120595(119886) = 119887 Hence 119866 is vertex transitive

Theorem 17 Every vertex transitive bipolar fuzzy graph isregular

Proof Let 119866 = (119881 119877) be any vertex transitive bipolar fuzzygraph Let 119906 V isin 119881 Then there is an automorphism 119891 on 119866

such that 119891(119906) = V Note that

ind (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119909 119906) 119877

120583119873 (119909 119906))

= sum

119909isin119881

(119877120583119875 (119891 (119909) 119891 (119906)) 119877

120583119873 (119891 (119909) 119891 (119906)))

= sum

119909isin119881

(119877120583119875 (119891 (119909) V) 119877

120583119873 (119891 (119909) V))

= sum

119909isin119881

(119877120583119875 (119910 V) 119877

120583119873 (119910 V))

= ind (V)

outd (119906) = sum

119909isin119881

119877 (119909 119906)

= sum

119909isin119881

(119877120583119875 (119906 119909) 119877

120583119873 (119906 119909))

= sum

119909isin119881

(119877120583119875 (119891 (119906) 119891 (119909)) 119877

120583119873 (119891 (119906) 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119891 (119909)) 119877

120583119873 (V 119891 (119909)))

= sum

119909isin119881

(119877120583119875 (V 119910) 119877

120583119873 (V 119910))

= outd (V) (10)

Hence 119866 is regular

Theorem 18 Cayley bipolar fuzzy graphs are regular

Proof Proof follows fromTheorems 16 and 17

Theorem 19 Let 119866 = (119881 119877) denote bipolar fuzzy graph Thenbipolar fuzzy relation 119877 is reflexive if and only if 120583119875

119860(1) = 1 and

120583119873

119860(1) = minus1

4 The Scientific World Journal

Table 1 119877(119886 119887) for Cayley bipolar fuzzy graph

119886 0 0 0 1 1 1 2 2 2119887 0 1 2 0 1 2 0 1 2(minus119886) + 119887 0 1 2 2 0 1 1 2 0119877 (119886 119887) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04)

(05 minus04)

(03 minus03)

(03 minus02)

(03 minus03)

(03 minus02)(03 minus02)

(05 minus04)

(03 minus02)(05 minus04)

0

1

2

G

Figure 1 Cayley bipolar fuzzy graph

Proof 119877 is reflexive if and only if 119877(119909 119909) = (1 minus1) for all 119909 isin

119881 Now

119877 (119909 119909) = (120583119875

119860(119909minus1

119909) 120583119873

119860(119909minus1

119909))

= (120583119875

119860(1) 120583

119873

119860(1)) forall119909 isin 119881

(11)

Hence 119877 is reflexive if and only if 120583119875119860(1) = 1 and 120583

119873

119860(1) =

minus1

Theorem 20 Let 119866 = (119881 119877) denote bipolar fuzzy graphThen bipolar fuzzy relation 119877 is symmetric if and only if(120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Proof Suppose that 119877 is symmetric Then for any 119909 isin 119881

(120583119875

119860(119909) 120583

119873

119860(119909)) = (120583

119875

119860(119909minus1

1199092

) 120583119873

119860(119909minus1

1199092

))

= 119877 (119909 1199092

) = 119877 (1199092

119909)

(since 119877 is symmetric)

= (120583119875

119860((1199092

)

minus1

119909) 120583119873

119860(1199092

)

minus1

119909)

= 120583119875

119860(119909minus2

119909) 120583119873

119860(119909minus2

119909)

= 120583119875

119860(119909minus1

) 120583119873

119860(119909minus1

)

(12)

Conversely suppose that (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

)

120583119873

119860(119909minus1

)) for all 119909 isin 119881 Then for all 119909 119910 isin 119881

119877 (119909 119910) = (120583119875

119860(119909minus1

119910) 120583119873

119860(119909minus1

119910))

= (120583119875

119860(119910minus1

119909) 120583119873

119860(119910minus1

119909))

= 119877 (119910 119909)

(13)

Hence 119877 is symmetric

Theorem21 Abipolar fuzzy relation119877 is antisymmetric if andonly if 119909 (120583

119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = (1 minus1)

Definition 22 Let (119878 lowast) be a semigroup Let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878Then119860 is said to be a bipolar fuzzysubsemigroup of 119878 if for all 119909 119910 isin 119878 120583119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 120583119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910)

Theorem23 Abipolar fuzzy relation119877 is transitive if and onlyif 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Proof Suppose that119877 is transitive and let 119909 119910 isin 119881Then1198772

le

119877 Now for any 119909 isin 119881 we have 119877(1 119909) = (120583119875

119860(119909) 120583119873

119860(119909))

This implies that 119877(1 119911) and 119877(119911 119909119910) 119911 isin 119881 = 1198772

(1 119909119910) le

119877(1 119909119910) That is or120583119875119860(119911) and 120583

119875

119860(119911minus1

119909119910) 119911 isin 119881 le 120583119875

119860(119909119910) and

and120583119873

119860(119911) or 120583

119873

119860(119911minus1

119909119910) 119911 isin 119881 ge 120583119873

119860(119909119910) Hence 120583

119875

119860(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119860(119909119910) le 120583

119875

119860(119909) or 120583

119873

119860(119910) Hence 119860 =

(120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Conversely suppose that 119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy

subsemigroup of (119881 lowast) That is for all 119909 119910 isin 119881120583119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Then for any

119909 119910 isin 119881

1198772

(119909 119910) = (1198772

120583119875

(119909 119910) 1198772

120583119873

(119909 119910))

