無三角形且含五邊形之距離正則圖

୯!ҥ!Ҭ!೯!ε!Ꮲ!

ᔈҔኧᏢس!

റ!γ!ፕ!Ў!

คΟف׎Ъ֖ϖᜐ׎ϐຯᚆ҅߾კ!

Triangle-free Distance-regular Graphs

with Pentagons

!

റ!γ!ғ

!

;

!

ዐ཰۸!

!

ࡰᏤ௲௤

!

;

!

શדЎ

!

௲௤!

ύ ๮ ҇ ୯ ΐ Μ Ύ ԃ Ϥ Д!

คΟف׎Ъ֖ϖᜐ׎ϐຯᚆ҅߾კ!

Triangle-free Distance-regular Graphs

with Pentagons

!

!

!

റ!γ!ғǺዐ཰۸! ! Student : Yeh-Jong Pan!

ࡰᏤ௲௤ǺશדЎ!!!!Advisor : Chih-Wen Weng!

!

୯!ҥ!Ҭ!೯!ε!Ꮲ!

ᔈҔኧᏢس!

റ!γ!ፕ!Ў!

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

In Partial Fulfillment of the Requirements

For the Degree of Doctor of Philosophy

In Applied Mathematics

June 2008

Hsinchu, Taiwan, Republic of China

_!

誌 謝

在職進修原本就是一件辛苦的事，對我而言，這條路尤其漫長而艱辛，別的

不說，光這幾年所走過的路，豈止萬里而已，估計大約可以環繞台灣 200 圈了，

現在總算可以稍事休息，待養精蓄銳後，再往下一個目標前進。

取得博士學位是我現階段的目標，而完成博士論文是這個階段的終點，但這

卻只是通往研究之路的起點，後面還有很長的一段路要走，回首這些日子的點點

滴滴，對於週遭曾幫助我的人，心中始終存著感激，僅以此文表達我誠摯的謝意。

首先要感謝的是我的指導教授翁志文教授，這幾年來，除了在研究課題上的

指引之外，其它如研究方法與研究態度、論文的寫作、投稿及對審稿者的回應等

方面，亦多所教導，而其間不管是投稿的論文、演講稿或博士論文，每每修改十

幾、二十幾次，他總是耐心的和我討論，給我很好的建議，我能夠順利的取得博

士學位，完全要歸功於翁老師的耐心指導。此外，我也非常感謝 Terwilliger 教授，

他提供了很多研究上的寶貴意見，讓我獲益良多。

其次要感謝的是黃大原教授，他是我讀碩士班時的指導教授。當初，我跟他

說要來報考博士班，他就自行影印歷屆考古題寄給我，並針對我的讀書計劃給我

很好的意見，甚至後來推薦我去找翁老師當指導教授，這些都對我有很大的幫

助，老師對我的支持與愛護，令我銘記於心。

傅恆霖教授一直是相當受學生歡迎的老師，從早期便是我非常敬佩與學習的

對象，我在教書的過程中，不管是當導師或者和學生互動，很多是受傅老師身教

的影響，雖然他不是我的指導教授，但卻是人生的導師，能夠兩度受教，真是幸

運。陳秋媛教授是很棒的老師，雖然我只旁聽過一學期的課，但從她課前的準備、

上課的認真詳細及對學生的關心，不難看出她也很受歡迎，而從這門課當中，我

除了獲得演算法的知識以外，也學到一些教學經驗及得到一些啟發。

此外，還要感謝很多幫助過我的朋友，例如 DavidGuo、飛黃、賓賓、貴弘、

嘉文、Taller、小培、Robin、亮銓、維展、明欣、國元、經凱、泰峰、惠蘭、祐

寧、貓頭等，可能還有一些遺漏掉的朋友，在此一併致謝。

最後，要謝謝我的母親、家人及親友的支持與鼓勵，感謝大家。

คΟف׎Ъ֖ϖᜐ׎ϐຯᚆ҅߾კ!

റγғǺዐ཰۸! ! !!!!!!!! ࡰᏤ௲௤ǺશדЎ!

୯ ҥ Ҭ ೯ ε Ꮲ!

ᔈ Ҕ ኧ Ꮲ س

!

ᄔ!!!!!ा!

Եቾ΋ঁڀԖ

܄፦ޑຯᚆ҅߾კ̳Ǵଷ೛̳ޑޔ৩

D ԿϿࣁ 4 Ъځ

࣬Ҭୖኧ!

a

>0 Ъ

a

ɫ0Ǵךॺஒ᛾ܴΠӈ(i)-(iii)ࢂ฻ሽޑǺ!

(i)!̳ڀԖ

܄፦Ъό֖ߏࡋࣁ 4 ޑѳՉѤᜐ׎Ƕ!

(ii)!̳ڀԖ

܄፦Ъό֖ҺՖߏࡋࣁ

i ޑѳՉѤᜐ׎Ǵځύ

Ƕ!

(iii)

̳ڀԖђڂୖኧ(

D-b-͉-͊)Ǵځύ b-͉-͊ࢂჴኧǴЪ b<-1Ƕ

Զ྽చҹ(i)-(iii) ԋҥਔǴךॺ᛾ள̳ڀԖ 3-bounded ܄፦ǶճҔ೭ঁ܄፦Ǵךॺ

ёа᛾ܴځ࣬Ҭୖኧ

c

฻ܭ 1 ܈ 2ǹЪӵ݀

c

>1Ǵ߾ (

b-͉-͊) = (-2, -2,

)

Ƕ!

Triangle-free Distance-regular Graphs

with Pentagons

!

!

!

Student : Yeh-Jong Pan

!!!!!

Advisor : Chih-Wen Weng

!

Department of Applied Mathematics Department of Applied Mathematics National Chiao Tung University National Chiao Tung University

!

!

Abstract

Let Γ denote a distance-regular graph withQ-polynomial property. Assume the diam-eter D of Γ is at least 3 and the intersection numbers a₁ = 0 and a₂ = 0. We show the following (i)-(iii) are equivalent.

(i) Γ isQ-polynomial and contains no parallelograms of length 3.

(ii) Γ isQ-polynomial and contains no parallelograms of any length i for 3 ≤ i ≤ D. (iii) Γ has classical parameters (D, b, α, β) for some real constants b, α, β with b < −1. When (i)-(iii) hold, we show that Γ has 3-bounded property. Using this property we prove that the intersection number c₂ is either 1 or 2, and if c₂ = 1 then (b, α, β) = (−2, −2, ((−2)D+1− 1)/3).

Table of Contents

Abstract (in Chinese) i

Abstract iv

Table of Contents v

1 Introduction 1

2 Preliminaries 4

2.1 Distance-regular Graphs . . . 5

2.2 D-bounded Distance-regular Graphs . . . 9

2.3 Q-polynomial Property . . . 12

2.4 Classical Parameters . . . 14

2.5 Block Designs . . . 16

3 A Combinatorial Characterization of Distance-regular Graphs with Classical Parameters 18 3.1 Counting 4-vertex Configurations . . . 18

3.2 Combinatorial Characterization . . . 20

4 An Upper Bound of c2 23 4.1 Results from Simple Computations . . . 23

4.2 Multiplicity Technique . . . 25

5 3-bounded Property 27 5.1 Weak-geodetically Closed with respect to a Vertex . . . 27

6 A Constant Bound of c2 32

6.1 Preliminary Lemmas . . . 32 6.2 An Application of 3-bounded Property . . . 35

Chapter 1

Introduction

Distance-regular graphs were introduced by Biggs as a combinatorial generalization of distance-transitive graphs in 1970. They became a popular topic after that Desarte studied P -polynomial schemes [5], which are exactly the distance-regular graphs, mo-tivated by problems of coding theory in his thesis. After that, Leonard proved that the dual eigenvalues of a Q-polynomial distance-regular graph satisfy a recurrence relation and derived explicit formulae of the intersection numbers [12]. With these formulae it sheds light on the classification of Q-polynomial distance-regular graphs, as also stated in the book of Eiichi Bannai and Tatsuro Ito on Algebraic Combinatorics I : Association Schemes [1].

