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Fuzzy Norm on Fuzzy Autocatalytic Set (FACS) of Fuzzy Graph Type-3

1Umilkeram Qasim Obaid and 2Tahir Ahmad

1Department of Mathematics, College of Science, AL-Mustansiriya University, Baghdad, Iraq. 2Department of Mathematical Science and Centre for Sustainable Nano Materials, IbnuSina Institute for Scientific and Industrial Research, Universiti Teknologi Malaysia, 81310 Skudai, Johor, Malaysia.

Address For Correspondence: Umilkeram Qasim Obaid, Department of Mathematics, College of Science, AL-Mustansiriya University, Baghdad, Iraq. E-mail: [email protected]

ToCite ThisArticle:UmilkeramQasimObaid and 1Tahir Ahmad. Fuzzy Norm on Fuzzy Autocatalytic Set (FACS) of Fuzzy Graph Type-3. Mathematics and Statistics Journal, 3(1): 12-24, 2017

A B S T R A C T

Fuzzy Autocatalytic Set (FACS) of fuzzy graph Type-3 particularly on the fuzziness of normed structure

of FACS and its relation to cycles in FACS is explored. In other words, a fuzzy norm on a FT3-cycle

space of FACS which will called a fuzzy normed cycle space of FACSis given and interpreted the catalytic

chain reaction between two cycles in FT3-cycle space of FACS. We investigate some of its

important properties such as the notions of convergence, Cauchyness and completeness infuzzy normed

cycle space of FACS. Then, the application of the fuzzy normed cycle space of FACS to the clinical

incineration process is elaborated.

Keywords:FT3-cycle space of FACS, fuzzy autocatalytic set, fuzzy graph, incineration process, normed space

of FACS

INTRODUCTION

Fuzzy theory has been studied for quite extensively in conjunction with problems in mathematics subjects

including set theory, graph theory, measure theory, control theory, differential equations, topology and most

recently functional analysis. The fuzzy characteristics that were mainly introduced by Zadeh in 1965 were

merged into the crisp graph. In this regard, Rosenfeld (1975)unveiled his design for the basic structure of fuzzy

graph. Ever since that, fuzzy graph has been promptly expanded and implemented in diverse fields. The study

on a modeling of clinical waste incineration process in Malacca (Sabariah, 2005; Tahir et al., 2010) is a

significant example of the implementation of fuzzy graph theory.

The notion of an Autocatalytic Set (ACS) as initiated by Jain and Krishna (1998; 2003) has been integrated

with graph theoretical concepts in 1998. Nevertheless, ACS was deficient in explaining the incineration process

(Sabariahet al., 2009). This led to the development of the modeling work of the process and adapted the fuzzy

logic as an integral part of the system. In other words, fuzzy graph had succeeded to apply the modeling of this

process as presented in Figure 1.Six main variables specified in the process were modelled as vertices (nodes)

and the catalytic relationships were presented as edges (links).

Representing the process as fuzzy graph made amalgamation three notions which are graph, autocatalytic

set and fuzzy. This study was focused on fuzzy graph of type-3 and innovated a fresh notion called Fuzzy

Autocatalytic Set (FACS) of fuzzy graph type-3. It was shown that the modeling fuzzy work (FACS) of the

incineration process was more precise and suitable in describing the dynamics of this process (Tahir et al.,

2010).

http://www.iwnest.com/MSJ/

13 UmilkeramQasimObaid and Tahir Ahmad, 2017

Mathematics and Statistics Journal, 3(2) April 2017, Pages: 12-24

Fig. 1: FACS of fuzzy graph Type-3 for the incineration process (Tahir etal., 2010)

The structure of FACS with its relation to normed space and its cycles was taken into consideration in more

investigation on FACS of fuzzy graph Type-3 and the richness of directed graph through functional analysis

advantages (Umilkeram and Tahir, 2016). More precisely, a new concept namely normed space of FACS of

fuzzy graph Type-3 was defined and implemented in the modelling of the incineration process.

This paper focuses on the representation of FACS in fuzziness of normed structure of FACS by the

construction a fuzzy normed cycle space of FACS that will be presented in this paper. In fact, a notion of a

fuzzy norm in a fuzzy graph theory setting is established and interpreted the catalytic chain reaction between

two cycles in this structure of FACS and employed in the incineration process that have different catalytic chain

reaction because of its fuzzy norm structure. In addition, some important results would be proven involving

convergence, Cauchyness and completeness of this structure of FACS.