1198772

120583119875

(119909 119910) = or 119877120583119875 (119909 119911) and 119877

120583119875 (119911 119910) 119911 isin 119881

= or 120583119875

119860(119909minus1

119911) and 120583119875

119860(119911minus1

119910) 119911 isin 119881

le 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

1198772

120583119873

(119909 119910) = and 119877120583119873 (119909 119911) or 119877

120583119873 (119911 119910) 119911 isin 119881

= and 120583119873

119860(119909minus1

119911) or 120583119873

119860(119911minus1

119910) 119911 isin 119881

ge 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910)

(14)

The Scientific World Journal 5

Hence 1198772

120583119875

(119909 119910) le 119877120583119875(119909 119910) and 119877

2

120583119873

(119909 119910) ge 119877120583119873(119909 119910)

Hence 119877 is transitive

We conclude that

Theorem 24 A bipolar fuzzy relation 119877 is a partial order ifand only if 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of

(119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1

Theorem 25 A bipolar fuzzy relation 119877 is a linear order ifand only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1(iii) 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 = 119881

Proof Suppose 119877 is a linear order Then by Theorem 24conditions (i) (ii) and (iii) are satisfied For any 119909 isin 119881(119877 or 119877

minus1

)(1 119909) gt 0 This implies that 119877(1 119909) or 119877(119909 1) gt 0Hence 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0Conversely suppose that conditions (i) (ii) and (iii) hold

Then by Theorem 24 119877 is partial order Now for any 119909 119910 isin

119881 we have (119909minus1

119910) (119910minus1119909) isin 119881 Then by condition (iv) 119909

120583119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 Therefore 119877 islinear order

Theorem 26 A bipolar fuzzy relation 119877 is a equivalencerelation if and only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup

of (119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) (120583119875119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Theorem 27 119866 is a Hasse diagram if and only if for anycollection 119909

1 1199092 1199093 119909

119899of vertices in 119881 with 119899 ge 2 and

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we have

120583119875

119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0

Proof Suppose 119866 is a Hasse diagram and let 1199091 1199092 119909

119899

be vertices in 119881 with 119899 ge 2 and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt

0 for 119894 = 1 2 3 119899 Then it is obvious that 119877(11990911199092sdot sdot sdot

119909119894minus1

11990911199092sdot sdot sdot 119909119894) = (120583

119875

119860(119909119894) 120583119873

119860(119909119894)) for 119894 = 1 2

119899 where 1199090

= 1 Therefore (1 1199091 11990911199092 119909

11199092sdot sdot sdot 119909119899)

is a path from 1 to 11990911199092sdot sdot sdot 119909119899 Since119866 is a Hasse diagram we

have 119877(1 11990911199092sdot sdot sdot 119909119899) = 0 This implies that 120583119875

119860(11990911199092sdot sdot sdot 119909119899) =

0 and 120583119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Conversely suppose that for

any collection 1199091 1199092 119909119899of vertices in 119881 with 119899 ge 2

and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we

have 120583119875119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Let

(1199090 1199091 1199092 119909119899) be a path in 119866 from 119909

0to 119909119899with 119899 ge

2 Then 119877(119909119894minus1

119909119894) gt 0 for 119894 = 1 2 119899 Therefore

120583119875

119860(119909minus1

119894minus1119909119894) gt 0 120583

119873

119860(119909minus1

119894minus1119909119894) lt 0 for 119894 = 1 2 119899

Now consider the elements 119909minus1

01199091 119909minus1

11199092 119909

minus1

119899minus1119909119899in 119881

Then by assumption 120583119875

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 and

120583119873

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 That is 120583119875

119860(119909minus1

0119909119899) = 0 and

120583119873

119860(119909minus1

0119909119899) = 0 Hence 119877(119909

0 119909119899) = 0 Thus 119866 is a Hasse

diagram

Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is connected (weakly connected semiconnected locallyconnected or quasi-connected) if and only if the inducefuzzy graph (119881 119877

+

0) is connected (weakly connected semi-

connected locally connected or quasi-connected)

Definition 28 Let (119878 lowast) be a semigroup and let 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then the subsemigroup gener-ated by119860 is the meeting of all bipolar fuzzy subsemigroups of119878 which contains 119860 It is denoted by ⟨119860⟩

Lemma 29 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then bipolar fuzzy subset ⟨119860⟩ isprecisely given by ⟨120583119875

119860⟩(119909) = or120583

119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

119909 = 11990911199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 ⟨120583119873

119860⟩(119909) =

and120583119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878

Proof Let1198601015840 = (119875

119860 119873

119860) be a bipolar fuzzy subset of 119878 defined

by 119875119860(119909) = or120583

119875

119860(1199091) and120583119875

119860(1199092) and sdot sdot sdot and 120583

119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 119873

119860(119909) = and120583

119873

119860(1199091) or

120583119873

119860(1199092)orsdot sdot sdotor120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119873

119860(119909119894) lt 0 for 119894 =

1 2 119899 for any 119909 isin 119878 Let 119909 119910 isin 119878 If 120583119875119860(119909) = 0 or 120583119875

119860(119910) =

0 then 120583119875

119860(119909) and 120583

119875

119860(119910) = 0 and 120583

119873

119860(119909) = 0 or 120583119873

119860(119910) = 0 and

then 120583119873

119860(119909) or 120583

119873

119860(119910) = 0 Therefore 119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Again if 120583119875