Brouwer, Cohen, and Neumaier found that the intersection numbers of most known families of distance-regular graphs could be described in terms of four parameters (D, b, α, β) [3, p. ix, p193]. They invented the term classical to describe such graphs. The class of distance-regular graphs which have classical parameters is a special case of distance-regular graphs with the Q-polynomial property [3, Corollary 8.4.2]. Note that the converse is not true, since an ordinary n-gon has the Q-polynomial property, but does not have classical parameters [3, Table 6.6]. Many authors proved the con-verse under various additional assumptions. Let Γ denote a distance-regular graph with diameter D ≥ 3 (See Chapter 2 for formal definitions.). Indeed assume Γ is Q-polynomial. Then Brouwer, Cohen, Neumaier in [3, Theorem 8.5.1] show that if Γ is a near polygon, with the intersection number a₁ = 0, then Γ has classical parameters. Weng generalizes this result with a weaker assumption, without kites of length 2 or 3 in Γ, to replace the near polygon assumption [23, Lemma 2.4]. For the complement case

a1 = 0, Weng shows that Γ has classical parameters if (i) Γ contains no parallelograms

of length 3 and no parallelograms of length 4; (ii) Γ has the intersection numbera₂ = 0; and (iii) Γ has diameter d ≥ 4 [25, Theorem 2.11]. We improve the above result by showing Theorem 3.2.1 in chapter 3.

Many authors study distance-regular graph Γ with a₁ = 0 and other additional assumptions. For example, Miklaviˇc assumes Γ is Q-polynomial and shows Γ is 1-homogeneous [13]; Koolen and Moulton assume Γ has degree 8, 9 or 10 and show that there are finitely many such graphs [11]; Juriˇsi´c, Koolen and Miklaviˇc assume Γ has an eigenvalue with multiplicity equal to the valency, a₂ = 0, and the diameter d ≥ 4 to show a₄ = 0 and Γ is 1-homogeneous [10].

In this thesis we aim at distance-regular graphs which have classical parameters (D, b, α, β) and intersection numbers a₁ = 0 anda₂ = 0. Since b < −1 [14], our work is a part of the classification of classical distance-regular graphs of negative type [27]. It worths to mention that all classical distance-regular graphs with b = 1 are classified by Y. Egawa, A. Neumaier and P. Terwilliger independently (See [3, p195] for details). Let Γ be a distance-regular graph which has classical parameters (D, b, α, β) and a₁ = 0,

a2 = 0, and D ≥ 3. It was previously known that Γ has 2-bounded property [26, 19].

By applying this to a strongly regular subgraph of Γ, we find an upper bound of c₂ in terms of an expression of b in chapter 4. After that we prove the 3-bounded property of Γ in chapter 5. Finally we use the 3-bounded property to conclude thatc₂ = 1 or 2.

The following preprints and papers are included in this thesis:

1. Y. Pan, M. Lu, and C. Weng, Triangle-free distance-regular graphs, J. Algebr.

Comb., 27(2008), 23-34.

2. Y. Pan and C. Weng, 3-bounded Property in a Triangle-free Distance-regular Graph, European Journal of Combinatorics, 29(2008), 1634-1642.

3. Y. Pan and C. Weng, A note on triangle-free distance-regular graphs witha₂ = 0, preprint (2007), submitted to Journal of Combinatorial Theory, Series B.

In Chapter 2 we introduce definitions, terminologies and some results concerning distance-regular graphs and block designs.

In Chapter 3 we discuss a combinatorial property of distance-regular graphs which have classical parameters.

In Chapter 4 we work on distance-regular graphs with classical parameters and use the multiplicity technique to find an upper bound of c₂.

In Chapter 5 we prove the 3-bounded property of the distance-regular graphs. In Chapter 6 we use the 3-bounded property and Fisher’s inequality to show the upper boundc₂ ≤ 2 of c₂. This upper bound rules out almost all the graphs of our target in the classification. Also we find that ifc₂ = 1, then (b, α, β) = (−2, −2,(−2)D+1₃ −1).

Chapter 2

Preliminaries

In this chapter we review some definitions, basic concepts and some previous results concerning distance-regular graphs and block designs. See Bannai and Ito [1] or Ter-williger [20] for more background information of distance-regular graphs and van Lint and Wilson [22] for block designs.

Let Γ=(X, R) denote a finite undirected, connected graph without loops or multiple edges with vertex setX, edge set R, distance function ∂, and diameter D:=max{ ∂(x, y) |

x, y ∈ X}. By a pentagon, we mean a 5-tuple x1x2x3x4x5 consisting of vertices of Γ

such that ∂(x_i, x_i+1) = 1 for 1≤ i ≤ 4, ∂(x₅, x₁) = 1 and no other edges between two distinct vertices.

For a vertex x ∈ X and an integer 0 ≤ i ≤ D, set Γ_i(x) := { z ∈ X | ∂(x, z) = i}. The valency k(x) of a vertex x ∈ X is the cardinality of Γ₁(x). The graph Γ is called

regular (with valency k) if each vertex in X has valency k.

An incidence structure is a triple (P, B, I), where P and B are two sets and I ⊆

P×B. The elements of P and B are called points and blocks respectively. If (p, B)∈ I,

then we say point p and block B are incident.

A t-(v, κ, λ) design is an incidence structure (P, B, I), where |P| = v, satisfying the following conditions:

• For each block B ∈ B, there are exactly κ points incident with B.

is not incident withB.

• For any set T of t points, there are exactly λ blocks incident with all points of T .

It is easy to prove that the number of blocks incident with any fixed point p of

P is the same [22, Theorem 19.3] and is called the replication number of the design.

Actually the number is λv−1_t−1/k−1_t−1.

2.1 Distance-regular Graphs

A graph Γ = (X, R) is said to be distance-regular whenever for all integers 0 ≤ h, i, j ≤

D, and all vertices x, y ∈ X with ∂(x, y) = h, the number ph

ij =|Γi(x) ∩ Γj(y)|

is independent of x, y. The constants ph_ij are known as the intersection numbers of Γ. Let Γ=(X, R) be a distance-regular graph. For two vertices x, y ∈ X with ∂(x, y) =

i, set B(x, y) := Γ1(x) ∩ Γi+1(y), C(x, y) := Γ1(x) ∩ Γi−1(y), A(x, y) := Γ1(x) ∩ Γi(y). Note that |B(x, y)| = pi 1 i+1, |C(x, y)| = pi 1 i−1, |A(x, y)| = pi 1 i are independent of x, y.

For convenience, set c_i :=pi_{1 i−1} for 1≤ i ≤ D, a_i :=pi_{1 i} for 0≤ i ≤ D, b_i :=pi_{1 i+1} for 0≤ i ≤ D − 1, k_i :=p0_{i i} for 0≤ i ≤ D, and set b_D := 0, c₀ := 0, k := b₀. Note that

k is the valency of Γ. It follows immediately from the definition of ph

ij that bi = 0 for

0≤ i ≤ D − 1 and c_i = 0 for 1 ≤ i ≤ D. Moreover

and

ki = b0_c· · · bi−1

1· · · ci for 1≤ i ≤ D. (2.1.2)

A strongly regular graph is a distance-regular graph with diameter 2. We quote a couple of Lemmas about strongly regular graphs which will be used in Chapter 4 and Chapter 6.

