저 시-비 리- 경 지 2.0 한민

는 아래 조건 르는 경 에 한하여 게

l 저 물 복제, 포, 전송, 전시, 공연 송할 수 습니다.

다 과 같 조건 라야 합니다:

l 하는, 저 물 나 포 경 , 저 물에 적 된 허락조건 명확하게 나타내어야 합니다.

l 저 터 허가를 면 러한 조건들 적 되지 않습니다.

저 에 른 리는 내 에 하여 향 지 않습니다.

것 허락규약(Legal Code) 해하 쉽게 약한 것 니다.

Disclaimer

저 시. 하는 원저 를 시하여야 합니다.

비 리. 하는 저 물 리 목적 할 수 없습니다.

경 지. 하는 저 물 개 , 형 또는 가공할 수 없습니다.

교육학석사학위논문

On the matrix sequence{Γ(Am)}∞m=1 for a Boolean matrix A

whose digraph is linearly connected(Bool행렬 A의그래프가선형연결그래프일때,

행렬열 {Γ(Am)}∞m=1에대한연구)

2014년 8월

서울대학교대학원

수학교육과

최지훈

On the matrix sequence{Γ(Am)}∞m=1 for a Boolean matrix A

whose digraph is linearly connected(Bool행렬 A의그래프가선형연결그래프일때,

행렬열 {Γ(Am)}∞m=1에대한연구)

지도교수김서령

이논문을교육학석사학위논문으로제출함

2014년 7월

서울대학교대학원

수학교육과

최지훈

최지훈의교육학석사학위논문을인준함

2014년 7월

위 원 장 조 한 혁 (인)

부위원장 유 연 주 (인)

위 원 김 서 령 (인)

On the matrix sequence

{Γ(Am)}∞m=1 for a Boolean matrix A

whose digraph is linearly connected

A dissertation

submitted in partial fulfillment

of the requirements for the degree of

Master of Science in Mathematics Education

to the faculty of the Graduate School of

Seoul National University

by

Jihoon Choi

Dissertation Director : Professor Suh-Ryung Kim

Department of Mathematics EducationSeoul National University

August 2014

Abstract

In this thesis, we extend the results given by Parket al. [12] by studying the

convergence of the matrix sequence{Γ(Am)}∞m=1 for a matrixA ∈ Bn the digraph

of which is linearly connected with an arbitrary number of strong components. In

the process for generalization, we concretize ideas behindtheir arguments. We

completely characterizeA for which {Γ(Am)}∞m=1 converges. Then we find its

limit when all of the irreducible diagonal blocks are of order at least two. We go

further to characterizeA for which the limit of{Γ(Am)}∞m=1 is aJ block diagonal

matrix. All of these results are derived by studying them-step competition graph

of the digraph ofA.

Key words: irreducible Boolean(0, 1)-matrices; powers of Boolean(0, 1)-matrices;

linearly connected digraphs; index of imprimitivity;m-step competition graphs;

graph sequence; powers of digraphs.

Student Number: 2012-21424

Remark: The text of this thesis is a reprint of the material as it appears inLinear Al-

gebra Appl. (2014). The coauthor listed in this publication directed and supervised

research which forms the basis for the thesis.

i

Contents

Abstract i

1 Introduction 1

1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A preview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Convergence of{Γ(Am)}∞m=1 7

3 The limit of {Γ(Am)}∞m=1 13

3.1 The limit of{Γ(Am)}∞m=1 . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Limit of a particular form: the disjoint union of complete subgraphs 26

4 Conclusions and closing remarks 32

Abstract (in Korean) 35

ii

Chapter 1

Introduction

1.1 Preliminaries

In this section, we introduce some basic notions in combinatorial matrix theory,

which shall be used in this thesis. We assume that we are already familiar with the

basic notions in graph theory such as graphs, digraphs, neighbor, degree, adjacency

matrix, connectedness and components.

Let B = {0, 1} denote the two-element Boolean algebra with addition (+) and

multiplication (·) defined by

0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 1,

0 · 0 = 0, 0 · 1 = 0, 1 · 0 = 0, 1 · 1 = 1.

We denote byBn the set of alln × n matrices overB. We define a matrix

operatorΓ : Bn → Bn by Γ(A) = (γij) where

γij =

0 if i = j;

0 if i 6= j and the inner product of rowi and rowj of A is 0;

1 if i 6= j and the inner product of rowi and rowj of A is not0.

1

Given a matrixA ∈ Bn, there exists a unique digraph whose adjacency matrix isA.

We call such a digraph thedigraph ofA and denote it byD(A).

The greatest common divisor of all lengths of directed cycles in a nontrivial

digraphD is called theindex of imprimitivity ofD. A digraphD is said to be

primitive if D is strongly connected and has the index of imprimitivity1. LetA be a

matrix inBn. We call the index of imprimitivity ofD(A) the index of imprimitivity

of A. If D(A) is primitive, then we say thatA is primitive. If D(A) is strongly

connected, then we sayA is irreducible. For undefined terms in this paper, one may

refer to [1].

Given a digraphD, thecompetition graphC(D) of D has the same vertex set

asD and has an edge between verticesu andv if and only if there exists a common

prey ofu andv in D. If (u, v) is an arc of a digraphD, then we callv apreyof u (in

D) and callu apredatorof v (in D). A graphG is called therow graphof a matrix

M if the rows ofM are the vertices ofG, and two vertices are adjacent inG if and

only if their corresponding rows have a nonzero entry in the same column ofM .

This notion was studied by Greenberget al. [6]. As noted in [6], the competition

graph of a digraphD is the row graph of its adjacency matrix. Thus it can easily be

checked that the adjacency matrix of the competition graph of a digraphD is Γ(A)

whereA is the adjacency matrix ofD.

The notion of competition graph is due to Cohen [5] and has arisen from ecol-

ogy. Competition graphs also have applications in coding, radio transmission, and

modeling of complex economic systems. (See [14] and [15] fora summary of these

applications.)

It is well-known that for an irreducible matrixA in Bn, the matrix sequence

{Am}∞m=1 converges if and only ifA is primitive. Yet, a matrix sequence{Γ(Am)}∞m=1

might converge even if the matrixA is not primitive. For example, themth power

of the matrixA given in Figure 1.1 does not converge asm increases since it is not

primitive. However, the sequence{Γ(Am)}∞m=1 converges toA′ sinceΓ(Am) = A′

2

A =

0 1 0 1

0 0 1 0

1 0 0 0

0 0 1 0

A2 =

0 0 1 0

1 0 0 0

0 1 0 1

1 0 0 0

A3 =

1 0 0 0

0 1 0 1

0 0 1 0

0 1 0 1

A4 =

0 1 0 1

0 0 1 0

1 0 0 0

0 0 1 0

· · ·

A′ =

0 0 0 0

0 0 0 1

0 0 0 0

0 1 0 0

Figure 1.1: An example given by Parket al. [12]. We note thatA, A2, A3 are all

distinct andA4 = A. Thus{Am}∞m=1 does not converge. HoweverΓ(Am) = A′ for

each positive integerm.

for any positive integerm.

Parket al. [12] showed that{Γ(Am)}∞m=1 converges asm increases for any ir-

reducible Boolean matrixA and its limit is a block diagonal matrix each of whose

blocks consists of all1s except in the main diagonal and all0s in the main diagonal

up to conjugation by simultaneous permutation of rows and columns. They also

completely characterized a matrixA ∈ Bn whose digraph has exactly two strongly

connected components (in short, strong components) and forwhich {Γ(Am)}∞m=1

converges, and found the limit of{Γ(Am)}∞m=1 in terms of its digraph when it con-

verges. They derived these facts in terms of the competitiongraph of the digraph of

A.