Preliminaries:

Some of the basic concepts and results that included definitions and theorems pertaining to this research are

introduced. The theoretical foundations of FACS are also presented.A directed graph is a certain trends of a

(undirected) graph which the lines (edges) are labeled from points (vertices) to other as shown in Figure 2.

Henceforward, this kind of graph will be studied throughout this study.

Fig. 2:Examples of Graphs. (a) Directed (b) Undirected

A directed graph is introduced by a relation on a set where denotes the set of

vertices and denotes the set of its edges (Epp, 1993). A directed graph is also called a crisp graph if all the

values of edges are 1 or 0 and it is called a fuzzy graph if its values is between 0 and 1. In other words, a fuzzy

graph is with a vertex set as the underlying set with a pair of functions such that and is a fuzzy relation on with ≤ for all and denotes the minimum of and (Rosenfeld, 1975). A path : , , ,..., , from a

vertex to a vertex in a fuzzy graph if its sequence of distinct vertices and edges starting from and ending

at such that the membership value for . If and coincide in a path then we

call as a cycle. Note that the underlying crisp graph of the fuzzy graph is referred to .

(a) (b)

V1: Waste (particularly clinical waste)

V2: Fuel

V3: Oxygen

V4: Carbon Dioxide

V5: Carbon Monoxide

V6: Other Gases including Water



Moreover, Blue et al. (1997; 2002) have given five interpretations of the fuzziness of graph that led to study

the choice of one of its type to best describe the model of incineration process. Thereby, Tahir et al. (2010)

found that Type-3 is to be the most convenient for the assignment of improving the model of the process as

explained in (Sabariahet al., 2009). The Type-3 of fuzzy graph is defined as having crisp vertices and edges with

fuzzy edge connectivity of the edges i.e. the edge has fuzzy head and tail. With this consideration, the

following definition was adopted in obtaining the fuzzy graph representing the clinical incineration process. The

formal definition of FACS is given as follows: Fuzzy autocatalytic set (FACS) is a subgraph where each of

whose nodes has at least one incoming link with membership value (Tahir et al., 2010).

Aforementioned, Figure 1 is a fuzzy graphical for the modelof the incineration process with 6 variables

(namely waste ( ), fuel ( ), oxygen ( ), carbon dioxide ( ), carbon monoxide ( ), and other gases

including water ( )) and 15 edges which based on the catalytic relationship between the variables. Thus,

is the set of vertices and is the set of edges where for and . Hence, fifteen edges are the connection between these variables in the

process and the membership values of each fuzzy connectivity of edge are given as follows:

, ,

, ,

, ,

, ,

, ,

, ,

, .

,

In this context, a cycle in a fuzzy graph is a directed closed path of its vertices such that each edge and each

vertex (except starting point) is visited only once. Then, FACS of fuzzy graph Type-3 particularly on the

structures of normed spaces and its relations to cycles in FACS has been studied by Umilkeram and Tahir

(2016). Several new notions namely FT3-fuzzy detour cycles of FACS, a FT3-cycle space of FACS as a vector

space, and normed space of FACS of fuzzy graph Type-3 were presented as follows.

Definition 1:

Let be a FACS of fuzzy graph Type-3. The FT3-cycle is a closed path of distinct

vertices , , . . ., (except = ) such that the membership value , , for each fuzzy edge connectivity of FACS and n is the number of edges in this cycle. The length of

FT3-cycle in FACS is calculated by

=

such that each edge in this cycle

is traversed in the right direction (see Figure 3).

Definition 2:

Let n be the number of all edges in the FT3-cycle containing an edge in FACS. A length of FT3-cycle

containing the edge with n, denoted by , in FACS of fuzzy graph Type-3 is defined as the maximum

length of any FT3-cycle that goes through the edge with the number of all edges in these FT3-cycles is the

same and equal to n. A FT3-cycle containing the edge of length is called FT3-fuzzy detour cycle

such that

.

Fig. 3: FT3-fuzzy detour cycle

in FACS



Definition 3:

For a graph , let denote the set of all FT3-fuzzy detour cycles and edge-disjoint

unions of these cycles in FACS, and an empty graph which means all vertices separate from each other.