Lemma 2.1.1. [22, Theorem 21.1] Suppose Ω is a strongly regular graph with

inter-section numbers a_i, b_i, c_i, where 0 ≤ i ≤ 2. Let v = |Ω| and k = b₀. Suppose that r ≥ s are the eigenvalues other than k. Let f and g be the multiplicities of r and s respectively. Then f = 1 2(v − 1 + (v − 1)(c₂− a₁)− 2k  (c₂− a₁)2+ 4(k − c₂)) (2.1.3) and g = 1 2(v − 1 − (v − 1)(c₂− a₁)− 2k  (c₂− a₁)2+ 4(k − c₂)) (2.1.4)

are nonnegative integers.

Proof. Let A be the adjacency matrix of Ω, J be the v by v all-one matrix, and j be

the v by 1 all-one vector. We have AJ = kJ, Aj = kj, and A2 =kI + a₁A + c₂(J −

I − A) by direct computation. Note that k is an eigenvalue of A with eigenvector j

whose multiplicity is one since Ω is connected. Suppose that x is an eigenvalue with eigenvector orthogonal to j. Then

x2_{+ (c}

2− a1)x + (c2− k) = 0. (2.1.5)

Equation (2.1.5) has two solutions

r, s = 1

2(a1 − c2± 

(a₁− c₂)2+ 4(k − c₂) ). (2.1.6) Since f and g are multiplicities of r and s respectively, we have the following two equations.

1 +f + g = v (2.1.7) and

Solving (2.1.7) and (2.1.8) forf, g by (2.1.6), we have (2.1.3) and (2.1.4). It is obvious that f and g are nonnegative integers.

Lemma 2.1.2. [2, p. 276, Theorem 19] Let Ω be a strongly regular graph with valency

b0 =k, a1 = 0, and c2 = 1. Then k ∈ {2, 3, 7, 57}. 

Proof. Note thatc₁ = 1 andb₁ =k−a₁−c₁ =k−1. Then v := |Ω| = 1+k₁+k₂ = 1+k2. Substitutingv, c₂ and a₁ into (2.1.3) we have

f = 1

2(k

2_{+ √}k2− 2k

4k − 3 ). (2.1.9) Equation (2.1.9) implies k2 − 2k = 0 or 4k − 3 = s2 for some integer s since f is a nonnegative integer. If k2 − 2k = 0 then k = 2. Suppose 4k − 3 = s2, then

k = s2+ 3

4 . (2.1.10)

Substituting (2.1.10) into (2.1.9) yields

s5_+s4_{+ 6s}3_{− 2s}2_{+ (9− 32f)s = 15. (2.1.11)}

Hence s is a factor of 15. The result follows from substituting s into k and deleting the case k = 1.

Example 2.1.3. The Petersen graph shown in Figure 2.1 is a strongly regular graph

with intersection numbers a₁ = 0, a₂ = 2, c₁ =c₂ = 1, b₀ = 3, b₁ = 2.

s 0 s 1 s 2 s3 s4 s₅ s 6 s 7 s8 s 9Q_Q Q Q Q H H H H H A A A A  SSS     C C C C CC PP



 

Figure 2.1: Petersen graph.

Example 2.1.4. [3, p. 285](Hermitian forms graph Her2(D)) Let U denote a finite

vector space over GF (2) consisting of the Hermitian forms on U. Thus f ∈ H if and only if f(u, v) is linear in v, and f(v, u) = f(u, v) for all u, v ∈ U. Pick f ∈ H. We define

rk(f) = dim(U \ Rad(f)), where

Rad(f) = {u ∈ U | f(u, v) = 0 for all v ∈ U}.

Set X = H, and xy ∈ R if and only if rk(x − y) = 1 for all x, y ∈ X. Then Γ = (X, R) is a distance-regular graph with diameter D and intersection numbers

ci = 2 i−1₍₂i_{− (−1)}i₎ 3 (1≤ i ≤ D), (2.1.12) bi = 2 2D_{− 2}2i 3 (0≤ i ≤ D). (2.1.13) By (2.1.1), (2.1.12) and (2.1.13) we have ai = 2 2i−1_{+ (−1)}i₂i−1_{− 1} 3 (1≤ i ≤ D). (2.1.14) Note that a₁ = 0 and a₂ = 3. It was shown in [9] that Γ is the unique distance-regular graph with intersection numbers satisfying (2.1.12) and (2.1.13).

Example 2.1.5. [3, p. 372](Gewirtz graph) Suppose (P, B, I) is a 3-(22, 6, 1) design,

where I = {(p, B) | p ∈ P, B ∈ B, and p ∈ B}. Fix an element p of P. Let

X = {B ∈ B | p ∈ B} and R = {B1B2 | B1, B2 ∈ X and B1 ∩ B2 = ∅}. Then

Γ = (X, R) is a distance-regular graph which is known as Gewirtz graph. It is a strongly regular graph with intersection numbers a₁ = 0, a₂ = 8, c₁ = 1, c₂ = 2,

b0 = 10, and b1 = 9. It was shown in [6] and [7] that Γ is the unique strongly regular

graph with intersection numbers satisfying b₀ = 10, b₁ = 9, c₁ = 1, and c₂ = 2.

Example 2.1.6. [3, Theorem 11.4.2](Witt graph M23) Suppose (P, B, I) is a 5-(24, 8, 1)

design whereI = {(p, B) | p ∈ P, B ∈ B, and p ∈ B}. Fix a point σ ∈ P, and let B be the collection of 506 blocks in B missing σ. Then (P \ {σ}, B) is a 4-(23, 8, 4) design. LetX = B and R = {B₁B₂ | B₁∩ B₂ =∅ for distinct B₁, B₂ ∈ X}. Then Γ = (X, R) is a distance-regular graph which is known as Witt graph M₂₃. It has diameter D = 3 and intersection numbers a₁ = 0,a₂ = 2,a₃ = 6,c₁ =c₂ = 1,c₃ = 9, b₀ = 15, b₁ = 14

and b₂ = 12. It was shown in [3, Theorem 11.4.2] that Γ is the unique distance-regular graph of diameter 3 with intersection numbers satisfying b₀ = 15, b₁ = 14, b₂ = 12,

c1 =c2 = 1, and c3 = 9.

Throughout this chapter we assume Γ=(X, R) is a distance-regular graph.

Definition 2.1.7. Pick an integer 2 ≤ i ≤ D. By a parallelogram of length i in Γ, we

mean a 4-tuple xyzw of vertices of X such that

∂(x, y) = ∂(z, w) = 1, ∂(x, z) = i, ∂(x, w) = ∂(y, w) = ∂(y, z) = i − 1.

For a parallelogram of length i, see Figure 2.2.

i − 1 1 @ @ @ @ @ @@ i − 1 i − 1 1 t t t t y z w x

Figure 2.2: A parallelogram of length i.

2.2 D-bounded Distance-regular Graphs

Assume Γ = (X, R) is distance-regular with diameter D ≥ 3. Recall that a sequence

x, y, z of vertices of Γ is geodetic whenever

∂(x, y) + ∂(y, z) = ∂(x, z).

Definition 2.2.1. A sequence x, y, z of vertices of Γ is weak-geodetic whenever

∂(x, y) + ∂(y, z) ≤ ∂(x, z) + 1.

Definition 2.2.2. A subset Ω ⊆ X is weak-geodetically closed if for any weak-geodetic

sequence x, y, z of Γ,

Weak-geodetically closed subgraphs are called strongly closed subgraphs in [18]. We refer the readers to [17, 4, 9, 19, 26, 8] for information on weak-geodetically closed subgraphs.

We make one more definition which will be used later.