Given a digraphD and a positive integerm, a vertexy is anm-step preyof a

vertexx if and only if there exists a directed walk fromx to y of lengthm. Given

a digraphD and a positive integerm, the digraphDm has the same vertex set asD

and has an arc(u, v) if and only if v is anm-step prey ofu. It is well-known that a

3

digraphD is primitive if and only ifDm equals the digraph which has all possible

arcs for anym ≥ N for some positive integerN (we call the smallest such integer

N the exponentof D). Motivated by this, Parket al. [12] introduced the notion

of convergence of{Gn}∞

n=1 as follows: A graph sequence{Gn}∞

n=1 (resp. digraph

sequence)convergesif there exists a positive integerN such thatGn is equal toGN

for anyn ≥ N . They called the graphGN the limit of the graph sequence (resp.

digraph sequence). Then they translated their goals described above into competi-

tion graph version and showed that{C(Dm)}∞m=1 converges to a graph with only

complete components asm increases ifD is strongly connected, completely char-

acterized a digraphD with exactly two strong components for which{C(Dm)}∞m=1

converges, and found the limit of{C(Dm)}∞m=1 when{C(Dm)}∞m=1 converges.

Given a positive integerm, them-step competition graphof a digraphD, de-

noted byCm(D), has the same vertex set asD and has an edge between verticesu

andv if and only if there exists anm-step common prey ofu andv. The notion of

m-step competition graph is introduced by Choet al. [4] and one of the important

variations (see the survey articles by Kim [10] and Lundgren[13] for the variations

which have been defined and studied by many authors since Cohen introduced the

notion of competition). Since its introduction, it has beenextensively studied (see

for example [2, 3, 7–9, 11, 16]). Choet al. [3] showed that for any digraphD and

a positive integerm, Cm(D) = C(Dm). Thus the limit of the graph sequence

{C(Dm)}∞m=1, if it exists, is the same as that of the graph sequence{Cm(D)}∞m=1.

Consequently studying the graph sequence{C(Dm)}∞m=1 is actually studying the

sequence ofm-step competition graphs ofD. In other words, studying the ma-

trix sequence{Γ(Am)}∞m=1 is actually studying the graph sequence{Cm(D)}∞m=1,

4

which can be seen by the following commutative digram:

Am

D(Am)

Γ(Am)

C(Dm(A))

�

Γ

C

1.2 A preview of thesis

In this thesis, we extend the results given by Parket al.[12] by studying the conver-

gence of{Γ(Am)}∞m=1 for a matrixA ∈ Bn satisfying

PAP T =

A11 A12 O O · · · O

O A22 A23 O · · · O

O O A33 A34 · · · O...

......

. . ....

...

O O O O · · · Aηη

(1.1)

for a permutation matrixP of ordern, irreducible matricesA11, A22, . . ., Aηη, and

nonzero matricesA12, A23, . . ., Aη−1,η such that if a diagonal blockAii is of order

at least two andκi is the index of imprimitivity of the digraph ofAii, then

Aii =

O A(ii)12 O · · · O O

O O A(ii)23 · · · O O

O O O · · · O O...

......

. . ....

...

O O O · · · O A(ii)κi−1,κi

A(ii)κi,1

O O · · · O O

.

5

We completely characterizeA for which {Γ(Am)}∞m=1 converges in Chapter 2.

Then, in Chapter 3, we find its limit when each ofA11, A22, . . ., Aηη is of order

at least two. In this case, the convergence of{Γ(Am)}∞m=1 is guaranteed by one of

their results:

Theorem 1.2.1(See [12]). If a digraphD is trivial or each vertex ofD has an

out-neighbor, then{C(Dm)}∞m=1 converges.

We go further to generalize one of their results to characterize a matrixA with

A11, A22, . . ., Aηη of order at least two such that the limit of{Γ(Am)}∞m=1 is aJ

block diagonal matrix. Adopting the notion defined by Parket al.[12], we mean by

a J block diagonal(for shortJBD) matrix a block diagonal matrix each of whose

blocks consists of all1s except in the main diagonal and all0s in the main diagonal.

We derive our results by studying the convergence of{C(Dm)}∞m=1 for the di-

graphD of a matrixA satisfying (1.1). The digraphD has the property that it is

weakly connected and has strong componentsD1, . . ., Dη such that there is an arc

going fromDi toDj for some distincti, j ∈ {1, . . . , η} only if j = i+1. A digraph

with this property shall be said to belinearly connected. We note that a weakly

connected digraph with two strong components is linearly connected.

Given a linearly connected digraphD with η strong components, unless other-

wise stated, we mean byD1, . . ., Dη the strong components ofD such that there is

an arc going fromDi toDj for some distincti, j ∈ {1, . . . , η} only if j = i+1 and

by Di,i+1 the subdigraph ofD induced byV (Di) ∪ V (Di+1) for eachi = 1, . . .,

η − 1. We denote byκ(Di) (κi for short) the index of imprimitivity ofDi and the

sets of imprimitivity ofDi by U(i)1 , U (i)

2 , . . ., U (i)κi for i = 1, . . ., η.

In this paper, all the graphs and digraphs are assumed to be simple.

6

Chapter 2

Convergence of{Γ(Am)}∞m=1

In this chapter, we completely characterize a matrixA ∈ Bn in the form given in

(1.1) whose digraph is linearly connected and for which{Γ(Am)}∞m=1 converges.

We denote byℓ(W ) the length of a walkW in a graph or digraph.

Lemma 2.0.2.Let D be a linearly connected digraph withη strong components.

Suppose that a vertexx has anm-step prey in a nontrivial componentDi for some

positive integerm and i ∈ {1, 2, . . . , η}. Thenx has anm′-step prey inDi for all

m′ ≥ m. Furthermore, every vertex inDi is a k-step prey ofx for some positive

integerk.

Proof. Let z ∈ V (Di) be anm-step prey ofx. Take an integerm′ satisfyingm′ ≥

m. SinceDi is nontrivial and strongly connected, there exists a closeddirected walk

of a positive length containing all the vertices inDi. By traversing such a walk, we

may find a vertexz′ in Di such that there exists a directed(z, z′)-walk of length

m′ −m in Di. Thenz′ is anm′-step prey ofx.

Take any vertexw ∈ V (Di). SinceDi is strongly connected, there exists a

directed(z, w)-walk W . Sincez is anm-step prey ofx, w is ak-step prey ofx for

7

k = m+ ℓ(W ).

The following corollary is immediately true by the above theorem.

Corollary 2.0.3. LetD be a linearly connected digraph withη strong components.

Suppose that any two verticesx and y have anm-step common prey belonging

to a nontrivial strong component for some positive integerm. Then there exists a

positive integerN such thatx andy are adjacent inC(Dm) for any integerm ≥ N .

Corollary 2.0.3 implies that, when two verticesx andy have anα-step common

prey z in a nontrivial strong component of a linearly connected digraphD for a

positive integerα, the adjacency ofx andy in the limit of {C(Dm)}∞m is determined

by not the value ofα but the fact thatx andy have a ‘step common prey’z in a

nontrivial strong component. In this context, we sometimesomit ‘α’ and just say

thatx andy have a ‘step common prey’.

The following lemma shall be frequently quoted in the rest ofthis paper.

Lemma 2.0.4(Lemma 3.4.3, [1]). Let D be a nontrivial strongly connected di-

graph, andU1, U2, . . ., Uκ(D) be the sets of imprimitivity ofD. Then there exists a

positive integerN such that ifx andy are vertices belonging respectively toUi and

Uj , then there are directed(x, y)-walks of every lengthj − i+ tκ(D) with t ≥ N .

For a digraphD and a vertexv of D, N−

D (v) denotes the set of all in-neighbors

of v.

Given a linearly connected digraphD, suppose thatDp is a nontrivial strong

component whileDp+1 is a trivial component consisting of vertexv. Let

Λ(D) :={

i | U(p)i ∩N−

D(v) 6= ∅}

= {k1, . . . , ks}.

Then, by Lemma 2.0.4, it is easy to check that for eachj ∈ Zκpandx ∈ U

(p)j , there

is a positive integerNp such that there exist(x, v)-walks of lengths(k1−j)+1+tκp,

8

U(1)1

U(1)2

U(2)1

U(2)2

U(2)3

U(2)4

v w

L1→v = {0, 1, 2}L2→v = {0, 1, 3}

L3→v = {0, 2, 3} L4→v = {1, 2, 3}

D

Figure 2.1: It is easy to see that(Li→v ∩ Lj→v) ∪ ((1 + Li→v) ∩ (1 + Lj→v)) ∪

((2 + Li→v) ∩ (2 + Lj→v)) = {0, 1, 2, 3} for any i, j, 1 ≤ i < j ≤ 4. Therefore

{C(Dm)}∞m=1 converges by Theorem 2.0.5. However, ifD′ = D−w, then(L1→v∩

L2→v) ∪ ((1 +L1→v) ∩ (1 + L2→v)) = {0, 1, 2} and{C(D′m)}∞m=1 diverges by the

theorem.