Then, is called the FT3-cycle space of . Thereafter,Umilkeram and Tahir (2016) proved that under the sum operation and

multiplicationoperation (see Theorem 1) forms a vector space over {0, 1} (with addition and multiplication

modulo 2), where the vector addition and the scalar multiplication are given by

(1)

and for each

, (2)

Theorem 1:

The FT3-cycle space of , , is a vector space over {0, 1} (with addition and

multiplication modulo 2).

Then, the normed space of FACSwas constructed using the notion of the FT3-cycle space of FACS as a

vector space over together with the following norm.

= max

(3)

This shows that the concept of the norm of a vector in the FT3-cycle space of , is a

generalization of the concept of length in . In other words, a FT3-cycle space of forms

a normed space with the function as presented by Eq. (3). Then, a new type of normed space (see Theorem 2)

which is the normed space of FACS of fuzzy graph Type-3 was presented.

Theorem 2:

Let be a FT3-cycle space of over and = max

be a real-valued function on . Then, is a normed space.

Furthermore, the basic FT3-fuzzy detour cycles for each edge in FACS of the incineration

processwith respect to this norm are determined in this process represented by the FT3-cycle space associated

with a graph of FACS of the incineration process. Consequently, the basic FT3-fuzzy detour cycles with respect

to the norm as presented in Eq. (3) when interpret physically means that each fuzzy connectivity of edge in a

FT3-fuzzy detour cycle has a certain proportion of the chemical interaction with other edges to the greatest

extent norm .

Now, we look at the idea of fuzzy norm on a FT3-cycle space of and its relation with the crisp

norm which is presented in Theorem 2. The novelty of this notion is the incorporation of a couple important

definitions in our work that are fuzzy graph represented by FACS of fuzzy graph Type-3 and fuzziness of norm

represented by fuzzy norm in our sense which is close to the adaptation of Bag and Samanta type (2013).

However, the modern notion is defined in such a way that the corresponding metric fuzziness is fuzzy quasi

(pseudo) FT3-metric of FACS as shown inUmilkeram and Tahir (2014).

RESULTS AND DISCUSSION

Fuzzy Norm on a FT3-Cycle Space of FACS:

In this section, a concept of fuzzy norm is given on a FT3-cycle space of which

defined in Definition 3. It begins with taking a function defined on as given in Eq. (4),

followed by discussion on its properties. We see how this equation can be specialized for the idea of fuzzy

norm on a FT3-cycle space as follows.

(4)

It is noted that the above function has the following properties given in the following remarks.

Remark 1:

It is clear that for all .

Clearly, for all .

The following statement is not satisfied that (for all if and only if ) for

Eq. (4), due to the second condition of the norm of (see Theorem 2). i.e. if and only if the cycle

and which means the number of all edges in this cycle is . This is a contradiction since .



Evidently, for all and ,

if (i.e. if

).

is nondecreasing function on .

If and , then it is obvious that . Now,

if , then

=

=

Hence, clearly, .

If , then .

.

In what follows, it could be also shown that as presented in Eq. (4) fulfills the main property of the

idea of fuzzy norm on a FT3-cycle space in the following theorem.

Theorem 3:

Suppose is a FT3-cycle space of over and is

a norm on . Then, defined in Eq. (4) satisfies the following relation:

for all , and .

Proof:

The relation is proved as follows:

If (1) with ; ;

(2)

(3) with ; , then it is easily verified the above relation.

If (4) with , then by the function as presented in Eq. (4) and

the fourth condition of the norm of , we have

(5)

Now, note that

=

=

=

=

(6)

since

Hence, by Eq.(5) and Eq.(6), we have the relation

and this concludes the proof.

Thus, at this placement, combining these properties with known notions on fuzzy norm in functional

analysis, especially Bag and Samanta type (2013), this brings to the notion of a fuzzy normed cycle space of

FACS as given in the following.

Definition 4:

Let be a FT3-cycle space of over and be a

norm on . Suppose is a fuzzy set defined by ,

Wheretis the number of all edges in the FT3-fuzzy detour cycle and a continuous t-norm is

theusual multiplication for all and is satisfied the following properties, for all

.



1. for all ,

2. for all ,

3. for all , ,

4. is a nondecreasing function on

5. , .