Definition 2.2.3. Let Ω be a subset of X, and pick any vertex x ∈ Ω. Ω is said to be

weak-geodetically closed with respect to x, whenever for all z ∈ Ω and for all y ∈ X, x, y, z are weak-geodetic =⇒ y ∈ Ω. (2.2.1) Note that Ω is weak-geodetically closed with respect to a vertex x ∈ Ω if and only if

C(z, x) ⊆ Ω and A(z, x) ⊆ Ω for all z ∈ Ω

[26, Lemma 2.3]. Also Ω is weak-geodetically closed if and only if for any vertexx ∈ Ω, Ω is weak-geodetically closed with respect to x. The following theorems will be used later in this thesis.

Theorem 2.2.4. [26, Theorem 4.6] Let Γ be a distance-regular graph with diameter

D ≥ 3. Let Ω be a regular subgraph of Γ with valency γ and set d := min{i | γ ≤ ci+ai}.

Then the following (i),(ii) are equivalent.

(i) Ω is weak-geodetically closed with respect to at least one vertex x ∈ Ω. (ii) Ω is weak-geodetically closed with diameter d.

In this case γ = c_d+a_d.

Suppose (i) and (ii) hold. Then Ω is distance-regular, with diameter d, and inter-section numbers

ci(Ω) = ci(Γ), (2.2.2)

ai(Ω) = ai(Γ) (2.2.3)

Lemma 2.2.5. ([19, Lemma 2.6]) Let Γ be a distance-regular graph with diameter 2,

and let x be a vertex of Γ. Suppose a₂ = 0. Then the subgraph induced on Γ₂(x) is

connected of diameter at most 3. 

Definition 2.2.6. Γ is said to be i-bounded whenever for all x, y ∈ X with ∂(x, y) ≤ i,

there is a regular weak-geodetically closed subgraph of diameter∂(x, y) containing x, y. The properties ofD-bounded distance-regular graphs were studied in [24], and these properties were used in the classification of classical distance-regular graphs of negative type [27].

Theorem 2.2.7. ([26, Proposition 6.7],[19, Theorem 1.1]) Let Γ be a distance-regular

graph with diameter D ≥ 3. Suppose a₁ = 0, a₂ = 0 and Γ contains no parallelograms

of length 3. Then Γ is 2-bounded. 

Theorem 2.2.8. ([26, Lemma 6.9],[19, Lemma 4.1]) Let Γ be a distance-regular graph

with diameter D ≥ 3. Suppose a₁ = 0, a₂ = 0 and Γ contains no parallelograms of any

length. Let x be a vertex of Γ, and let Ω be a weak-geodetically closed subgraph of Γ with diameter 2. Suppose there exists an integer i and a vertex u ∈ Ω ∩ Γ_i−1(x), and

suppose Ω∩ Γ_i+1(x) = ∅. Then for all t ∈ Ω, we have ∂(x, t) = i − 1 + ∂(u, t). 

Theorem 2.2.9. ([24, Corollary 2.2]) Let Γ = (X, R) denote a distance-regular graph

with diameter D. Suppose that Γ is D-bounded. For two distinct vertices x, y ∈ X, there exists a unique regular weak-geodetically closed subgraph Δ(x, y) containing x and y with diameter ∂(x, y). Furthermore, Δ(x, y) is a distance-regular graph.  Let Γ = (X, R) denote a distance-regular graph with diameter D. Suppose that Γ isD-bounded. For two distinct vertices x, y ∈ X, we use Δ(x, y) to denote the unique weak-geodetically closed subgraph containing x and y with diameter ∂(x, y).

Theorem 2.2.10. ([24, Lemma 2.6]) Let Γ denote a distance-regular graph with

di-ameter D. Suppose that Γ is D-bounded. Then

bi > bi+1 (0≤ i ≤ D − 1). (2.2.4)

Proof. For 0≤ i ≤ D − 1, pick x, y with ∂(x, y) = i + 1. Then Δ(x, y) is a

distance-regular graph with diameteri+1 by Theorem 2.2.9. Note that b_i(Δ(x, y)) = b_i−b_i+1= 0. The result follows immediately.

2.3 Q-polynomial Property

Let Γ = (X, R) denote a distance-regular graph with diameter D ≥ 3. Let R denote the real number field. Let Mat_X(R) denote the algebra of all the matrices over R with the rows and columns indexed by the elements of X. For 0 ≤ i ≤ D let A_i denote the matrix in Mat_X(R), defined by the rule

(A_i)_xy = ⎧ ⎪ ⎨ ⎪ ⎩ 1, if ∂(x, y) = i; 0, if ∂(x, y) = i for x, y ∈ X. We call A_i the distance matrices of Γ. We have

A0 =I, (2.3.1)

A0+A1 +· · · + AD =J (J = all 1s matrix), (2.3.2)

At

i =Ai for 0≤ i ≤ D (Ati means the transpose of Ai), (2.3.3)

AiAj = D h=0 ph ijAh for 0≤ i, j ≤ D, (2.3.4) AiAj =AjAi for 0 ≤ i, j ≤ D. (2.3.5)

Let M denote the subspace of Mat_X(R) spanned by A₀, A₁, . . . , A_D. Then M is a commutative subalgebra of Mat_X(R), and is known as the Bose-Mesner algebra of Γ.

By [3, p. 59, 64], M has a second basis E₀, E₁, . . . , E_D such that E0 =|X|−1J, (2.3.6) EiEj =δijEi for 0≤ i, j ≤ D, (2.3.7) E0+E1+· · · + ED =I, (2.3.8) Et i =Ei for 0≤ i ≤ D. (2.3.9)

The E₀, E₁, . . . , E_D are known as the primitive idempotents of Γ, and E₀ is known as the trivial idempotent. Let E denote any primitive idempotent of Γ. Then we have

E = |X|−1 D i=0

θ∗

iAi (2.3.10)

for some θ∗₀, θ∗₁, . . . , θ∗_D ∈ R, called the dual eigenvalues associated with E.

Set V = R|X| (column vectors), and view the coordinates ofV as being indexed by

X. Then the Bose-Mesner algebra M acts on V by left multiplication. We call V the standard module of Γ. For each vertex x ∈ X, set

ˆ

x = (0, 0, . . . , 0, 1, 0, . . . , 0)t_{, (2.3.11)}

where the 1 is in coordinate x. Also, let  ,  denote the dot product

u, v = ut_{v for u, v ∈ V. (2.3.12)}

Then referring to the primitive idempotentE in (2.3.10), we compute from (2.3.9)-(2.3.12) that for x, y ∈ X,

Eˆx, Eˆy = |X|−1_θ∗

i, (2.3.13)

where i = ∂(x, y).

Let ◦ denote the entry-wise multiplication in Mat_X(R). Then

Ai◦ Aj =δijAi for 0≤ i, j ≤ D,

so M is closed under ◦. Thus there exists q_ijk ∈ R for 0 ≤ i, j, k ≤ D such that

Ei◦ Ej =|X|−1 D k=0 qk ijEk for 0 ≤ i, j ≤ D.

Γ is said to be Q-polynomial with respect to the given ordering E₀, E₁,. . ., E_D of the primitive idempotents, if for all integers 0 ≤ h, i, j ≤ D, qh_ij = 0 (resp. q_ijh = 0) whenever one of h, i, j is greater than (resp. equal to) the sum of the other two. Let

E denote any primitive idempotent of Γ. Then Γ is said to be Q-polynomial with

respect to E whenever there exists an ordering E₀, E₁ = E,. . ., E_D of the primitive idempotents of Γ, with respect to which Γ isQ-polynomial. If Γ is Q-polynomial with respect toE, then the associated dual eigenvalues are distinct [20, p. 384].