. . ., (ks − j) + 1 + tκp for everyt ≥ Np. We put

Lj→v := {(kr − j) + 1 (modκp) | r = 1, 2, . . . , s}.

(See Figure 2.1 for an illustration.) In general, for a nonnegative integeri, we set

i+ Lj→v := {(kr − j) + i+ 1 (modκp) | r = 1, 2, . . . , s}

whereLj→v = 0 + Lj→v. Obviously, ifDp+1, . . ., Dp+ζ are trivial components of

D, then, fori = 1, . . ., ζ ,

l ∈ (i+ Lj1→v) ∩ (i+ Lj2→v) if and only if the vertex ofDp+i+1

is an(l + tκp)-step common prey of any vertex inU (p)j1

and any

vertex inU(p)j2

for any t greater than or equal to some positive

integerNi.

(∗)

9

Theorem 2.0.5.LetD be a linearly connected digraph with exactlyη strong com-

ponents. Then{C(Dm)}∞m=1 converges if and only if one of the following is true:

(i) For eachi = 1, . . ., η, Di is trivial.

(ii) Dη is nontrivial.

(iii) Dη is trivial whereas there is a nontrivial strong component inD, and if

p is the largest index for whichDp is a nontrivial strong component, then⋃η−p−1

i=0 ((i + Lj1→v) ∩ (i + Lj2→v)) = ∅ or Zκpfor any j1, j2 in Zκp

where

V (Dp+1) = {v} andΛ(D) ={

i | U(p)i ∩N−

D(v) 6= ∅}

= {k1, . . . , ks} for

some integers, 1 ≤ s ≤ κp.

Proof. We show the ‘if’ part first. If (i) is true, thenC(Dm) is an edgeless graph

with the vertex setV (D) for any positive integerm and so{C(Dm)}∞m=1 converges.

If (ii) is true, then{C(Dm)}∞m=1 converges by Theorem 1.2.1.

Now we suppose that (iii) is true. Take two verticesx andy of D. If x andy

do not have anα-step common prey for any positive integerα, thenx andy are not

adjacent inC(Dα) for any positive integerα. Now consider the case wherex and

y have a step common prey. Ifx or y is a vertex of a trivial component appearing

afterDp, thenx andy cannot have anα-step common prey for any positive integer

α. Thusx and y belong to components appearing beforeDp+1. If they have a

step common prey inDq for q ≤ p, then they have a step common prey inDp

and so there exists an integerM such thatx andy are adjacent inC(Dm) for any

integerm ≥ M by Corollary 2.0.3. Suppose thatx andy have a step common

prey only in a trivial component appearing after the component Dp and letw be a

step common prey ofx andy. Then there exist a directed(x, w)-walk W1 and a

directed(y, w)-walkW2 of the same length. Deleting fromW1 andW2 the vertices

10

in trivial components appearing afterDp, we obtain a directed(x, w1)-walk and

a directed(y, w2)-walk of the same length wherew1 ∈ U(p)j1

andw2 ∈ U(p)j2

for

somej1, j2 ∈ Zκp. Thenw1 andw2 have a 1-step common preyv and so1 ∈

Lj1→v ∩ Lj2→v. ThereforeLj1→v ∩ Lj2→v 6= ∅. Then, as we assumed that (iii) is

true,⋃η−p−1

i=0 ((i + Lj1→v) ∩ (i + Lj2→v)) = Zκp. By (∗), there exists a positive

integerNκpsuch thatx andy have an(l + tκp)-step common prey for anyl ∈ Zκp

andt ≥ Nκp, which implies thatx andy are adjacent inC(Dα) for any integerα

greater than or equal toNκpκp.

We show the ‘only if’ part by verifying the contrapositive. Suppose thatDη is

trivial whereas there is a nontrivial strong component inD and that for the largest

indexp such thatDp is a nontrivial strong component,∅ ⋃η−p−1

i=0 ((i + Lj1→v) ∩

(i + Lj2→v)) Zκpfor somej1, j2 ∈ Zκp

. Then(i1 + Lj1→v) ∩ (i1 + Lj2→v) 6= ∅

for somei1 ∈ {0, . . . , η− p− 1}. Therefore, by (∗), for somel ∈ Zκp, the vertex of

Dp+i1+1 is a common prey of any vertex inU (p)j1

and any vertex inU (p)j2

in Dl+tκp for

any integert greater than or equal to some positive integerNi1 . Thus every vertex in

U(p)j1

and every vertex inU (p)j2

are adjacent inC(Dl+tκp) for any integert ≥ Ni1 . On

the other hand, since⋃η−p−1

i=0 ((i+Lj1→v)∩ (i+Lj2→v)) 6= Zκp, there is an element

l′ in Zκpsuch thatl′ 6∈ (i+Lj1→v)∩(i+Lj2→v) for eachi = 0, . . ., η−p−1. Then,

by (∗), for anyi = 0, . . ., η − p − 1 and for any positive integerN , there exists an

integert ≥ N such that the vertex ofDp+i+1 is not an(l′ + tκp)-step common prey

of some vertex inU (p)j1

and some vertex inU (p)j2

. That is, for any positive integerN ,

there exists an integerβ ≥ N such that some vertex inU (p)j1

and some vertex inU (p)j2

are not adjacent inC(Dl′+βκp). However, we have shown the existence ofNi1 such

that every vertex inU (p)j1

and every vertex inU (p)j2

are adjacent inC(Dl+tκp) for any

integert ≥ Ni1 . Hence we can conclude that{C(Dm)}∞m=1 diverges.

For a matrixA in the form given in (1.1),p, q = 1, . . ., η; i = 1, . . ., κp; j = 1,

11

. . ., κp+1, we let

A(pq)ij denote the submatrix ofPAP T induced by the rows of

PAP T intersectingA(pp)i,i+1 and the columns ofPAP T intersect-

ingA(qq)j−1,j, whereA(pp)

κp,κp+1 andA(qq)0,1 are identified withA(pp)

κp,1 and

A(qq)κq ,1, resepctively.

(§)

We note thatA(pq)ij is a zero matrix if|p− q| ≥ 2.

Suppose that, forp ≤ η − 1, App is the last diagonal block of order at least two

in PAP T . Then the set

Λ(D) ={

i | U(p)i ∩N−

D (v) 6= ∅}

used in Theorem 2.0.5 corresponds to the set

Λ(A) :={i | The column ofPAP T intersecting the trivial blockAp+1,p+1

and a row ofPAP T intersectingA(pp)i,i+1 meet at1.}

FurthermoreLj→v used in the same theorem corresponds to

Lj→(p+1) := {(kr − j) + 1 (mod κp) | r = 1, 2, . . . , s}

whereΛ(A) = {k1, k2, . . . , ks}.

Now we are ready to translate Theorem 2.0.5 into matrix version.

Corollary 2.0.6. Suppose thatA ∈ Bn is a matrix in the form given in (1.1). Then

{Γ(Am)}∞m=1 converges if and only if one of the following holds:

(i) For eachi = 1, . . ., η, Aii is of order one.

(ii) Aηη is of order at least two.

(iii) Aηη is of order one whereas there is a diagonal block of order at least two of

PAP T , and ifp is the largest index such thatApp is of order at least two, then⋃η−p−1

i=0 ((i + Lj1→(p+1)) ∩ (i + Lj2→(p+1))) = ∅ or Zκpfor anyj1, j2 in Zκp

whereΛ(A) = {k1, . . . , ks} for some integers, 1 ≤ s ≤ κp.

12

Chapter 3

The limit of {Γ(Am)}∞m=1

3.1 The limit of {Γ(Am)}∞m=1

In this section, we find the limit of{Γ(Am)}∞m=1 for a matrix in the form given

in (1.1) whenA11, A22, . . ., Aηη are of order at least two. The convergence of

{Γ(Am)}∞m=1 for such a matrixA is guaranteed by Theorem 1.2.1 or Theorem 2.0.5.