Then, an ordered pair ( , ) is said to be fuzzy normed cycle space of FACS and is

called a fuzzy norm induced by a norm on , see Figure 4.

Fig. 4: Graphical representation ofafuzzy norm induced by a norm on FACS

It is to be noted here thata fuzzy norm on in the sense that relates the greatest extent norm

to andtis the number of all edges in this cycle . Moreover, an important

observation in Definition 4 that the function is considered as the degree of nearness

, where is the number of

all edges in the FT3-fuzzy detour cycle in FACS.

= max

, where is the number of vertices

and is one of these FT3-fuzzy detour cycle

in FACS.

wheretis the number of all edges in the FT3-fuzzy

detour cycle in FACS.

If satisfies the mentioned conditions

1,2,3,4,5 in Definition 4 then isfuzzy norm

induced by norm on FACS.



between two cycles in . It means this function can be interpreted the catalytic chain

reaction between two cycles infuzzy normed cycle space of FACS.

Therefore, one can talk about a sequence of cycles in fuzzy normed cycle space of FACS and discuss

the concepts of a convergent sequence and Cauchy sequence in ( , ) in the following subsection.

Convergence in Fuzzy Normed Cycle Space of FACS:

Suppose that is FT3-fuzzy detour cycle in , then from Definition 4, has a fuzzy

norm such that the cycle containing the edge with which is the number of all edges in this

cycle. Then, take another cycle containing the edge with which is the number of all edges in this cycle

and hence, has a fuzzy norm . Consequently, the degree of nearness between two cycles

of the catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACS ( , )

is . Now, by the same argument, another cycle containing the edge with has a

fuzzy norm , then the degree of nearness between two cycles of the catalytic chain reaction

between these cycles in ( , ) is . Continuing in this way, we obtain a

sequence of FT3-fuzzy detour cycles in ( , ) which related to a fixed cycle with a fuzzy

norm as shown in Figure 5.

Fig. 5: Graphical illustration of a sequence of FT3-fuzzy detour cycles in a fuzzy normed cycle space

( , ) of FACS

Without loss of generality, it can replace by means of due to each element in

is its own negative (i.e. as usual writing , see Theorem 1). In other words, represents

inverse of the sum operation of symmetric difference as given by Eq.(1) in the vector space of FT3-cycle space

. Then, one can provide a notion of a convergent sequence of FT3-fuzzy detour cycles in

( , ) as follows.

Definition 5:

Let ( , ) be a fuzzy normed cycle space of FACS and is a fuzzy norm induced by a

norm on . Then, a sequence of FT3-fuzzy detour cycles in converges to (i.e.



) if satisfying a condition that for all . One calls a

convergent point ofa sequence , see Figure 5.

The above definition can be illustrated with the values of a function for the sequence of the

above cycles in which converges to other cycle . Thus, we could studythe degree of nearness

between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space

of FACS. In other words, we could study the relationship between these FT3-fuzzy detour cycles in FACS by a

notion of a convergent sequence for all cycles in .

Theorem 4:

The convergent point of a sequence is unique.

Proof:

Let and and assume that . Then, by Definition 5,

and for all .

Thus, we obtain

(by )

(by condition 3of Definition 4)

= .

Then, by taking limit, imply that

Thus, .

By condition 2 of Definition 4, yield that

and this is a contradiction. Therefore, and this

complete the proof.

Now, the notion of a Cauchy sequence can be given when we have many FT3-fuzzy detour cycles

in and it is complicated to find the catalytic chain reaction between these cycles that related to a fixed

cycle in ( , ).

Definition 6:

Suppose ( , ) is a fuzzy normed cycle space of FACS and is a fuzzy norm induced by a

norm on . Then, a sequence of FT3-fuzzy detour cycles in is a Cauchy sequence

if satisfying a condition that for all and (see Figure 6).



Fig. 6: Graphical illustration of a Cauchy sequence of FT3-fuzzy detour cycles in a fuzzy normed cycle

space ( , ) of FACS

Theorem 5:

Let be a convergent sequence in a fuzzy normed cycle space of FACS ( , ). Then,

is also a Cauchy sequence.

Proof:

Since is a convergent sequence in ( , ), then byDefinition 5,

for all and is convergent point of a sequence . Thus, by

( ) and for all and , we obtain

=

Now, it is clear that each subsequence of converges to the same convergent point of a sequence

and by taking limit, imply that

for all and .