119860(119909) = 0 120583

119873

119860(119909) = 0

then by definition of 119875119860(119909) and

119873

119860(119909) we have

119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Hence (119875

119860 119873

119860)

is a bipolar fuzzy subsemigroup of 119878 containing (120583119875

119860 120583119873

119860)

Now let 119871 be any bipolar fuzzy subsemigroup of 119878 containing(120583119875

119860 120583119873

119860) Then for any 119909 isin 119878 with 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 we have 120583

119875

119871(119909119894) ge

120583119875

119871(1199091)and120583119875

119871(1199092)andsdot sdot sdotand120583

119875

119871(119909119899) ge 120583119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

and 120583119873

119871(119909119894) ge 120583

119873

119871(1199091) and 120583119873

119871(1199092) and sdot sdot sdot and 120583

119873

119871(119909119899) ge 120583

119873

119860(1199091) and

120583119873

119860(1199092)and sdot sdot sdotand120583

119873

119860(119909119899)Thus 120583119875

119871(119909) ge or120583

119875

119860(1199091)and120583119875

119860(1199092)and sdot sdot sdotand

120583119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899

120583119873

119871(119909) le and120583

119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878 Hence

120583119875

119871(119909) ge

119875

119860(119909) 120583119873

119871(119909) le

119873

119860(119909) for all 119909 isin 119878 Thus 119875

119860(119909) le

120583119875

119871(119909) 119873

119860(119909) ge 120583

119875

119860(119909) Thus 1198601015840 = (

119875

119860 119873

119860) is the meeting of

all bipolar fuzzy subsemigroups containing (120583119875

119860 120583119873

119860)

Theorem 30 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then for any 120572 isin [0 1](⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) and (⟨(120583

+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) =

(⟨120583119875

⟩

+

120572 ⟨120583119873

⟩

+

120572) where (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) denotes the subsemigroup

generated by (120583119875

120572 120583119873

120572) and ⟨(120583

119875

120583119873

)⟩ denotes bipolar fuzzysubsemigroup generated by (120583119875 120583119873)

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

Table 1 119877(119886 119887) for Cayley bipolar fuzzy graph

119886 0 0 0 1 1 1 2 2 2119887 0 1 2 0 1 2 0 1 2(minus119886) + 119887 0 1 2 2 0 1 1 2 0119877 (119886 119887) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04) (03 minus02) (03 minus02) (03 minus02) (05 minus04)

(05 minus04)

(03 minus03)

(03 minus02)

(03 minus03)

(03 minus02)(03 minus02)

(05 minus04)

(03 minus02)(05 minus04)

0

1

2

G

Figure 1 Cayley bipolar fuzzy graph

Proof 119877 is reflexive if and only if 119877(119909 119909) = (1 minus1) for all 119909 isin

119881 Now

119877 (119909 119909) = (120583119875

119860(119909minus1

119909) 120583119873

119860(119909minus1

119909))

= (120583119875

119860(1) 120583

119873

119860(1)) forall119909 isin 119881

(11)

Hence 119877 is reflexive if and only if 120583119875119860(1) = 1 and 120583

119873

119860(1) =

minus1

Theorem 20 Let 119866 = (119881 119877) denote bipolar fuzzy graphThen bipolar fuzzy relation 119877 is symmetric if and only if(120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Proof Suppose that 119877 is symmetric Then for any 119909 isin 119881

(120583119875

119860(119909) 120583

119873

119860(119909)) = (120583

119875

119860(119909minus1

1199092

) 120583119873

119860(119909minus1

1199092

))

= 119877 (119909 1199092

) = 119877 (1199092

119909)

(since 119877 is symmetric)

= (120583119875

119860((1199092

)

minus1

119909) 120583119873

119860(1199092

)

minus1

119909)

= 120583119875

119860(119909minus2

119909) 120583119873

119860(119909minus2

119909)

= 120583119875

119860(119909minus1

) 120583119873

119860(119909minus1

)

(12)

Conversely suppose that (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

)

120583119873

119860(119909minus1

)) for all 119909 isin 119881 Then for all 119909 119910 isin 119881

119877 (119909 119910) = (120583119875

119860(119909minus1

119910) 120583119873

119860(119909minus1

119910))

= (120583119875

119860(119910minus1

119909) 120583119873

119860(119910minus1

119909))

= 119877 (119910 119909)

(13)

Hence 119877 is symmetric

Theorem21 Abipolar fuzzy relation119877 is antisymmetric if andonly if 119909 (120583

119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = (1 minus1)

Definition 22 Let (119878 lowast) be a semigroup Let 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878Then119860 is said to be a bipolar fuzzysubsemigroup of 119878 if for all 119909 119910 isin 119878 120583119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 120583119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910)

Theorem23 Abipolar fuzzy relation119877 is transitive if and onlyif 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Proof Suppose that119877 is transitive and let 119909 119910 isin 119881Then1198772

le

119877 Now for any 119909 isin 119881 we have 119877(1 119909) = (120583119875

119860(119909) 120583119873

119860(119909))

This implies that 119877(1 119911) and 119877(119911 119909119910) 119911 isin 119881 = 1198772

(1 119909119910) le

119877(1 119909119910) That is or120583119875119860(119911) and 120583

119875

119860(119911minus1

119909119910) 119911 isin 119881 le 120583119875

119860(119909119910) and

and120583119873

119860(119911) or 120583

119873

119860(119911minus1

119909119910) 119911 isin 119881 ge 120583119873

119860(119909119910) Hence 120583

119875

119860(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119860(119909119910) le 120583