To characterize the limit of{Γ(Am)}∞m=1 for a matrixA ∈ Bn whose digraph

has exactly two strong components and for which{Γ(Am)}∞m=1 converges, Parket

al. [12] introduced the notionBD for a weakly connected digraphD with exactly

two strong componentsD1 andD2 whereD2 is nontrivial.

Definition 3.1.1 ( [12]). We take a weakly connected digraphD with exactly two

strong componentsD1 andD2 whereD2 is nontrivial. LetI(D) = {(k, l) | (x, y) ∈

A(D) for somex ∈ U(1)k , y ∈ U

(2)l }. LetBD =

(

Zκ(D1),Zκ(D2)

)

be the bipartite

graph defined as follows. IfD1 is nontrivial, thenBD has an edge(i, j) if and only

if i ≡ k+1+ p (mod κ(D1)) andj ≡ l+ p (mod κ(D2)) for some(k, l) ∈ I(D)

and some integerp. If D1 is trivial, thenBD has an edge(i, j) if and only ifj ≡ l−1

13

(mod κ(D2)) for some(1, l) ∈ I(D), which is obtained by substitutingp = −1 and

k(D1) = 1 in the nontrivial case.

They described the limit of{C(Dm)}∞m=1 using a notion of ‘expansion’ ofBD.

Definition 3.1.2 ( [12]). Given a bipartite graphB = (X, Y ), we construct a su-

pergraph ofB as follows. We write each edge ofB in the arc form(x, y) to make

clear thatx ∈ X andy ∈ Y . Then we replace each vertexv with a complete graph

Gv (of any size) so thatGv andGw are vertex-disjoint ifv 6= w, and join each vertex

of Gx and each vertex ofGy whenever either(x, y) is an edge ofB or there exists

z ∈ Y such that(x, z) and(y, z) are edges ofB. We say that the resulting graph is

an expansion ofB.

Theorem 3.1.3(See [12]). LetD be a weakly connected digraph with two strong

componentsD1 andD2 such that no arc goes fromD2 to D1, D2 is nontrivial, and

{C(Dm)}∞m=1 converges. Then the limit of{C(Dm)}∞m=1 is an expansion of the

bipartite graphBD defined in Definition 3.1.1.

Given a linearly connected digraphD with η strong componentsD1, . . ., Dη,

we recall thatDp,p+1 denotes the subdigraph ofD induced byV (Dp) ∪ V (Dp+1)

for eachp = 1, . . ., η − 1. Noting thatDp,p+1 is a weakly connected digraph with

two strong components, we extend Definition 3.1.1 as followsto take care of the

general case given in (1.1).

Definition 3.1.4. Let D be a linearly connected digraph withη nontrivial strong

components and let

I(Dp,p+1) := {(k, l) | (x, y) ∈ A(D) for somex ∈ U(p)k andy ∈ U

(p+1)l }.

We define thecompetition skeleton graph(CS-graph for short) ofD as theη-partite

graphPT(η)D = (V1, V2, . . . , Vη) with Vi = Zκi

, 1 ≤ i ≤ η and an edge(i, j) if

14

U(1)1

U(1)2

U(2)1 U

(2)2 U

(2)3 U

(2)4

U(3)1 U

(3)2

1

1

1

2

2

2

3 4

D PT(3)D

Figure 3.1: A strongly connected digraphD and its CS-graphPT(3)D .

and only if forp ∈ {1, . . . , η − 1}, (i, j) ∈ E(BDp,p+1). (See Figure 3.1 for an

illustration.)

In the following, for convenience sake, we writeBp,p+1 for BDp,p+1. If {i, j} is

an edge ofBp,p+1 wherei ∈ Zκpandj ∈ Zκp+1

, we denote it by(i, j) instead of

{i, j}.

We state a few useful lemmas.

Lemma 3.1.5.LetD be a weakly connected digraph with exactly two strong com-

ponentsD1 andD2 both of which are nontrivial. Then the following are equivalent:

(a) There exists a directed(u, v)-walk of length2sκ1κ2 for some verticesu ∈

U(1)i andv ∈ U

(2)j and for some positive integers.

(b) For any verticesx ∈ U(1)i andy ∈ U

(2)j , there exists a positive integerN(x, y)

such that, for anyt ≥ N(x, y), there exists a directed(x, y)-walk of length

2tκ1κ2.

15

Proof. To show (a) implies (b), suppose that there exists a directed(u, v)-walk of

length2sκ1κ2 for some verticesu ∈ U(1)i andv ∈ U

(2)j and for some positive integer

s. Take any verticesx ∈ U(1)i andy ∈ U

(2)j . Then, by Lemma 2.0.4, there exist

positive integersN1 andN2 such that there are directed(x, u)-walks inD1 of every

lengths1κ1 with s1 ≥ N1 and there are directed(v, y)-walks inD2 of every length

s2κ2 with s2 ≥ N2. LetN = s+N1 +N2. Fix anyt ≥ N and choose two positive

integerss1 ≥ N1, s2 ≥ N2 satisfyingt = s + s1 + s2. Then there exist a directed

(x, u)-walk Q of length2s1κ1κ2 and a directed(v, y)-walk R of length2s2κ1κ2.

Thus we obtain a directed(x, y)-walkQPR of length2(s+s1+s2)κ1κ2 = 2tκ1κ2.

The statement (a) immediately follows from (b) by taking anyverticesu ∈ U(1)i

andv ∈ U(2)j ands = N(u, v).

By Lemma 3.1.5, the following two lemmas are logically equivalent:

Lemma 3.1.6(See [12]). Let D be a weakly connected digraph with exactly two

strong componentsD1 andD2 both of which are nontrivial. Then(i, j) is an edge

of BD if and only if there exists a directed(u, v)-walk of length2sκ1κ2 for some

verticesu ∈ U(1)i andv ∈ U

(2)j and for some positive integers.

Lemma 3.1.7.LetD be a weakly connected digraph with exactly two strong com-

ponentsD1 andD2 both of which are nontrivial. Then(i, j) is an edge ofBD if

and only if, for any verticesx ∈ U(1)i andy ∈ U

(2)j , there exists a positive integer

N(x, y) such that, for anyt ≥ N(x, y), there exists a directed(x, y)-walk of length

2tκ1κ2.

We shall generalize Lemma 3.1.6. To do so, we need the following lemma.

Lemma 3.1.8.Let D be a linearly connected digraph with only nontrivial strong

components as many asη ≥ 3. Let x ∈ U(p)i and z ∈ U

(q)k for positive integers

16

p and q satisfyingp + 2 ≤ q ≤ η. If there exists a directed(x, z)-walk of length

sκp+1κp+2 · · ·κq for some positive integers, then, for somej ∈ Zκp+1and for some

y ∈ U(p+1)j , there exist

(i) a positive integerK and a directed(x, y)-walk of lengthkκp+1 for each inte-

gerk ≥ K, in particular, a directed(x, y)-walk of length2Kκpκp+1;

(ii) a positive integerK ′ and a directed(y, z)-walk of lengthk′κp+1 for each

integerk′ ≥ K ′, in particular, directed(y, z)-walks of lengths2K ′κp+1κp+2

andK ′κp+1κp+2 · · ·κq, respectively.

Proof. Let W be a directed(x, z)-walk of lengthsκp+1κp+2 · · ·κq. SinceD is

linearly connected, there is a vertexy1 onW belonging toDp+1. Theny1 ∈ U(p+1)j1

for somej1 ∈ Zκp+1. We denote byW1 andW2 the(x, y1)-section ofW and(y1, z)-

section ofW , respectively, and then, byj the element inZκp+1satisfying

j ≡ j1 − ℓ(W1) (mod κp+1),

or

ℓ(W1) = j1 − j + tκp+1 (3.1)

for some integert.

Now we take a vertex inU (p+1)j and call ity. By Lemma 2.0.4, there exists

L ∈ N such that, for any integerk ≥ L, there exists a directed(y1, y)-walk of

(j − j1) + kκp+1 and so a directed(x, y)-walk of lengthℓ(W1) + (j − j1) + kκp+1,

which equals(t+ k)κp+1 by (3.1). We letK = max{L, L+ t} and (i) follows.