By condition 2 of Definition 4, yield that

.

Hence, from Definition 6, is a Cauchy sequence in ( , ).

Common Characteristics of Fuzzy Norm and Norm on FT3-Cycle Space of FACS:

Consider the FT3-cycle space over of FACS and the norm defined on by

. Thus, from Theorem 2,

. . .

. . .



( , ) is a normed space. Also, consider the fuzzy norm induced by a norm on

and defined by ,

Thus, from Definition 4, ( , ) is a fuzzy normed cycle space of FACS.

In this section, some properties of the norm on a FT3-cycle space can be participated with

the fuzzy norm . The question that comes to our mind is to know which of these characteristics can be

generalized to this fuzzy norm. The following two theorems try to answer the question.

Theorem 6:

Let( , ) be a fuzzy normed cycle space of FACS and is a fuzzy norm deduced of the

norm on . Then, is a Cauchy sequence in ( , ) if and only if is a Cauchy

sequence in ( , ).

Proof:

Suppose is a Cauchy sequence ina normed space ( , ). It means the sequence

of FT3-fuzzy detour cycles in satisfies a condition, as usual,

for all (7)

Then, for all imply that

.

By taking limit, imply that

=

= 1 (by (7))

Therefore, is a Cauchy sequence in ( , ).

Conversely, let be a Cauchy sequence in ( , ). Then,

for all and . Thus, we have

= 1

. Hence, is a Cauchy sequence in ( , ).

Theorem 7:

Let ( , ) be a fuzzy normed cycle space of FACS and is a fuzzy norm deduced of the

norm on . Then, is a convergent sequence in ( , ) if and only if is a

convergent sequence in ( , ).

Proof:

Let is a convergent sequence in ( , ). From Definition 5, for all , imply

that

= 1

= 1

in ( , ). i.e. the sequence of FT3-fuzzy detour

cycles in is converges to in a normed space ( , ).

Thus, from previous two theorems, it is noticed that if there exists a normed space ( , ) that is

not complete, then a fuzzy normed cycle space of FACS ( , ) is also not complete that the fuzzy

norm induced by the crisp norm .

Implementation:

Implementation of Fuzzy Normed Cycles Space of FACS to Clinical Incineration Process:

The idea of a fuzzy normed cycle space of FACS is used to give a fuzzy norm associated with a graph of

FACS for the incineration process. Thus, it is easily verified that the function in Eq. (4) is satisfied the five

conditions in Definition 4 with a graph of FACS for the incineration process.Then, the fuzzy norm

of each FT3-fuzzy detour cycle in Figure 3 with respect to t which is the number of all edges in this cycle

is computed as follows.



1. The FT3-fuzzy detour cyclethat contained the edge is with norm

, then

2. The FT3-fuzzy detour cyclethat contained the edge is with norm

, then

.

3. The FT3-fuzzy detour cycle that contained the edge is withnorm

, then

.

4. The FT3-fuzzy detour cycle that contained the edge is with norm

, then

.


, then

.

6. The FT3-fuzzy detour cycle that contained the edge is

with norm , then

.

7. The FT3-fuzzy detour cycle that contained the edge is

with norm , then


, then

.


, then

.


, then

.


, then

.


, then

.

13. The FT3-fuzzy detour cycles that contained the edge is with norm

, then


, then


, then

Thus, each FT3-fuzzy detour cycle in fuzzy normed cycle space of FACS for the incineration process have

different catalytic chain reaction because of its fuzzy norm structure. Besides that, a chain reaction means here

a chemical reaction in which the products themselves causes additional reactions to take place and enhance the

reaction which under specific conditions may quicken. However, the function is

considered as the degree of nearness between two cycles of the catalytic chain of chemical reaction

between these cycles in fuzzy normed cycle space of FACS for the incineration process as explained in the

following subsection.

It was shown in Umilkeram and Tahir (2016) that the basic FT3-fuzzy detour cycles with respect to the

norm given in Eq. (3) are ten cycles due to the fact that and

Then, we attempt to explain the function as the catalytic chain reaction between two

cycles in fuzzy normed cycle space of FACS for the incineration process in the following details.

From third condition in Definition 4, note that for all , ,

. Then, we consider a FT3- fuzzy detour cycle that contained

the edge with which is the number of all edges in this cycle and observe that

and



=

.

and

=

.