119875

119860(119909) or 120583

119873

119860(119910) Hence 119860 =

(120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

Conversely suppose that 119860 = (120583119875

119860 120583119873

119860) is a bipolar fuzzy

subsemigroup of (119881 lowast) That is for all 119909 119910 isin 119881120583119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and 120583

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Then for any

119909 119910 isin 119881

1198772

(119909 119910) = (1198772

120583119875

(119909 119910) 1198772

120583119873

(119909 119910))

1198772

120583119875

(119909 119910) = or 119877120583119875 (119909 119911) and 119877

120583119875 (119911 119910) 119911 isin 119881

= or 120583119875

119860(119909minus1

119911) and 120583119875

119860(119911minus1

119910) 119911 isin 119881

le 120583119875

119860(119909minus1

119910)

= 119877120583119875 (119909 119910)

1198772

120583119873

(119909 119910) = and 119877120583119873 (119909 119911) or 119877

120583119873 (119911 119910) 119911 isin 119881

= and 120583119873

119860(119909minus1

119911) or 120583119873

119860(119911minus1

119910) 119911 isin 119881

ge 120583119873

119860(119909minus1

119910)

= 119877120583119873 (119909 119910)

(14)

The Scientific World Journal 5

Hence 1198772

120583119875

(119909 119910) le 119877120583119875(119909 119910) and 119877

2

120583119873

(119909 119910) ge 119877120583119873(119909 119910)

Hence 119877 is transitive

We conclude that

Theorem 24 A bipolar fuzzy relation 119877 is a partial order ifand only if 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of

(119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1

Theorem 25 A bipolar fuzzy relation 119877 is a linear order ifand only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1(iii) 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 = 119881

Proof Suppose 119877 is a linear order Then by Theorem 24conditions (i) (ii) and (iii) are satisfied For any 119909 isin 119881(119877 or 119877

minus1

)(1 119909) gt 0 This implies that 119877(1 119909) or 119877(119909 1) gt 0Hence 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0Conversely suppose that conditions (i) (ii) and (iii) hold

Then by Theorem 24 119877 is partial order Now for any 119909 119910 isin

119881 we have (119909minus1

119910) (119910minus1119909) isin 119881 Then by condition (iv) 119909

120583119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 Therefore 119877 islinear order

Theorem 26 A bipolar fuzzy relation 119877 is a equivalencerelation if and only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup

of (119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) (120583119875119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Theorem 27 119866 is a Hasse diagram if and only if for anycollection 119909

1 1199092 1199093 119909

119899of vertices in 119881 with 119899 ge 2 and

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we have

120583119875

119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0

Proof Suppose 119866 is a Hasse diagram and let 1199091 1199092 119909

119899

be vertices in 119881 with 119899 ge 2 and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt

0 for 119894 = 1 2 3 119899 Then it is obvious that 119877(11990911199092sdot sdot sdot

119909119894minus1

11990911199092sdot sdot sdot 119909119894) = (120583

119875

119860(119909119894) 120583119873

119860(119909119894)) for 119894 = 1 2

119899 where 1199090

= 1 Therefore (1 1199091 11990911199092 119909

11199092sdot sdot sdot 119909119899)

is a path from 1 to 11990911199092sdot sdot sdot 119909119899 Since119866 is a Hasse diagram we

have 119877(1 11990911199092sdot sdot sdot 119909119899) = 0 This implies that 120583119875

119860(11990911199092sdot sdot sdot 119909119899) =

0 and 120583119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Conversely suppose that for

any collection 1199091 1199092 119909119899of vertices in 119881 with 119899 ge 2

and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we

have 120583119875119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Let

(1199090 1199091 1199092 119909119899) be a path in 119866 from 119909

0to 119909119899with 119899 ge

2 Then 119877(119909119894minus1

119909119894) gt 0 for 119894 = 1 2 119899 Therefore

120583119875

119860(119909minus1

119894minus1119909119894) gt 0 120583

119873

119860(119909minus1

119894minus1119909119894) lt 0 for 119894 = 1 2 119899

Now consider the elements 119909minus1

01199091 119909minus1

11199092 119909

minus1

119899minus1119909119899in 119881

Then by assumption 120583119875

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 and

120583119873

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 That is 120583119875

119860(119909minus1

0119909119899) = 0 and

120583119873

119860(119909minus1

0119909119899) = 0 Hence 119877(119909

0 119909119899) = 0 Thus 119866 is a Hasse

diagram

Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is connected (weakly connected semiconnected locallyconnected or quasi-connected) if and only if the inducefuzzy graph (119881 119877

+

0) is connected (weakly connected semi-

connected locally connected or quasi-connected)

Definition 28 Let (119878 lowast) be a semigroup and let 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then the subsemigroup gener-ated by119860 is the meeting of all bipolar fuzzy subsemigroups of119878 which contains 119860 It is denoted by ⟨119860⟩

Lemma 29 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then bipolar fuzzy subset ⟨119860⟩ isprecisely given by ⟨120583119875

119860⟩(119909) = or120583

119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

119909 = 11990911199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 ⟨120583119873

119860⟩(119909) =

and120583119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878

Proof Let1198601015840 = (119875

119860 119873

119860) be a bipolar fuzzy subset of 119878 defined

by 119875119860(119909) = or120583

119875

119860(1199091) and120583119875

119860(1199092) and sdot sdot sdot and 120583

119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 119873

119860(119909) = and120583

119873

119860(1199091) or

120583119873

119860(1199092)orsdot sdot sdotor120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119873