The following theorem about theQ-polynomial property will be used in this thesis.

Theorem 2.3.1. [21, Theorem 3.3] Let Γ be Q-polynomial with respect to a primitive

idempotent E, and let θ∗₀, . . . , θ∗_D denote the corresponding dual eigenvalues. Then the following (i), (ii) hold.

(i) For all integers 1 ≤ h ≤ D, 0 ≤ i, j ≤ D and for all x, y ∈ X such that ∂(x, y) = h, z∈X ∂(x,z)=i ∂(y,z)=j Eˆz − z∈X ∂(x,z)=j ∂(y,z)=i Eˆz = ph ij θ∗ i − θj∗ θ∗ 0 − θh∗ (Eˆx − Eˆy). (2.3.14)

(ii) For an integer 3≤ i ≤ D, θ∗

i−2− θ∗i−1=σ(θi−3∗ − θi∗) (2.3.15)

for an appropriate σ ∈ R \ {0}. 

2.4 Classical Parameters

A distance-regular graph Γ is said to have classical parameters (D, b, α, β) whenever the intersection numbers of Γ satisfy

ci = i 1  1 +α i − 1 1  for 0 ≤ i ≤ D, (2.4.1) bi =  D 1  − i 1   β − α i 1  for 0 ≤ i ≤ D, (2.4.2)

where i 1  := 1 +b + b2+· · · + bi−1. (2.4.3) Suppose Γ has classical parameters (D, b, α, β). Combining (2.4.1)-(2.4.3) with (2.1.1), we have ai = i 1  β − 1 + α  D 1  − i 1  − i − 1 1  = i 1  a1+α  1− i 1  − i − 1 1  for 0 ≤ i ≤ D. (2.4.4)

Example 2.4.1. Petersen graph shown in Figure 2.1 is a distance-regular graph which

has classical parameters (D, b, α, β) with D = 2, b = −2, α = −2 and β = −3, which satisfies a₁ = 0, a₂ = 0 and 1 = c₂ < b(b + 1) = 2.

Example 2.4.2. [9] Hermitian forms graph Her2(D) is a distance-regular graph with

classical parameters (D, b, α, β) with b = −2, α = −3 and β = −((−2)D + 1), which satisfies a₁ = 0, a₂ = 0 and c₂ =b(b + 1) = 2.

Example 2.4.3. [22, p. 237] Gewirtz graph is a distance-regular graph which has

classical parameters (D, b, α, β) with D = 2, b = −3, α = −2, β = −5, which satisfies

a1 = 0, a2 = 0 and 2 = c2 < b(b + 1) = 6.

Example 2.4.4. [3, Table 6.1] Witt graph M23 is a distance-regular graph which has

classical parameters (D, b, α, β) with D = 3, b = −2, α = −2, β = 5, which satisfies

a1 = 0, a2 = 0 and 1 = c2 < b(b + 1) = 2.

We list the parameters of the above examples in the following table for summary.

name D b α β a₁ a₂ c₂

Petersen graph 2 −2 −2 −3 0 2 1

Hermitian forms graph Her₂(D) D −2 −3 −((−2)D + 1) 0 3 2

Gewirtz graph 2 −3 −2 −5 0 8 2

The following theorem characterizes the distance-regular graphs with classical pa-rameters in an algebraic way.

Theorem 2.4.5. ([21, Theorem 4.2]) Let Γ denote a distance-regular graph with

di-ameter D ≥ 3. Choose b ∈ R \ {0, −1}. Then the following (i)-(ii) are equivalent. (i) Γ is Q-polynomial with associated dual eigenvalues θ₀∗, θ∗₁, . . . , θ∗_D satisfying

θ∗ i − θ0∗ = (θ1∗− θ∗0) i 1  b1−i _{for 1≤ i ≤ D. (2.4.5)}

(ii) Γ has classical parameters (D, b, α, β) for some real constants α, β. 

2.5 Block Designs

In this section we introduce some results of block designs which will be used in the proof of Theorem 6.2.1.

Lemma 2.5.1. Let (P, B, I) be a 2-(v, κ, λ) design. Suppose |B| = b and r is the

replication number. Then bκ = vr.

Proof. Counting in two ways the number of pairs (x, B) ∈ I, where x ∈ P and B ∈ B,

the equality follows immediately.

The following famous theorem is known as Fisher’s inequality.

Theorem 2.5.2. [22, Theorem 19.6] For a 2-(v, κ, λ) design with b blocks and v > κ

we have b ≥ v. 

Proof. Let r denote the replication number and N denote the v × b incidence matrix

of the design. Then

NNt_{= (r − λ)I + λJ, (2.5.1)}

where J is the v × v all-one matrix. Note that J has eigenvalues v and 0 with multi-plicities 1 and v − 1 respectively. Hence the eigenvalues of NNt are λv + (r − λ) and

r − λ with multiplicities 1 and v − 1 respectively. This implies

where det(NNt) denotes the determinant of NNt. Observe that

r = λ(v − 1)_{k − 1} > λ. (2.5.3) By (2.5.2) and (2.5.3), NNt is invertible and has rankv. Note that

rank(NNt) ≤ rank(N) ≤ min{v, b}. The assertion of the theorem follows immediately.

Corollary 2.5.3. For a 2-(v, κ, λ) design with replication number r we have r ≥ κ.

Chapter 3

A Combinatorial Characterization

of Distance-regular Graphs with

Classical Parameters

The following theorem was shown in [25, Theorem 2.11].

Theorem 3.0.4. [25, Theorem 2.11] Let Γ = (X, R) denote a distance-regular graph

with diameter D ≥ 4 and intersection numbers a₁ = 0, a₂ = 0. Suppose Γ is

Q-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. Then Γ has classical parameters (D, b, α, β) with b < −1.

In this chapter we show the same result holds for the case D = 3. Theorem 3.2.1 is the main result of this chapter.

3.1 Counting 4-vertex Configurations

To prove Theorem 3.2.1, our main theorem in this chapter, we need a couple of lemmas. The first lemma is essentially given in [13, Theorem 5.2(i)], a proof is given here for completeness.

Lemma 3.1.1. [13, Theorem 5.2(i)] Let Γ denote a Q-polynomial distance-regular

2≤ i ≤ D and three vertices x, y, z such that

∂(x, y) = 1, ∂(y, z) = i − 1, ∂(x, z) = i. Then the quantity

si(x, y, z) := |Γi−1(x) ∩ Γi−1(y) ∩ Γ1(z)| (3.1.1)

is equal to

ai−1

(θ₀∗− θ∗_i−1)(θ∗₂− θ∗_i)− (θ∗₁− θ_i−1∗ )(θ₁∗− θ_i∗)

(θ₀∗− θ∗_i−1)(θ∗_i−1− θ_i∗) . (3.1.2)

In particular (3.1.1) is independent of the choice of the vertices x, y, z. Proof. Let s_i(x, y, z) denote the expression in (3.1.1) and set

i(x, y, z) = |Γi(x) ∩ Γi−1(y) ∩ Γ1(z)|. Observe si(x, y, z) + i(x, y, z) = ai−1. (3.1.3) By (2.3.14) we have w∈X ∂(y,w)=i−1 ∂(z,w)=1 E ˆw − w∈X ∂(y,w)=1 ∂(z,w)=i−1 E ˆw = ai−1θ ∗ i−1− θ1∗ θ∗ 0− θi−1∗ (Eˆy − Eˆz). (3.1.4) Taking the inner product of (3.1.4) with ˆx using (2.3.13) and the assumption a₁ = 0, we obtain

si(x, y, z)θi−1∗ +i(x, y, z)θ∗i − ai−1θ2∗ =ai−1θ ∗

i−1− θ∗1

θ∗

0− θi−1∗

(θ₁∗− θ∗_i). (3.1.5) Solving s_i(x, y, z) by using (3.1.3) and (3.1.5), we get (3.1.2).