=

.

=

.

=

.

=

.

=

.

=

.

=

.

Thus, the catalytic chain of chemical reaction between two cycles is more nearness between two

cycles among other cycles that associated to and which is equal to = .

Now, by the same argument, one can take another cycle that contained the edge

with which is the number of all edges in this cycle and hence, has a fuzzy norm

. Consequently, the degree of nearness between two cycles of the catalytic chain reaction

between these cycles in a fuzzy normed cycle space of FACS is and is computed as bellow.

=

.

=

.

=

.

=

.

=

.

=

.

=

.

=

.

=

.



Now, by taking another cycle that contained the edge with which is the number

of all edges in this cycle and hence, has a fuzzy norm . Then, the degree of nearness

between two cycles of the catalytic chain reaction between these cycles in a fuzzy normed cycle space

of FACS is and is computed as bellow.

=

.

=

.

=

.

=

.

=

.

=

.

=

.

=

.



=

.



Continuing in this way, we obtain a sequence of the degree of nearness between two cycles of the

catalytic chain reaction between these cycles in a fuzzy normed cycle space of FACSfor the incineration

process.

Conclusion:

In this paper, we introduce a notion of a fuzzy norm in a graph theory setting, namely, a fuzzy normed cycle

space of FACS. Indeed, constructing fuzzy norm based on the FT3-cycle space of FACS is established and

interpreted the catalytic chain reaction between two cycles in thisspace. Consequently, each FT3-fuzzy detour

cycle in fuzzy normed cycle space of FACS for the incineration process have different catalytic chain reaction

because of its fuzzy norm structure.In addition, some important results have been proven involving

convergence, Cauchyness and completeness of the fuzzy normed cycle space of FACS.

REFERENCES

Blue, M., B. Bush and J. Puckett, 1997.

Applications of Fuzzy Logic to Graph Theory. Los

Alamos National Laboratory, LA-UR-96-4792.

Blue, M., B. Bush, and J. Puckett, 2002. Unified

Approach to Fuzzy Graph Problems. Fuzzy Sets and

Systems, 125(3): 355-368.

Epp, S.S., 1993. Discrete Mathematics with

Applications. Boston: PWS Publishing Company.

Jain, S. and S. Krishna, 1998. Autocatalytic Sets

and Growth of Complexity in an Evolutionary

Model. Physical Review Letters, 81(25): 5684-5687.

Jain, S. and S. Krishna, 2003. Graph Theory and

the Evolution of Autocatalytic Networks. In:

Handbook of Graphs and Networks. Bornholdt, S.

and H. G. Schuster (Eds.). Berlin: John Wily and

VCH Publishers, pp: 355-395.

Rosenfeld, A., 1975. Fuzzy Graphs. In: Fuzzy

Sets and Their Applications to Cognitive and

Decision Processes. Zadeh, L. A., K.S. Fu and M.

Shimura (Eds.). New York: Academic Press.

Sabariah, B., 2005. Thinking Mathematical:

Modeling of a Clinical Waste Incineration Process

using Novel Fuzzy Autocatalytic Set, PhD. Thesis,

UniversitiTeknologi Malaysia-Johore, Malaysia.

Sabariah, B., A. Tahir, and M. Rashid, 2009.

Relationship between Fuzzy Edge Connectivity and

the Variables in Clinical Waste Incineration Process.

MATEMATIKA, 25(1): 31-38.

Tahir, A., B. Sabariah and A.A. Khairil, 2010.

Modeling a Clinical Incineration Process Using

Fuzzy Autocatalytic Set. Journal of Mathematical

Chemistry, 47(4): 1263-1273.

Umilkeram, Q.O. and A. Tahir, 2014. Fuzzy

quasi-metric spaces of Fuzzy Autocatalytic Set

(FACS) of fuzzy graph Type-3. Life Science Journal,

11(9): 738-746.

Umilkeram, Q.O. and A. Tahir, 2016. Novel

Structures of Normed Spaces with Fuzzy

Autocatalytic Set (FACS) of FuzzyGraph Type-3.

Research Journal of Applied Sciences, Engineering

and Technology, 12(5): 562-573.

Zadeh, L.A., 1965. Fuzzy sets. Information and

Control, 8: 338-353.

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