119860(119909119894) lt 0 for 119894 =

1 2 119899 for any 119909 isin 119878 Let 119909 119910 isin 119878 If 120583119875119860(119909) = 0 or 120583119875

119860(119910) =

0 then 120583119875

119860(119909) and 120583

119875

119860(119910) = 0 and 120583

119873

119860(119909) = 0 or 120583119873

119860(119910) = 0 and

then 120583119873

119860(119909) or 120583

119873

119860(119910) = 0 Therefore 119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Again if 120583119875

119860(119909) = 0 120583

119873

119860(119909) = 0

then by definition of 119875119860(119909) and

119873

119860(119909) we have

119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Hence (119875

119860 119873

119860)

is a bipolar fuzzy subsemigroup of 119878 containing (120583119875

119860 120583119873

119860)

Now let 119871 be any bipolar fuzzy subsemigroup of 119878 containing(120583119875

119860 120583119873

119860) Then for any 119909 isin 119878 with 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 we have 120583

119875

119871(119909119894) ge

120583119875

119871(1199091)and120583119875

119871(1199092)andsdot sdot sdotand120583

119875

119871(119909119899) ge 120583119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

and 120583119873

119871(119909119894) ge 120583

119873

119871(1199091) and 120583119873

119871(1199092) and sdot sdot sdot and 120583

119873

119871(119909119899) ge 120583

119873

119860(1199091) and

120583119873

119860(1199092)and sdot sdot sdotand120583

119873

119860(119909119899)Thus 120583119875

119871(119909) ge or120583

119875

119860(1199091)and120583119875

119860(1199092)and sdot sdot sdotand

120583119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899

120583119873

119871(119909) le and120583

119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878 Hence

120583119875

119871(119909) ge

119875

119860(119909) 120583119873

119871(119909) le

119873

119860(119909) for all 119909 isin 119878 Thus 119875

119860(119909) le

120583119875

119871(119909) 119873

119860(119909) ge 120583

119875

119860(119909) Thus 1198601015840 = (

119875

119860 119873

119860) is the meeting of

all bipolar fuzzy subsemigroups containing (120583119875

119860 120583119873

119860)

Theorem 30 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then for any 120572 isin [0 1](⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) and (⟨(120583

+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) =

(⟨120583119875

⟩

+

120572 ⟨120583119873

⟩

+

120572) where (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) denotes the subsemigroup

generated by (120583119875

120572 120583119873

120572) and ⟨(120583

119875

120583119873

)⟩ denotes bipolar fuzzysubsemigroup generated by (120583119875 120583119873)

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Hence 1198772

120583119875

(119909 119910) le 119877120583119875(119909 119910) and 119877

2

120583119873

(119909 119910) ge 119877120583119873(119909 119910)

Hence 119877 is transitive

We conclude that

Theorem 24 A bipolar fuzzy relation 119877 is a partial order ifand only if 119860 = (120583

119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of

(119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1

Theorem 25 A bipolar fuzzy relation 119877 is a linear order ifand only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup of (119881 lowast)

satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) 119909 (120583119875

119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) = 1 minus1(iii) 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 = 119881

Proof Suppose 119877 is a linear order Then by Theorem 24conditions (i) (ii) and (iii) are satisfied For any 119909 isin 119881(119877 or 119877

minus1

)(1 119909) gt 0 This implies that 119877(1 119909) or 119877(119909 1) gt 0Hence 119909 120583

119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0Conversely suppose that conditions (i) (ii) and (iii) hold

Then by Theorem 24 119877 is partial order Now for any 119909 119910 isin

119881 we have (119909minus1

119910) (119910minus1119909) isin 119881 Then by condition (iv) 119909

120583119875

119860(119909) or 120583

119875

119860(119909minus1

) gt 0 120583119873

119860(119909) and 120583

119873

119860(119909minus1

) lt 0 Therefore 119877 islinear order

Theorem 26 A bipolar fuzzy relation 119877 is a equivalencerelation if and only if (120583119875

119860 120583119873

119860) is a bipolar fuzzy subsemigroup

of (119881 lowast) satisfying

(i) 120583119875119860(1) = 1 and 120583

119873

119860(1) = minus1

(ii) (120583119875119860(119909) 120583119873

119860(119909)) = (120583

119875

119860(119909minus1

) 120583119873

119860(119909minus1

)) for all 119909 isin 119881

Theorem 27 119866 is a Hasse diagram if and only if for anycollection 119909

1 1199092 1199093 119909

119899of vertices in 119881 with 119899 ge 2 and

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we have

120583119875

119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0

Proof Suppose 119866 is a Hasse diagram and let 1199091 1199092 119909

119899

be vertices in 119881 with 119899 ge 2 and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt

0 for 119894 = 1 2 3 119899 Then it is obvious that 119877(11990911199092sdot sdot sdot

119909119894minus1

11990911199092sdot sdot sdot 119909119894) = (120583

119875

119860(119909119894) 120583119873

119860(119909119894)) for 119894 = 1 2

119899 where 1199090

= 1 Therefore (1 1199091 11990911199092 119909

11199092sdot sdot sdot 119909119899)

is a path from 1 to 11990911199092sdot sdot sdot 119909119899 Since119866 is a Hasse diagram we

have 119877(1 11990911199092sdot sdot sdot 119909119899) = 0 This implies that 120583119875