By Lemma 3.1.1, s_i(x, y, z) is a constant for any vertices x, y, z with ∂(x, y) = 1,

∂(y, z) = i − 1, ∂(x, z) = i. Let si denote the expression in (3.1.1). Note thatsi = 0 if

and only if Γ contains no parallelograms of length i.

Lemma 3.1.2. Let Γ denote a distance-regular graph which has classical parameters

Proof. Since a₁ = 0 and a₂ = 0, from (2.4.3) and (2.4.4) we have

−α(b + 1)2 _=a

2− (b + 1)a1 =a2 > 0. (3.1.6)

Hence

α < 0. (3.1.7) By direct computation from (2.4.1), we get

(c₂− b)(b2+b + 1) = c₃ > 0. (3.1.8) Since

b2_{+b + 1 > 0,}

(3.1.8) implies

c2 > b. (3.1.9)

Using (2.4.1) and (3.1.9), we get

α(1 + b) = c2− b − 1 ≥ 0. (3.1.10)

Hence b < −1 by (3.1.7) and b = −1.

3.2 Combinatorial Characterization

The following theorem characterizes the distance-regular graphs with classical parame-ters and a₁ = 0, a₂ = 0 in a combinatorial way.

Theorem 3.2.1. Let Γ denote a distance-regular graph with diameter D ≥ 3 and

intersection numbers a₁ = 0, a₂ = 0. Then the following (i)-(iii) are equivalent.

(i) Γ is Q-polynomial and contains no parallelograms of length 3.

(ii) Γ is Q-polynomial and contains no parallelograms of any length i for 3 ≤ i ≤ D. (iii) Γ has classical parameters (D, b, α, β) for some real constants b, α, β with b < −1.

Proof. (ii)⇒(i) This is clear.

(iii)⇒(ii) Suppose Γ has classical parameters. Then Γ is Q-polynomial with asso-ciated dual eigenvalues θ₀∗, θ∗₁,. . . , θ∗_D satisfying

θ∗ i − θ0∗ = (θ1∗− θ∗0) i 1  b1−i _{for 1≤ i ≤ D. (3.2.1)}

We need to prove s_i = 0 for 3≤ i ≤ D. To compute s_i in (3.1.2), observe from (3.2.1) that

θ∗

i−1− θi∗ = (θ0∗− θ1∗)b1−i for 1≤ i ≤ D. (3.2.2)

Summing (3.2.2) for consecutive i, we find

(θ₁∗− θ_i∗) = (θ∗₀ − θ₁∗)(b−1+b−2+· · · + b1−i), (3.2.3) (θ∗₁− θ_i−1∗ ) = (θ∗₀− θ₁∗)(b−1+b−2+· · · + b2−i), (3.2.4) (θ₂∗− θ_i∗) = (θ∗₀ − θ₁∗)(b−2+b−3+· · · + b1−i), (3.2.5) (θ∗₀− θ∗_i−1) = (θ₀∗− θ∗₁)(b0 +b−1+· · · + b2−i) (3.2.6) for 3≤ i ≤ D. Evaluating (3.1.2) by using (3.2.2)-(3.2.6), we find s_i = 0 for 3≤ i ≤ D. (i)⇒(iii) Observe s₃ = 0. Then by setting i = 3 in (3.1.2) and using the assumption

a2 = 0, we find (θ₀∗− θ₂∗)(θ₂∗− θ₃∗)− (θ₁∗− θ∗₂)(θ₁∗− θ∗₃) = 0. (3.2.7) Set b := θ1∗− θ∗0 θ∗ 2 − θ∗1. (3.2.8) Then θ∗ 2 =θ∗0+ (θ∗₁− θ₀∗)(b + 1) b . (3.2.9)

Eliminating θ∗₂, θ₃∗ in (3.2.7) using (3.2.9) and (2.3.15), we have

−(θ∗

1− θ0∗)2(σb2+σb + σ − b)

σb2 = 0 (3.2.10)

for an appropriate σ ∈ R \ {0}. Since θ₁∗ = θ∗₀,

and hence

σ−1 ₌ b2+b + 1

b . (3.2.11)

By Theorem 2.4.5, to prove that Γ has classical parameters, it suffices to prove that

θ∗ i − θ∗0 = (θ∗1− θ0∗) i 1  b1−i _{for 1 ≤ i ≤ D. (3.2.12)}

We prove (3.2.12) by induction on i. The case i = 1 is trivial and the case i = 2 is from (3.2.9). Now suppose i ≥ 3. Then (2.3.15) implies

θ∗

i =σ−1(θ∗i−1− θi−2∗ ) +θ∗i−3 for 3≤ i ≤ D. (3.2.13)

Evaluating (3.2.13) using (3.2.11) and the induction hypothesis, we find that θ∗_i − θ₀∗ is as in (3.2.12). Therefore, Γ has classical parameters (D, b, α, β) for some scalars α, β. Note that b < −1 from Lemma 3.1.2.

Chapter 4

An Upper Bound of c

₂

In this chapter we assume that Γ has classical parameters and intersection numbers

a1 = 0, a2 = 0 to obtain the following theorem.

Theorem 4.0.2. Let Γ denote a distance-regular graph with diameter D ≥ 3 and

intersection numbers a₁ = 0, a₂ = 0. Suppose Γ has classical parameters (D, b, α, β).

Then each of b(b + 1)2_{(b + 2)} c2 , (b − 2)(b − 1)b(b + 1) 2 + 2b − c₂ (4.0.1) is an integer. Moreover c2 ≤ b(b + 1). (4.0.2)

Note that the bound in (4.0.2) will be improved toc₂ ≤ 2 in Chapter 6.

4.1 Results from Simple Computations

Theorem 4.1.1. [26, Proposition 6.7, Theorem 4.6] Let Γ = (X, R) denote a

distance-regular graph with diameter D ≥ 3. Assume that the intersection numbers a₁ = 0 and

a2 = 0. Suppose that Γ contains no parallelograms of length 3. Then for each pair

of vertices v, w ∈ X at distance ∂(v, w) = 2, there exists a weak-geodetically closed subgraph Ω of diameter 2 in Γ containing v, w. Furthermore Ω is strongly regular with

intersection numbers

ai(Ω) = ai(Γ), (4.1.1)

ci(Ω) = ci(Γ), (4.1.2)

bi(Ω) = a2(Γ) +c2(Γ)− ai(Ω)− ci(Ω) (4.1.3)

for 0≤ i ≤ 2. 

Corollary 4.1.2. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β), where D ≥ 3. Assume Γ has intersection numbers a₁ = 0 and a₂ = 0.

Then there exists a weak-geodetically closed subgraph Ω of diameter 2. Furthermore the intersection numbers of Ω satisfy

b0(Ω) = (1 +b)(1 − αb), (4.1.4) b1(Ω) = b(1 − α − αb), (4.1.5) c2(Ω) = (1 +b)(1 + α), (4.1.6) a2(Ω) = −(1 + b)2α, (4.1.7) |Ω| = (1 +b)(bα − 2)(bα − 1 − α) (1 +α) . (4.1.8)

Proof. Observeb < −1 by Lemma 3.1.2 and Γ contains no parallelograms of length 3 by

Theorem 3.2.1. Hence there exists a weak-geodetically closed subgraph Ω of diameter 2 by Theorem 2.2.7. By applying (2.4.1), (2.4.2) and (2.4.4) to (4.1.1)-(4.1.3), we have (4.1.4)-(4.1.7) immediately. Observe that |Ω| = 1 + k(Ω) + k(Ω)b₁(Ω)/c₂(Ω). (4.1.8) follows from this and (4.1.4)-(4.1.6).