As W is decomposed intoW1 andW2,

sκp+1κp+2 · · ·κq = ℓ(W1) + ℓ(W2)

17

By Lemma 2.0.4 again, there existsL′ ∈ N such that, for each integerk′ ≥ L′,

there exists a directed(y, y1)-walk of length(j1 − j) + k′κp+1 and so a directed

(y, z)-walk of length(j1 − j) + k′κp+1 + ℓ(W2). Then for each integerk′ ≥ L′,

(j1 − j) + k′κp+1 + ℓ(W2)

= (j1 − j) + k′κp+1 + (sκp+1κp+2 · · ·κq − ℓ(W1))

= k′κp+1 + (sκp+1κp+2 · · ·κq − tκp+1)

= (k′ + sκp+2 · · ·κq − t)κp+1.

We letK ′ = max{L′, L′ + sκp+2 · · ·κq − t}, which satisfies (ii).

Corollary 3.1.9. LetD be a linearly connected digraph with only nontrivial strong

components as many asη ≥ 3 and letx ∈ U(p)i andz ∈ U

(p+2)k for a positive integer

p satisfyingp + 2 ≤ η. If there exists a directed(x, z)-walk of lengthsκp+1κp+2

for some positive integers, then there existsj ∈ Zκp+1such that(i, j) ∈ E(Bp,p+1)

and(j, k) ∈ E(Bp+1,p+2).

Proof. By the hypothesis, there existsy inDp+1 satisfying (i) and (ii) of Lemma 3.1.8.

Then, by Lemma 3.1.6, (i) and (ii) imply(i, j) ∈ E(Bp,p+1) and(j, k) ∈ E(Bp+1,p+2),

respectively, which is the desired conclusion.

The following lemma is a generalization of Lemma 3.1.6 and Corollary 3.1.9.

Lemma 3.1.10.LetD be a linearly connected digraph with only nontrivial strong

components as many asη ≥ 3. Let x ∈ U(p)i and z ∈ U

(q)k for positive integers

p and q satisfyingp + 2 ≤ q ≤ η. If there exists a directed(x, z)-walk of length

sκp+1κp+2 · · ·κq for some positive integers, then there exists an(i, k)-path in the

CS-graph ofD defined in Definition 3.1.4.

18

Proof. We proceed by induction onm = q − p + 1. The statement is true for

m = 3 by Corollary 3.1.9. Suppose that the statement is true form − 1 (m ≥ 4).

Let W be a directed(x, z)-walk of lengthsκp+1κp+2 · · ·κq for somes ∈ N. Then,

by Lemma 3.1.8, for somej ∈ Zκp+1and for a vertexy ∈ U

(p+1)j , there exist a

directed(x, y)-walk of length2Kκpκp+1 for someK ∈ N and a directed(y, z)-

walk of lengthK ′κp+1κp+2 · · ·κq for someK ′ ∈ N. Since there exists a directed

(x, y)-walk of length2Kκpκp+1, by Lemma 3.1.6,(i, j) ∈ E(Bp,p+1) ⊂ E(PT(η)D ).

Since there is a directed(y, z)-walk of lengthK ′κp+1κp+2 · · ·κq, by the induc-

tion hypothesis, there exists a(j, k)-path inPT(η)D , which is concatenated together

with the edge(i, j) to form an(i, k)-path inPT(η)D .

The next theorem plays a key role in describing the limit of{C(Dm)}∞m=1.

Theorem 3.1.11.LetD be a linearly connected digraph with only nontrivial strong

components as many asη ≥ 3. Suppose thatx ∈ U(p)i andy ∈ U

(q)j for integersp,

q, i, j with 1 ≤ p ≤ q ≤ η, i ∈ Zκp, j ∈ Zκq

, respectively. Then, for any integer

r ≥ max{p+1, q}, x andy have a step common prey inDr if and only if there exist

an (i, k)-path and a(j, k)-path in the CS-graph ofD for somek ∈ Zκrsatisfying

the property thatk = j if and only ifq = r.

Proof. To show the ‘if’ part, suppose that for an integerr ≥ max{p + 1, q} and an

elementk ∈ Zκr, there exist an(i, k)-pathP and a(j, k)-pathQ wherek = j if and

only if q = r. Let i0 = i, j0 = j, iℓ(P ) = jℓ(Q) = k, andP andQ denote directed

paths

i0 → i1 → · · · → iℓ(P )−1 → iℓ(P ) and j0 → j1 → · · · → jℓ(Q)−1 → jℓ(Q),

respectively. Sincer ≥ p+1, by the definition of a CS-graph, we may take a vertex

xt ∈ U(p+t)it

, especiallyx0 = x and, in caseq = r, xℓ(P ) = y, for eacht = 0, 1,

19

. . ., ℓ(P ). We setz := xℓ(P ). We will show thatz is anm-step common prey of

x andy for somem ∈ N. For simplicity, letλ = κ1κ2 · · ·κη. By Lemma 3.1.7,

there exists (a sufficiently large)N1 ∈ N such that, for anys ≥ N1, there exists

a directed(xt, xt+1)-walk Wt+1(s) of length2sλ for eacht = 0, 1, . . . , ℓ(P ) − 1.

For any integers ≥ N1, the concatenation ofW1(s), . . ., Wℓ(P )(s) results in a di-

rected(x, z)-walk of length2sℓ(P )λ. Supposer = q. Thenk = j andz = y

by the assumption. By Lemma 2.0.4, there exists a positive integerN2 such that,

for any integers ≥ N2, there exists a directed(y, y)-walk of length2sℓ(P )λ. If

we let m = 2ℓ(P )λmax{N1, N2}, theny is anm-step common prey ofx and

y. Now suppose thatr ≥ q + 1. Then we may take a vertexyt ∈ U(q+t)jt

, espe-

cially y0 = y andyℓ(Q) = z, for t = 0, . . . , ℓ(Q). By Lemma 3.1.7, there exists

(a sufficiently large)N3 ∈ N such that, for any integers ≥ N3, there exists a

directed(yt, yt+1)-walk W ′

t+1(s) of length 2sλ for eacht = 0, 1, . . . , ℓ(Q) − 1.

For any integers ≥ N3, the concatenation ofW ′

1(s), . . ., W′

ℓ(Q)(s) results in a

directed(y, z)-walk of length2sℓ(Q)λ. As there exists a directed(x, z)-walk of

length2sℓ(P )λ for any integers ≥ N1, z is anm-step common prey ofx andy for

m := 2ℓ(P )ℓ(Q)λmax{N1, N3}.

To show the ‘only if’ part, suppose thatx andy have anm-step common prey

in Dr for somer ≥ max{p + 1, q} andm ∈ N. Now we takes ∈ N such that

2sκp+1κp+2 · · ·κq ≥ m. Then, by Lemma 2.0.2,x andy have a2sκp+1κp+2 · · ·κq-

step common preyz in Dr. Sincez belongs toDr, z ∈ U(r)k for somek ∈ Zκr

. If

r ≥ p + 2, then there is an(i, k)-path inPT(η)D by Lemma 3.1.10 and ifr = p + 1,

then there is an edge(i, k) in Bp,p+1 by Lemma 3.1.6, which is an(i, k)-path in

PT(η)D . Thus we have shown that there is an(i, k)-path inPT

(η)D .

If r ≥ q+2, then, by Lemma 3.1.10, there is a(j, k)-path inPT(η)D . If r = q+1,

then there is an edge(j, k) in Bq,q+1 by Lemma 3.1.6, which is a(j, k)-path in

20

PT(η)D .

Now suppose thatr = q. Sincez is a step common prey ofx andy, there exist

a directed(x, z)-walk W1 and a directed(y, z)-walk W2 of the same length. Since

y andz belong to the same strong component, there exists a directed(z, y)-walk

W3. ThenW1W3 andW2W3 are a directed(x, y)-walk and a directed(y, y)-walk,

respectively, of the same length, which implies thaty is a step common prey ofx

andy. In particular, by Lemma 2.0.4, there exists a positive integerN such that, for

anys ≥ N , y is a2sκp+1κp+2 · · ·κq-step common prey ofx andy. Then there is

an(i, j)-path inPT(η)D by Lemma 3.1.6 or Lemma 3.1.10 depending upon whether

q = p+1 or q ≥ p+2. We take the(j, j)-path forj ∈ Zκq, which is trivial, and set

k := j.