119860(11990911199092sdot sdot sdot 119909119899) =

0 and 120583119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Conversely suppose that for

any collection 1199091 1199092 119909119899of vertices in 119881 with 119899 ge 2

and 120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 3 119899 we

have 120583119875119860(11990911199092sdot sdot sdot 119909119899) = 0 and 120583

119873

119860(11990911199092sdot sdot sdot 119909119899) = 0 Let

(1199090 1199091 1199092 119909119899) be a path in 119866 from 119909

0to 119909119899with 119899 ge

2 Then 119877(119909119894minus1

119909119894) gt 0 for 119894 = 1 2 119899 Therefore

120583119875

119860(119909minus1

119894minus1119909119894) gt 0 120583

119873

119860(119909minus1

119894minus1119909119894) lt 0 for 119894 = 1 2 119899

Now consider the elements 119909minus1

01199091 119909minus1

11199092 119909

minus1

119899minus1119909119899in 119881

Then by assumption 120583119875

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 and

120583119873

119860(119909minus1

01199091119909minus1

11199092sdot sdot sdot 119909minus1

119899minus1119909119899) = 0 That is 120583119875

119860(119909minus1

0119909119899) = 0 and

120583119873

119860(119909minus1

0119909119899) = 0 Hence 119877(119909

0 119909119899) = 0 Thus 119866 is a Hasse

diagram

Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is connected (weakly connected semiconnected locallyconnected or quasi-connected) if and only if the inducefuzzy graph (119881 119877

+

0) is connected (weakly connected semi-

connected locally connected or quasi-connected)

Definition 28 Let (119878 lowast) be a semigroup and let 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then the subsemigroup gener-ated by119860 is the meeting of all bipolar fuzzy subsemigroups of119878 which contains 119860 It is denoted by ⟨119860⟩

Lemma 29 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then bipolar fuzzy subset ⟨119860⟩ isprecisely given by ⟨120583119875

119860⟩(119909) = or120583

119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

119909 = 11990911199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 ⟨120583119873

119860⟩(119909) =

and120583119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878

Proof Let1198601015840 = (119875

119860 119873

119860) be a bipolar fuzzy subset of 119878 defined

by 119875119860(119909) = or120583

119875

119860(1199091) and120583119875

119860(1199092) and sdot sdot sdot and 120583

119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119875

119860(119909119894) gt 0 for 119894 = 1 2 119899 119873

119860(119909) = and120583

119873

119860(1199091) or

120583119873

119860(1199092)orsdot sdot sdotor120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119873

119860(119909119894) lt 0 for 119894 =

1 2 119899 for any 119909 isin 119878 Let 119909 119910 isin 119878 If 120583119875119860(119909) = 0 or 120583119875

119860(119910) =

0 then 120583119875

119860(119909) and 120583

119875

119860(119910) = 0 and 120583

119873

119860(119909) = 0 or 120583119873

119860(119910) = 0 and

then 120583119873

119860(119909) or 120583

119873

119860(119910) = 0 Therefore 119875

119861(119909119910) ge 120583

119875

119860(119909) and 120583

119875

119860(119910)

and 119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Again if 120583119875

119860(119909) = 0 120583

119873

119860(119909) = 0

then by definition of 119875119860(119909) and

119873

119860(119909) we have

119875

119861(119909119910) ge

120583119875

119860(119909) and 120583

119875

119860(119910) and

119873

119861(119909119910) le 120583

119873

119860(119909) or 120583

119873

119860(119910) Hence (119875

119860 119873

119860)

is a bipolar fuzzy subsemigroup of 119878 containing (120583119875

119860 120583119873

119860)

Now let 119871 be any bipolar fuzzy subsemigroup of 119878 containing(120583119875

119860 120583119873

119860) Then for any 119909 isin 119878 with 119909 = 119909

11199092sdot sdot sdot 119909119899with

120583119875

119860(119909119894) gt 0 120583

119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 we have 120583

119875

119871(119909119894) ge

120583119875

119871(1199091)and120583119875

119871(1199092)andsdot sdot sdotand120583

119875

119871(119909119899) ge 120583119875

119860(1199091)and120583119875

119860(1199092)andsdot sdot sdotand120583

119875

119860(119909119899)

and 120583119873

119871(119909119894) ge 120583

119873

119871(1199091) and 120583119873

119871(1199092) and sdot sdot sdot and 120583

119873

119871(119909119899) ge 120583

119873

119860(1199091) and

120583119873

119860(1199092)and sdot sdot sdotand120583

119873

119860(119909119899)Thus 120583119875

119871(119909) ge or120583

119875

119860(1199091)and120583119875

119860(1199092)and sdot sdot sdotand

120583119875

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899with 120583

119875

119860(119909119894) gt 0 for 119894 = 1 2 119899

120583119873

119871(119909) le and120583

119873

119860(1199091) or 120583119873

119860(1199092) or sdot sdot sdot or 120583

119873

119860(119909119899) 119909 = 119909

11199092sdot sdot sdot 119909119899

with 120583119873

119860(119909119894) lt 0 for 119894 = 1 2 119899 for any 119909 isin 119878 Hence

120583119875

119871(119909) ge

119875

119860(119909) 120583119873

119871(119909) le

119873

119860(119909) for all 119909 isin 119878 Thus 119875

119860(119909) le

120583119875

119871(119909) 119873

119860(119909) ge 120583

119875

119860(119909) Thus 1198601015840 = (

119875

119860 119873

119860) is the meeting of

all bipolar fuzzy subsemigroups containing (120583119875

119860 120583119873

119860)

Theorem 30 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860)

be a bipolar fuzzy subset of 119878 Then for any 120572 isin [0 1](⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) and (⟨(120583