Proposition 4.1.3. [26, Proposition 3.2] Let Γ denote a distance-regular graph with

diameter D ≥ 3. Suppose there exists a weak-geodetically closed subgraph Ω of Γ with diameter 2. Then the intersection numbers of Γ satisfy the following inequality

a3 ≥ a2(c2− 1) + a1. (4.1.9)

Corollary 4.1.4. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β), where D ≥ 3. Suppose the intersection numbers a₁ = 0 and a₂ = 0. Then

c2 ≤ b2+b + 2. (4.1.10)

Proof. Applyinga₁ = 0 in (2.4.4), we havea₃ =−α(b2+b+1)(b+1)2. Then by applying (4.1.9) using Lemma 3.1.2, (4.1.1), and (4.1.7), the result follows immediately.

4.2 Multiplicity Technique

We will improve the upper bound ofc₂ in (4.1.10). We need the following lemma.

Lemma 4.2.1. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β), where D ≥ 3. Assume the intersection numbers a₁ = 0 and a₂ = 0. Let Ω be a weak-geodetically closed subgraph of diameter 2 in Γ. Let r > s denote the

nontrivial eigenvalues of the strongly regular graph Ω. Then the following (i), (ii) hold: (i) The multiplicity of r is

f = (bα − 1)(bα − 1 − α)(bα − 1 + α)

(α − 1)(α + 1) . (4.2.1)

(ii) The multiplicity of s is

g = −b(bα − 1)(bα − 2)

(α − 1)(α + 1) . (4.2.2)

Proof. Let v=|Ω| and k be the valency of Ω. Note that c₂(Ω) = (1 +b)(1 + α) by (2.4.1),k(Ω) = (1 + b)(1 − αb) by (4.1.4), and v = (1 + b)(bα − 2)(bα − 1 − α)/(1 + α) by (4.1.8). Now (4.2.1) and (4.2.2) follow from (2.1.3) and (2.1.4).

Corollary 4.2.2. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β), where D ≥ 3. Assume Γ has intersection numbers a₁ = 0 anda₂ = 0. Then

b(b + 1)2_{(b + 2)}

c2

and

(b − 2)(b − 1)b(b + 1)

2 + 2b − c₂ (4.2.4)

are both integers.

Proof. Let f and g be as (4.2.1) and (4.2.2). Set ρ=α(1 + b) = c₂ − 1 − b being an

integer. Then both

f + g − (1 − 3b2_{− bρ + b}2_{ρ − b}3_{) =} 2b + 5b2+ 4b3+b4 1 +b + ρ = b(b + 1)2_{(b + 2)} c2 and f − g − (1 − 3b2 _{− bρ + b}2_{ρ + b}3_{) =} 2b − b2− 2b3+b4 −1 − b + ρ = (b − 2)(b − 1)b(b + 1) c2− 2 − 2b

are integers since f, g, b and ρ are integers.

Proposition 4.2.3. Let Γ denote a distance-regular graph which has classical

parame-ters (D, b, α, β), where D ≥ 3. Assume Γ has intersection numbers a₁ = 0 and a₂ = 0.

Then c₂ ≤ b(b + 1).

Proof. Recall c₂ ≤ b2+b + 2 by (4.1.10). First, suppose

c2 =b2+b + 2. (4.2.5)

Then the integral condition (4.2.3) becomes

b2_{+ 3b +} −4b

b2_{+b + 2}. (4.2.6)

Since 0 < −4b < b2 +b + 2 for b ≤ −5, we have −4 ≤ b ≤ −2. For b = −4 or −3, expression (4.2.6) is not an integer. The remaining case b=−2 implies α = −5 by (4.1.6),v = 28 by (4.1.8) and g = 6 by (4.2.2). This contradicts to v ≤ 1₂g(g + 3) [22, Theorem 21.4]. Hence c₂ = b2 +b + 2. Next suppose c₂ = b2 +b + 1. Then (4.2.4) becomes

−b2_{+b + 1 +} 1

b2_{− b − 1}. (4.2.7)

It fails to be an integer since b < −1.

Proof of Theorem 4.0.2:

Chapter 5

3-bounded Property

Let Γ denote a distance-regular graph which has classical parameters (D, b, α, β) and

D ≥ 3. Assume the intersection numbers a1 = 0 and a2 = 0. Note that Γ contains no

parallelograms of any length by Theorem 3.2.1. We have known that Γ is 2-bounded. We shall prove that Γ is 3-bounded in this chapter.

5.1 Weak-geodetically Closed with respect to a

Ver-tex

First we give a definition.

Definition 5.1.1. For any vertex x ∈ X and any subset C ⊆ X, define

[x, C] := {v ∈ X | there exists z ∈ C, such that ∂(x, v) + ∂(v, z) = ∂(x, z)}. Throughout this section, fix two vertices x, y ∈ X with ∂(x, y) = 3. Set

C := {z ∈ Γ3(x) | B(x, y) = B(x, z)}

and

Δ = [x, C]. (5.1.1)

We shall prove Δ is a regular weak-geodetically closed subgraph of diameter 3. Note that the diameter of Δ is at least 3. If D = 3 then C = Γ₃(x) and Δ = Γ is clearly a regular weak-geodetically closed graph. Thereafter we assume D ≥ 4. By referring to Theorem 2.2.4, we shall prove Δ is weak-geodetically closed with respect tox, and the subgraph induced on Δ is regular with valencya₃+c₃.

Lemma 5.1.2. For adjacent vertices z, z _{∈ Γ}

i(x), where i ≤ D, we have B(x, z) =

B(x, z_).

Proof. By symmetry, it suffices to show B(x, z) ⊆ B(x, z). Suppose contradictory there exists w ∈ B(x, z) \ B(x, z). Then ∂(w, z) = i + 1. Note that ∂(w, z) ≤

∂(w, x) + ∂(x, z_{) = 1 +i and ∂(w, z}_{)≥ ∂(w, z) − ∂(z, z}_{) =i. This implies ∂(w, z}_{) =i}

and wxzz forms a parallelogram of length i + 1, a contradiction.

It is known that Γ is 2-bound by Theorem 2.2.7. For two vertices z, s in Γ with

∂(z, s) = 2, let Ω(z, s) denote the regular weak-geodetically closed subgraph containing z, s of diameter 2.

Lemma 5.1.3. Suppose stuzw is a pentagon in Γ, where s, u ∈ Γ3(x) and z ∈ Γ2(x).

Pick v ∈ B(x, u). Then ∂(v, s) = 2.

Proof. Suppose contradictory ∂(v, s) = 2. Note ∂(z, s) = 1, since a₁ = 0. Note that

z, w, s, t, u ∈ Ω(z, s). Then s ∈ Ω(z, s) ∩ Γ2(v) and u ∈ Ω(z, s) ∩ Γ4(v) = ∅. Hence

∂(v, z) = ∂(v, s) + ∂(s, z) = 2 + 2 = 4 by Theorem 2.2.8. A contradiction occurs since ∂(v, x) = 1 and ∂(x, z) = 2.

Lemma 5.1.4. Suppose stuzw is a pentagon in Γ, where s, u ∈ Γ3(x) and z ∈ Γ2(x).

Then B(x, s) = B(x, u).