If k = j, thenq = r asj ∈ Zkq andk ∈ Zkr and so the theorem follows.

Based on the observations which we have made so far, we generalize the notion

of expansion given in Definition 3.1.2 as follows.

Definition 3.1.12. Let G = (V1, . . . , Vη) be anη-partite graph without edges be-

tweenVi andVj if |i− j| ≥ 2. FromG, we construct a supergraph ofG as follows

(see Figure 3.2 for an illustration):

(Step 1) We replace each vertexv with a complete graphGv (of any size) so that

Gv andGw are vertex-disjoint ifv 6= w.

(Step 2) For eachx ∈ Vi, y ∈ Vj wherei ≤ j, we join each vertex ofGx and each

vertex ofGy if there is an integerk ≥ j satisfying one of the following:

(i) k ≥ j + 1 and there are an(x, z)-path and a(y, z)-path inG for some

z ∈ Vk; (ii) j = k and there is an(x, y)-path.

We say that any graph resulting from this construction is anexpansion ofG.

21

x1 x2

Gx1Gx2

y1 y2 y3 y4 Gy1 Gy2Gy3 Gy4

z1 z2

Gz1 Gz2

G G∗

Figure 3.2: A graphG and an expansionG∗ of G.

We note that an expansion of anη-partite graphG is uniquely determined by the

vertex sets of complete graphs replacing the vertices ofG.

In order to describe the limit of the graph sequence{C(Dm)}∞m=1 for a linearly

connected digraphD with only nontrivial strong components, we need one more

lemma in the following.

Lemma 3.1.13(See [12]). LetD be a weakly connected digraph with exactly two

strong componentsD1 andD2. Then there exists an integerM such thatC(Dm)

contains complete graphs whose vertex sets areU(1)1 , . . ., U (1)

κ(D1), U (2)

1 , . . ., U (2)κ(D2)

,

respectively, as subgraphs form ≥ M .

Now we present the theorem describing the limit of the sequence{C(Dm)}∞m=1

for a linearly connected digraphD with only nontrivial strong components.

Theorem 3.1.14.LetD be a linearly connected digraph with only nontrivial strong

components as many asη ≥ 3. Then the limit graph of the sequence{C(Dm)}∞m=1

22

is an expansion of the CS-graph ofD.

Proof. Denote byG the limit graph of the sequence{C(Dm)}∞m=1. We take the

expansionG∗ of PT(η)D obtained by replacing the vertexi in Zκj

with the complete

graph whose vertex set is the set of imprimitivityU (s)l of Ds for l ∈ Zκs

ands ∈

{1, . . . , η}. ObviouslyV (G) = V (G∗). In the following, we show thatE(G) =

E(G∗).

Now take two verticesx andy of G. Thenx ∈ U(p)i andy ∈ U

(q)j for some

integersp, q, i, j satisfying1 ≤ p, q ≤ η, i ∈ Zκp, j ∈ Zκq

, respectively. Without

loss of generality, we may assumep ≤ q. First, suppose thatx andy are adjacent

in G. Thenx andy have a step common prey inDr for some integerr ≥ q. If

p = r, thenq = r and so, by one of well-known properties of sets of imprimitivity,

i = j. Then, by the definition of expansion,x andy are adjacent inG∗. Now we

assume thatp ≤ r + 1. Then, by Theorem 3.1.11, there existsk ∈ Zκrsuch that

there exist an(i, k)-path and a(j, k)-path inPT(η)D . By the definition of expansion,

all the vertices inU (p)i and all the vertices inU (q)

j are adjacent and sox andy are

adjacent inG∗. Hence we have shown thatE(G) ⊂ E(G∗).

To showE(G∗) ⊂ E(G), suppose that verticesu andv are adjacent inG∗.

Then, by the definition of expansion, one of the following holds:

(i) u andv belong to the same set of imprimitivity, that is,{u, v} ⊂ U(s)l for

somes ∈ {1, . . . , η} andl ∈ Zκs;

(ii) u andv belong to different setsU (s)l andU (s′)

l′ , respectively, of imprimitivity

for some integers1 ≤ s ≤ s′ ≤ η, l ∈ Zκs, l′ ∈ Zκs′

and either there exist

an(l, k)-path and an(l′, k)-path inPT(η)D for some integerr, s′ + 1 ≤ r ≤ η,

andk ∈ Zκror there is an(l, l′)-path.

23

Noting that them-step competition graph ofDi,i+1 is the subgraph of them-step

competition graph ofD for anym ∈ N andi ∈ {1, . . . , η−1}, we can conclude that

u andv are adjacent inG if (i) is true by Lemma 3.1.13. Consider the case (ii). Then,

by Theorem 3.1.11,u andv have a step common prey. Thus, by Corollary 2.0.3,u

andv are adjacent inG.

We consider a matrixA in the form given in (1.1) whereAii has order at least

two for eachi = 1, 2, . . ., η. Then for a blockA(p,p+1)i,j of Ap,p+1 defined in (§) for

p = 1, . . ., η − 1, it is easy to see thatA(p,p+1)ij 6= O if and only if (i, j) ∈ I(Dp,p+1)

whereD is the digraph ofA. Now Theorem 3.1.14 may be translated into matrix

version in the following way:

Corollary 3.1.15. LetA ∈ Bn be a matrix in the form given in (1.1) whereAii has

order at least two for eachi = 1, 2, . . ., η. Then{Γ(Am)}∞m=1 converges to a

(symmetric) matrixA′ such that

PA′P T =

C11 C12 C13 · · · C1,η−1 C1,η

C21 C22 C23 · · · C2,η−1 C2,η

C31 C32 C33 · · · C3,η−1 C3,η

......

.... . .

......

Cη−1,1 Cη−1,2 Cη−1,3 · · · Cη−1,η−1 Cη−1,η

Cη,1 Cη,2 Cη,3 · · · Cη,η−1 Cηη

where, for eachp, q = 1, . . . , η, Cpq satisfies the following:

(i) The order ofCpp is the same as the order ofApp;

(ii) Cpq = CTqp;

(iii) The blockC(pq)ij of Cpq has the order the same asA(pq)

ij and is located in the

position inPA′P T the same as the position whereA(pq)ij is located inPAP T .

24

Furthermore,

C(pq)ij =

J∗ if (P1) or (P2) is satisfied;

O otherwise.

(P1) p = q andi = j;

(P2) p 6= q or i 6= j, and either

∗ there exist positive integerst1 andt2 satisfyingp+ t1 = q+ t2 such

that, for eachr = 0, . . ., t1 and s = 0, . . ., t2, there exist some

positive integersir, js with i0 = i, j0 = j, and it1 = jt2 , some

integerskr, lr, ms, ns, and nonnegative integersar, bs for which

A(p+r,p+r+1)krlr

6= O, A(q+s,q+s+1)msns 6= O and which satisfyir ≡ kr +

ar+1 (mod κp+r), ir+1 ≡ lr+ar (mod κp+r+1), js ≡ ms+bs+1

(mod κq+s), js+1 ≡ ns + bs (mod κq+s+1), or

∗ p 6= q and, ifp < q, then there exists a positive integert satisfying

p + t = q such that for eachr = 0, . . ., t, there exist some positive

integer ir (i0 = i, iq = j), some integerskr, lr and nonnegative

integerar for whichA(p+r,p+r+1)krlr

6= O and which satisfyir ≡ kr +

ar + 1 (mod κp+r), ir+1 ≡ lr + ar (mod κp+r+1); if q < p, then

CTpq = J∗.

(We mean byO a zero matrix of any size and byJ∗ a square matrix of any

order such that all the diagonal entries are 0 and all the off-diagonal entries

are 1.)