+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) =

(⟨120583119875

⟩

+

120572 ⟨120583119873

⟩

+

120572) where (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) denotes the subsemigroup

generated by (120583119875

120572 120583119873

120572) and ⟨(120583

119875

120583119873

)⟩ denotes bipolar fuzzysubsemigroup generated by (120583119875 120583119873)

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

Proof

119909 isin (⟨120583119875

⟩120572

⟨120583119873

⟩120572

)

lArrrArr there exists 1199091 1199092 119909

119899in (120583119875

120572 120583119873

120572)

such that 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr there exists 1199091 1199092 119909

119899in 119878

such that 120583119875 (119909119894) ge 120572 120583

119873

(119909119894) le 120572

forall119894 = 1 2 119899 119909 = 11990911199092sdot sdot sdot 119909119899

lArrrArr ⟨120583119875

⟩ (119909) ge 120572 ⟨120583119873

⟩ (119909) le 120572

lArrrArr 119909 isin ⟨120583119875

⟩120572

119909 isin ⟨120583119873

⟩120572

(15)

Therefore (⟨120583119875

120572⟩ ⟨120583119873

120572⟩) = (⟨120583

119875

⟩120572 ⟨120583119873

⟩120572) Similarly we have

(⟨(120583+

)119875

120572⟩ ⟨(120583+

)119873

120572⟩) = (⟨120583

119875

⟩

+

120572 ⟨120583119873

⟩

+

120572)

Remark 31 Let (119878 lowast) be a semigroup and 119860 = (120583119875

119860 120583119873

119860) be

a bipolar fuzzy subset of 119878 Then by Theorem 30 we have⟨supp(119860) = 119860

+

⟩ = supp⟨119860⟩

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875 120583119873) Then we have the following results

Theorem 32 Let 119860 be any subset of 1198811015840 and 1198661015840

= (1198811015840

1198771015840

) bethe Cayley graph induced by (1198811015840 lowast 119860) Then 119866

1015840 is connected ifand only if ⟨119860⟩ supe 119881 minus V

1

Theorem 33 119866 is connected if and only if supp⟨119860⟩ supe 119881 minus V1

Theorem 34 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is weakly connected if and only if ⟨119860 cup119860minus1

⟩ supe 119881minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Definition 35 Let (119878 lowast) be a group and let 119860 be a bipolarfuzzy subset of 119878 Then we define 119860minus1 as bipolar fuzzy subsetof 119878 given by 119860

minus1

(119909) = 119860(119909minus1

) for all 119909 isin 119878

Theorem 36 119866 is weakly connected if and only if supp(⟨119860 cup

119860minus1

⟩) supe 119881 minus V1

Proof

119866 is weakly connected

lArrrArr (119881 119877+

0) is weakly connected

lArrrArr ⟨119860+

0cup (119860+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860) cup supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860 cup 119860minus1

⟩ supe 119881 minus V1

(16)

Theorem 37 Let 119860 be any subset of a set 1198811015840 and let 1198661015840 =

(1198811015840

1198771015840

) be the Cayley graph induced by the triplet (1198811015840 lowast 119860)Then 119866

1015840 is semiconnected if and only if ⟨119860⟩ cup ⟨119860minus1

⟩ supe 119881 minus V1

where 119860minus1 = 119909minus1

119909 isin 119860

Theorem 38 119866 is semi-connected if and only if supp(⟨119860⟩ cup

⟨119860minus1

⟩) supe 119881 minus V1

Proof

119866 is semiconnected

lArrrArr (119881 119877+

0) is semi connected

lArrrArr ⟨119860+

0⟩ cup ⟨(119860

+

0)

minus1

⟩ supe 119881 minus V1

lArrrArr ⟨supp (119860)⟩ cup ⟨supp (119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp ⟨119860⟩ cup ⟨(119860)minus1

⟩ supe 119881 minus V1

lArrrArr supp (⟨119860⟩cup⟨119860minus1

⟩) supe 119881 minus V1

(17)

Theorem 39 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860)Then119866

1015840 is locally connected if and onlyif ⟨119860⟩ = ⟨119860

minus1

⟩ where 119860minus1 = (119909minus1

119909 isin 119860)

Theorem 40 Let 119866 is locally connected if and only ifsupp(⟨119860⟩) = supp(⟨119860minus1⟩)

Proof

119866 is locally connected lArrrArr (119881 119877+

0) is locally connected

lArrrArr ⟨119860+

0⟩ = ⟨(119860

+

0)

minus1

⟩

lArrrArr ⟨supp (119860)⟩ = ⟨supp (119860)minus1

⟩

lArrrArr supp ⟨119860⟩ = supp ⟨119860minus1

⟩

(18)

Theorem 41 Let 1198661015840 = (1198811015840

1198771015840

) be the Cayley graph inducedby the triplet (1198811015840 lowast 119860) where 119881

1015840 is finite Then 1198661015840 is quasi-

connected if and only if it is connected

Theorem 42 A finite Cayley bipolar fuzzy graph 119866 is quasi-connected if and only if it is connected

Proof

119866 is quasi-connected lArrrArr (119881 119877+

0) is quasi-connected

lArrrArr (119881 119877+

0) is connected

lArrrArr 119866 is connected(19)

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

Definition 43 The 120583119875 strength of a path 119875 = V

1 V2 V

119899is

defined as min(1205831198752(V119894 V119895)) for all 119894 and 119895 and is denoted by