25

3.2 Limit of a particular form: the disjoint union of

complete subgraphs

We observe that the graphG∗ in Figure 3.2 is the disjoint union of two complete

componentsGx1∪Gy1 ∪Gy3 ∪Gz2 andGx2

∪Gy2 ∪Gy4 ∪Gz1 . Parket al.[12] char-

acterized a digraphD for which {C(Dm)}∞m=1 converges to the union of complete

subgraphs as follows:

Theorem 3.2.1(See [12]). LetD be a weakly connected digraph with exactly two

strong componentsD1 andD2 both of which are nontrivial and without arc from

D2 to D1. Suppose that{C(Dm)}∞m=1 converges to a graphG. ThenG is the

disjoint union of complete subgraphs if and only ifκ2 dividesκ1 andi− j ≡ i′ − j′

(mod κ2) for any(i, j), (i′, j′) ∈ I(D).

In the rest of this paper, we generalize the above theorem. Todo so, we need the

following lemma:

Lemma 3.2.2.Let D be a linearly connected digraph with only nontrivial strong

componentsD1, D2, . . ., Dη (η ≥ 2) such that there is an arc going fromDi to

Dj for some distincti, j ∈ {1, . . . , η} only if j = i + 1. Suppose that, in the

CS-graphPT(η)D of D, κη dividesκp and i − j ≡ i′ − j′ (mod κη) for any (i, j),

(i′, j′) ∈ I(Dp,p+1) for eachp = 1, . . ., η− 1. Then there exist an(x, k)-path and a

(y, k)-path for somek ∈ Zκηin PT

(η)D if and only ifx ≡ y (mod κη) for anyx, y

in Zκrand for anyr = 1, . . ., η.

Proof. To show the ‘only if’ part, suppose that, for somer ∈ {1, . . . , η}, there exist

an (x, k)-path and a(y, k)-path inPT(η)D for somek ∈ Zκη

andx, y in Zκr. If

r = η, thenx ≡ y (mod κη) and the lemma is trivially true. Now suppose thatr <

η. Then there exist an(x, k)-pathxsr+1 · · · sη−1k and a(y, k)-pathytr+1 · · · tη−1k

26

where{sm, tm} ⊂ Zκmfor m = r+1, . . ., η− 1. By the definition ofBD, for each

m = r, . . ., η − 1,

(sm − sm+1)− (gm − gm+1)− 1 ∈ 〈κm, κm+1〉 ⊂ 〈κη〉

and

(tm − tm+1)− (hm − hm+1)− 1 ∈ 〈κm, κm+1〉 ⊂ 〈κη〉

wheresr = x, tr = y, sη = tη = k, (gm, gm+1) ∈ I(Dm,m+1), (hm, hm+1) ∈

I(Dm,m+1), 〈κη〉 = {ακη | α ∈ Z}, and〈κm, κm+1〉 = {βκm+γκm+1 | β ∈ Z, γ ∈

Z}. By the hypothesis,(gm − gm+1)− (hm − hm+1) ∈ 〈κη〉 and so(sm − sm+1)−

(tm− tm+1) ∈ 〈κη〉 for eachm = r, . . ., η−1. Therefore,(x−k)− (y−k) ∈ 〈κη〉

and this completes the proof of the ‘only if’ part.

We prove the ‘if’ part by induction onη − p for p = 1, . . ., η − 1. Suppose

thatx ≡ y (mod κη) for x, y in Zκη−1. SinceD is linearly connected, there is an

element(k, l) in I(Dη−1,η). Sincex, y, andk belong toZκη−1, x ≡ k + m + 1

(mod κη−1) andy ≡ k +m′ + 1 (mod κη−1) for some integersm andm′. Then,

by definition ofBD, (x, z) ∈ Bη−1,η and (x, z′) ∈ Bη−1,η for a vertexz ∈ Zκη

satisfyingz ≡ l + m (mod κη) and a vertexz′ ∈ Zκηsatisfyingz′ ≡ l + m′

(mod κη). Sinceκη | κη−1, x ≡ k + m + 1 (mod κη) and y ≡ k + m′ + 1

(mod κη). Sincex ≡ y (mod κη), m ≡ m′ (mod κη) and soz ≡ z′ (mod κη).

Since bothz andz′ belong toZκη, z = z′ by the definition ofPT

(η)D and so the ‘if’

part is true forη − p = 1.

Suppose that the ‘if’ part is true forη − p (p ≥ 1). Then we assume thatx ≡ y

(mod κη) for x, y in Zκη−p−1. By applying a similar argument as above, we may

show that there exist verticesw andw′ in Zκη−psuch thatw ≡ w′ (mod κη) and

there are edges(x, w) and(y, w′) in PT(η)D . Then, by the induction hypothesis, there

exist a(w, z)-pathP and a(w′, z)-pathQ for z ∈ Zκη. AccordinglyxP andyQ

27

are an(x, z)-path and a(y, z)-path, respectively, and this completes the proof of the

‘if’ part.

Theorem 3.2.3.LetD be a linearly connected digraph with only nontrivial strong

componentsD1, D2, . . ., Dη (η ≥ 2) such that there is an arc going fromDi to Dj

for some distincti, j ∈ {1, . . . , η} only if j = i+ 1. Suppose that a graphG is the

limit of {C(Dm)}∞m=1. ThenG is the disjoint union of complete subgraphs if and

only ifκη dividesκp andi− j ≡ i′− j′ (mod κη) for any(i, j), (i′, j′) ∈ I(Dp,p+1)

for eachp = 1, . . ., η − 1.

Proof. By induction on the numberη of strong components of a linearly connected

digraph. By Theorem 3.2.1 and Lemma 3.2.2, the statement is true for η = 2.

Suppose that the statement is true forη − 1 (η ≥ 3). LetD be a linearly connected

digraph with only nontrivial strong componentsD1, D2, . . ., Dη such that there is

an arc going fromDi to Dj for some distincti, j ∈ {1, . . . , η} only if j = i + 1.

We deleteD1 from D to obtain a linearly connected digraphF with η − 1 strong

componentsD2, . . ., Dη. By the induction hypothesis,{C(Fm)}∞m=1 converges to

the disjoint union of complete subgraphs if and only ifκ(Dη) dividesκ(Dp) and

i− j ≡ i′ − j′ (mod κ(Dη)) for any(i, j), (i′, j′) ∈ I(Dp,p+1) for eachp = 2, . . .,

η − 1.

By the hypothesis,{C(Dm)}∞m=1 converges toG. By Theorem 3.1.14,G is an

expansion ofPT(η)D . Since there is no way to reach from a vertex inDj to a vertex

in Di for j > i, {C(Fm)}∞m=1 converges to a graph, sayG′, which is an expansion

of PT(η)D − Zκ1

. ThereforeG′ = G− V (D1) by the way in which the expansion is

constructed in the proof of Theorem 3.1.14.

To show the ‘only if’ part, suppose thatG is the disjoint union of complete sub-

graphs. Then, sinceG′ = G− V (D1), G′ is the disjoint union complete subgraphs.

28

Hence, by induction hypothesis,κη dividesκp andi− j ≡ i′− j′ (mod κη) for any

(i, j), (i′, j′) ∈ I(Dp,p+1) for eachp = 2, . . ., η − 1.

We first make the following observation:

If the complete graph replacingx and the complete graph replacingy are con-

tained in the same component inG for x andy in Zκ2, thenx ≡ y (mod κη).

(†)

To see why, suppose that the complete graph replacingx and the complete graph

replacingy are contained in the same component inG for x andy in Zκ2. Then, for

somer ∈ {2, . . . , η}, there are an(x, k′)-path and a(y, k′)-path for somek′ ∈ Zκr

by the definition of expansion. SinceD is linearly connected,PT(η)D is connected

and so there is a(k′, k)-path for somek ∈ Zκηin PT

(η)D . Then, by Lemma 3.2.2,

x ≡ y (mod κη).

Now we show thatκη | κ1. SinceD is linearly connected, there is an edge(a, b)

in B1,2 and so, by the definition ofBD, for some(i, j) ∈ I(D1,2) and for some

integerl,

a ≡ i+ l + 1 (mod κ1), b ≡ j + l (mod κ2).

Since(i, j) ∈ I(D1,2), it is true that(i+ l + κ1 + 1, j + l + κ1) is an edge ofB1,2.