119878119875

120583 The 120583

119873 strength of a path 119875 = V1 V2 V

119899is defined as

max(1205831198732(V119894 V119895)) for all 119894 and 119895 and is denoted by 119878

119873

120583

Definition 44 Let 119866 = (119881 120583119875

120583119873

) be a bipolar fuzzy graphThen 119866 is said to be

(1) 120572-connected if for every pair of vertices 119909 119910 isin 119866 thereis a path 119875 from 119909 to 119910 such that strength (119875) ge 120572

(2) weakly 120572-connected if a bipolar fuzzy graph (119881 119877 or

119877minus1

) is 120572-connected

(3) semi-120572-connected if for every 119909 119910 isin 119881 there is a pathof strength greater than or equal to 120572 from 119909 to 119910 orfrom 119910 to 119909 in 119866

(4) locally 120572-connected if for every pair of vertices 119909 and119910 there is a path119875 of strength greater than or equal to120572 from 119909 to 119910 whenever there is a path 119875

1015840 of strengthgreater than or equal to 120572 from 119910 to 119909

(5) quasi-120572-connected if for every pair 119909 119910 isin 119881 there issome 119911 isin 119881 such that there is directed path from 119911 to119909 of strength greater than or equal to 120572 and there is adirected path from 119911 to 119910 of strength greater than orequal to 120572

Remark 45 Let 119866 = (119881 119877) be any bipolar fuzzy graph then119866 is 120572-connected (weakly 120572-connected semi 120572-connectedlocally 120572-connected or quasi 120572-connected) if and only if theinduce fuzzy graph (119881 119877

+

0) is connected (weakly connected

semiconnected locally connected or quasi-connected)

Let 119866 denote the Cayley bipolar fuzzy graphs 119866 = (119881 119877)

induced by (119881 lowast 120583119875

120583119873

) Also for any 120572 isin [minus1 1] we havethe following results

Theorem 46 119866 is 120572-connected if and only if ⟨119860⟩120572supe 119881 minus V

1

Proof

119866 is connected lArrrArr (119881 119877120572) is connected

lArrrArr ⟨119860120572⟩ supe 119881 minus V

1

lArrrArr ⟨119860⟩120572supe 119881 minus V

1

(20)

Theorem 47 119866 is weakly 120572-connected if and only if⟨119860 cup 119860

minus1

⟩120572supe 119881 minus V

1

Proof

119866 is weakly connected lArrrArr (119881 119877120572) is weakly connected

lArrrArr ⟨119860120572cup (119860120572)minus1

⟩ supe 119881 minus V1

lArrrArr ⟨(119860 cup 119860minus1

)120572

⟩ supe 119881 minus V1

lArrrArr ⟨119860 cup (119860)minus1

⟩120572

supe 119881 minus V1

(21)

Theorem 48 119866 is semi-120572-connected if and only if (⟨119860⟩120572cup

⟨119860minus1

⟩120572) supe 119881 minus V

1

Theorem 49 Let 119866 be locally 120572-connected if and only if⟨119860⟩120572= ⟨119860minus1

120572⟩

Theorem 50 A finite Cayley bipolar fuzzy graph 119866 is quasi-120572-connected if and only if it is 120572-connected

4 Conclusions

Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim of reducing the differences between the traditionalnumerical models used in engineering and sciences and thesymbolic models used in expert systems A bipolar fuzzyset is a generalization of the notion of a fuzzy set Wehave introduced the notion of Cayley bipolar fuzzy graphsin this paper The natural extension of this research workis application of bipolar fuzzy digraphs in the area of softcomputing including neural networks decision making andgeographical information systems

Acknowledgments

This Project was funded by the Deanship of ScientificResearch (DSR) King Abdulaziz University Jeddah underGrant no 363-014-D1434 The authors therefore acknowl-edge with thanks DSR technical and financial support

References

[1] B Alspach andMMishna ldquoEnumeration of Cayley graphs anddigraphsrdquo Discrete Mathematics vol 256 no 3 pp 527ndash5392002

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Control vol 8 no 3pp 338ndash353 1965

[3] W-R Zhang ldquoBipolar fuzzy sets and relations a computationalframework for cognitive modeling and multiagent decisionanalysisrdquo in Proceedings of the 1st International Joint Con-ference of The North American Fuzzy Information ProcessingSociety Biannual Conference The Industrial Fuzzy Control andIntelligent Systems Conference and the NASA Joint TechnologyWorkshop on Neural Networks and Fuzzy Logic pp 305ndash309December 1994

[4] S Y Wu ldquoThe Compositions of fuzzy digraphsrdquo Journal ofResearch in Education Sciences vol 31 pp 603ndash628 1986

[5] A Kauffman Introduction a la Theorie des Sous-emsemblesFlous vol 1 Masson et Cie 1973

[6] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2nd edition 2001

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

[7] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011

[8] M Akram ldquoBipolar fuzzy graphs with applicationsrdquo KnowledgeBased Systems vol 39 pp 1ndash8 2013

[9] M Akram and W A Dudek ldquoRegular bipolar fuzzy graphsrdquoNeural Computing and Applications vol 21 no 1 pp 197ndash2052012

[10] M H Shahzamanian M Shirmohammadi and B DavvazldquoRoughness inCayley graphsrdquo Information Sciences vol 180 no17 pp 3362ndash3372 2010

[11] N M M Namboothiri V A Kumar and P T RamachandranldquoCayley fuzzy graphsrdquo Far East Journal of Mathematical Sci-ences vol 73 pp 1ndash15 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of