Sincei + l + κ1 + 1 ≡ a (mod κ1), it is true that(a, j + l + κ1) is an edge of

B1,2. Then, since(a, b) is also an edge ofPT(η)D , the complete graphs replacingb

andj + l + κ1 should be contained in the same component inG by the hypothesis.

By the observation (†), b ≡ j + l + κ1 (mod κη). Sinceb ≡ j + l (mod κ2) and

κη | κ2, we can conclude thatκη | κ1.

Take(i, j), (i′, j′) ∈ I(D1,2). Without loss of generality, we may assume that

i′ > i. Since(i, j) ∈ I(D1,2), by the definition ofBD, (i+(i′− i)+1, j+(i′− i)) is

an edge ofB1,2 and so is(i′+1, j+ i′− i). In addition,(i′+1, j′) is an edge ofB1,2

as(i′, j′) ∈ I(D1,2). Therefore both(i′ + 1, j + i′ − i) and(i′ + 1, j′) are edges of

29

B1,2. Then, by the hypothesis, the complete graphs replacingj+ i′− i andj′ should

be contained in the same component inG. By the observation (†), j + i′ − i ≡ j′

(mod κη). Thusi− j ≡ i′ − j′ (mod κη).

To show the ‘if’ part, suppose thatκη dividesκp andi− j ≡ i′ − j′ (mod κη)

for any (i, j), (i′, j′) ∈ I(Dp,p+1) for eachp = 1, . . ., η − 1. By the induction

hypothesis,G′ is the disjoint union of complete subgraphs. Take a vertexa ∈ Zκ1

of PT(η)D . If a has no neighbor inB1,2, then its degree is zero and the complete

graph representing it inG is a component ofG. Suppose thata has a neighbor in

B1,2. Let (a, b) and(a, c) are edges ofB1,2. Then, by definition, for some(i, j),

(i′, j′) ∈ I(D1,2) and for some integersl, l′,

a ≡ i+ l + 1 (mod κ1), b ≡ j + l (mod κ2),

a ≡ i′ + l′ + 1 (mod κ1), c ≡ j′ + l′ (mod κ2).

Sinceκη|κ1,

a ≡ i+ l + 1 ≡ i′ + l′ + 1 (mod κη),

and sol − l′ ≡ i′ − i (mod κη). Sinceκη|κ2,

b− c ≡ (j+ l)− (j′+ l′) ≡ (j− j′)+ (l− l′) ≡ (j− j′)+ (i′− i) ≡ 0 (mod κη).

Thus, by Lemma 3.2.2, there exist a(b, z)-path and a(c, z)-path for somez ∈ Zκη.

By the definition of expansion, the complete graph replacingb and the complete

graph replacingc induce the complete subgraph ofG. ThusG is the disjoin union

of complete subgraphs.

Theorem 3.2.3 is translated into matrix version as follows:

Corollary 3.2.4. Suppose thatA ∈ Bn is a matrix in the form given in (1.1) where

the blockAii has order at least two for eachi = 1, . . ., η. LetA′ be the limit of

{Γ(Am)}∞m=1. Then the following are equivalent:

30

(a) The matrixPA′P T is a JBD matrix.

(b) For eachp = 1, . . ., η − 1, κη dividesκp and i − j ≡ i′ − j′ (mod κη)

wheneverA(p,p+1)ij andA(p,p+1)

i′j′ defined in(§) are nonzero matrices.

31

Chapter 4

Conclusions and closing remarks

In this paper, we found the limit of the matrix sequence{Γ(Am)}∞m=1 for a matrix

A in the form given in (1.1) when all of the diagonal blocks are of order at least

two. We know from Theorem 2.0.5 that there are other cases where {Γ(Am)}∞m=1

converges. We suggest that the limit of{Γ(Am)}∞m=1 be computed forA satisfying

the condition for each of such cases.

In addition, Theorem 2.1 tells us the convergence of{Γ(Am)}∞m=1 only when

D(A) is linearly connected. We suggest to find the condition where{Γ(Am)}∞m=1

converges even ifD(A) is not linearly connected.

Finally, for a Boolean matrixA in the form given in (1.1) when all of the diag-

onal blocks are of order at least two, we can ask that what is the smallest powerN

such thatΓ(Am) equals the limit of{Γ(Am)}∞m=1 for anym ≥ N?

32

Bibliography

[1] R. A. Brualdi and H. J. Ryser:Combinatorial matrix theory, Cambridge, 1991.

[2] E. Belmont: A complete characterization of paths that arem-step competition

graphsDiscrete Appl. Math., 159(2011) 1381–1390.

[3] H. H. Cho and H. K. Kim, Competition indices of strongly connected digraphs

Bull. Korean Math. Soc., 48 (2011) 637–646.

[4] H. H. Cho, S. -R. Kim, and Y. Nam: Them-step competition graph of a di-

graph,Discrete Appl. Math., 105(2000) 115–127.

[5] J. E. Cohen: Interval graphs and food webs: a finding and a problem,RAND

Corporation Document 17696-PR, Santa Monica, California, 1968.

[6] H. J. Greenberg, J. R. Lundgren, and J. S. Maybee: Inverting graphs of rect-

angular matrices,Discrete Appl. Math., 8 (1984) 225–265.

[7] G. T. Helleloid, Connected triangle-freem-step competition graphs,Discrete

Appl. Math., 145(2005) 376–383.

[8] H. K. Kim, Competition indices of tournamentsBull. Korean Math. Soc., 45

(2008) 385–396.

33

[9] W. Ho: Them-step, same-step, and any-step competition graphs,Discrete

Appl. Math., 152(2005) 159–175.

[10] S. -R. Kim: The competition number and its variants, in J. Gimbel, J.W.

Kennedy, and L.V. Quintas (eds.),Quo Vadis Graph Theory?, Ann. Discrete

Math., 55 (1993) 313–325.

[11] B. Park, J. Y. Lee, and S. -R. Kim, Them-step competition graphs of doubly

partial orders,Appl. Math. Lett., 24 (2011) 811–816.

[12] W. Park, B. Park and S. -R. Kim, A matrix sequence{Γ(Am)}∞m=1 might con-

verge even if the matrixA is not primitive,Linear Algebra Appl., 438(2013)

2306–2319.

[13] J. R. Lundgren: Food webs, competition graphs, competition-common enemy

graphs, and niche graphs, in F. S. Roberts (ed.),Applications of Combinatorics

and Graph Theory in the Biological and Social Sciences, IMA Volumes in

Mathematics and its Applications,17, Springer-Verlag, New York, 1989, 221–

243.

[14] A. Raychaudhuri and F. S. Roberts: Generalized competition graphs and their

applications,Methods Oper. Res., 49 (1985) 295–311.

[15] F. S. Roberts: Competition graphs and phylogeny graphs, in L. Lovasz (ed.),

Graph Theory and Combinatorial Biology, Bolyai Math. Stud.7, J. Bolyai

Math. Soc., Budapest (1999) 333–362.

[16] Y. Zhao and G. J. Chang, Note on them-step competition numbers of paths

and cycles,Discrete Appl. Math., 157(2009) 1953–1958.

34

국문초록

이논문에서는성분의개수가임의로주어진선형연결그래프를그것의그래프로갖

는 Bool행렬 A ∈ Bn에대하여연구함으로써 Parket al. [12]의결과를확장하였다.일

반화의과정에서,그들이사용한증명의핵심적인아이디어를구체화하였는데,먼저

{Γ(Am)}∞m=1가수렴하게되는행렬 A의특징을완벽하게분석하였다. 또한모든대

각블록이 2차 이상의 기약행렬인 A 대하여 {Γ(Am)}∞m=1의 극한을 구하였다. 더 나

아가, {Γ(Am)}∞m=1의극한이 J 블록대각행렬이되는행렬 A를분석하였다. 이모든

결과들은 A의유향그래프의경쟁그래프를연구함으로써얻어졌다.

주요어휘: 기약 Bool (0,1)-행렬, Bool (0,1)-행렬의거듭제곱,선형연결그래프,비원

시성지수, m-걸음경쟁그래프,그래프열,그래프의거듭제곱

학번: 2012-21424