Proof. Since |B(x, s)| = |B(x, u)| = b₃, it suffices to show B(x, u) ⊆ B(x, s). By Lemma 5.1.3, B(x, u) ⊆ Γ3(s) ∪ Γ4(s). Suppose |B(x, u) ∩ Γ3(s)| = m, |B(x, u) ∩ Γ4(s)| = n. Then m + n = b3. (5.1.2)

By Theorem 2.3.1, r∈B(x,u) Eˆr − r_∈B(u,x) E ˆr _=b3θ1∗− θ4∗ θ∗ 0 − θ3∗ (Eˆx − Eˆu). (5.1.3) Observe B(u, x) ⊆ Γ₃(s), otherwise Ω(u, s) ∩ B(u, x) = ∅ and this leads to ∂(x, s) = 4 by Theorem 2.2.8, which contradicts to∂(x, s) = 3. Taking the inner product of s with both side of (5.1.3) and evaluating the result using (2.3.13), we have

mθ∗ 3+nθ4∗− b3θ∗3 =b3θ ∗ 1 − θ∗4 θ∗ 0 − θ∗3 (θ₃∗− θ∗₂). (5.1.4) Solve (5.1.2) and (5.1.4) to obtain

n = b3(θ ∗ 2 − θ∗3) (θ₃∗− θ∗₄) (θ∗₁− θ₄∗) (θ∗₀− θ₃∗). (5.1.5) Simplifying (5.1.5) using (2.4.5), we have n = b₃ and then m = 0 by (5.1.2). This implies B(x, u) ⊆ B(x, s) as required.

Lemma 5.1.5. Let z, u ∈ Δ. Suppose stuzw is a pentagon in Γ, where z, w ∈ Γ2(x)

and u ∈ Γ₃(x). Then w ∈ Δ.

Proof. Observe Ω(z, s) ∩ Γ₁(x) = ∅ and Ω(z, s) ∩ Γ₄(x) = ∅ by Theorem 2.2.8. Hence

s, t ∈ Γ2(x) ∪ Γ3(x). Observe s ∈ Γ3(x), otherwise w, s ∈ Ω(x, z), and this implies u ∈

Ω(x, z), a contradiction to that the diameter of Ω(x, z) is 2. Hence B(x, s) = B(x, u) by Lemma 5.1.4. Thens ∈ C and w ∈ Δ by construction.

Lemma 5.1.6. The subgraph Δ is weak-geodetically closed with respect to x.

Proof. Clearly C(z, x) ⊆ Δ for any z ∈ Δ. It suffices to show A(z, x) ⊆ Δ for any z ∈ Δ. Suppose z ∈ Δ. We discuss case by case in the following. The case ∂(x, z) = 1

is trivial since a₁ = 0. For the case ∂(x, z) = 3, we have B(x, y) = B(x, z) = B(x, w) for any w ∈ A(z, x) by definition of Δ and Lemma 5.1.2. This implies A(z, x) ⊆ Δ by the construction of Δ. For the remaining case∂(x, z) = 2, fix w ∈ A(z, x) and we shall prove w ∈ Δ. There exists u ∈ C such that z ∈ C(u, x). Observe that ∂(w, u) = 2 since a₁ = 0. Choose s ∈ A(w, u) and t ∈ C(u, s). Then stuzw is a pentagon in Γ. The result comes immediately from Lemma 5.1.5.

5.2 3-bounded Property

Theorem 5.2.1. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β) and D ≥ 3. Assume the intersection numbers a₁ = 0 and a₂ = 0. Then Γ

is 3-bounded.

Proof. By Theorem 2.2.4 and Lemma 5.1.6, it suffices to show that Δ defined in (5.1.1)

is regular with valencya₃+c₃. Clearly from the construction and Lemma 5.1.6,|Γ₁(z)∩ Δ| = a₃+c₃for anyz ∈ C. First we show |Γ₁(x)∩Δ| = a₃+c₃. Note thaty ∈ Δ∩Γ₃(x) by construction of Δ. For any z ∈ C(x, y) ∪ A(x, y),

∂(x, z) + ∂(z, y) ≤ ∂(x, y) + 1.

This impliesz ∈ Δ by Definition 2.2.3 and Lemma 5.1.6. Hence C(x, y) ∪ A(x, y) ⊆ Δ. Suppose B(x, y) ∩ Δ = ∅. Choose t ∈ B(x, y) ∩ Δ. Then there exists y ∈ Γ₃(x) ∩ Δ such that t ∈ C(x, y). Note thatB(x, y) = B(x, y). This leads to a contradiction to

t ∈ C(x, y_{). Hence B(x, y) ∩ Δ = ∅ and Γ}

1(x) ∩ Δ = C(x, y) ∪ A(x, y). Then we have

|Γ1(x) ∩ Δ| = a3+c3.

Since each vertex in Δ appears in a sequence of vertices x = x₀, x₁, x₂, x₃ in Δ, where ∂(x, x_j) =j and ∂(x_j−1, x_j) = 1 for 1≤ j ≤ 3, it suffices to show

|Γ1(xi)∩ Δ| = a3+c3 (5.2.1)

for 1≤ i ≤ 2. For each integer 0 ≤ i ≤ 2, we show

|Γ1(xi)\ Δ| ≤ |Γ1(xi+1)\ Δ|

by counting the number of pairs (s, z) for s ∈ Γ₁(x_i)\Δ, z ∈ Γ₁(x_i+1)\Δ and ∂(s, z) = 2 in two ways. For a fixed z ∈ Γ₁(x_i+1)\ Δ, we have ∂(x, z) = i + 2 by Lemma 5.1.6, so ∂(x_i, z) = 2 and s ∈ A(x_i, z). Hence the number of such pairs (s, z) is at most

|Γ1(xi+1)\ Δ|a2.

On the other hand, we show this number is exactly |Γ₁(x_i)\ Δ|a₂. Fix an s ∈ Γ₁(x_i)\ Δ. Observe ∂(x, s) = i + 1 by Lemma 5.1.6. Observe ∂(x_i+1, s) = 2 since

a1 = 0. Pick any z ∈ A(xi+1, s). We shall prove z ∈ Δ. Suppose contradictory z ∈ Δ

Case 1: i = 0.

Observe ∂(x, z) = 2, ∂(x, s) = 1 and ∂(x, w) = 2. This forces s ∈ Δ by Lemma 5.1.6, a contradiction.

Case 2: i = 1.

Observe ∂(x, z) = 3, otherwise z ∈ Ω(x, x₂) and this implies s ∈ Ω(x, x₂) ⊆ Δ by Lemma 2.2.5 and Lemma 5.1.6, a contradiction. This also implies s ∈ Δ by Definition 2.2.3 and Lemma 5.1.6, a contradiction.

Case 3: i = 2.

Observe ∂(x, z) = 2 or 3. Suppose ∂(x, z) = 2. Then B(x, x₃) =B(x, s) by Lemma 5.1.4 (with x₃ = u, x₂ = t). Hence s ∈ Δ, a contradiction. So z ∈ Γ₃(x). Note

∂(x, w) = 2, 3, otherwise s ∈ Δ by Lemma 5.1.4 and Lemma 5.1.6 respectively. Hence ∂(x, w) = 4. Then by applying Ω = Ω(x2, w) in Theorem 2.2.8 we have ∂(x2, z) = 1, a

contradiction to a₁ = 0.

From the above counting, we have

|Γ1(xi)\ Δ|a2 ≤ |Γ1(xi+1)\ Δ|a2 (5.2.2)

for 0≤ i ≤ 2. Eliminating a₂ from (5.2.2), we find

|Γ1(xi)\ Δ| ≤ |Γ1(xi+1)\ Δ|, (5.2.3)

or equivalently

|Γ1(xi)∩ Δ| ≥ |Γ1(xi+1)∩ Δ| (5.2.4)

for 0≤ i ≤ 2. We have known previously |Γ₁(x₀)∩ Δ| = |Γ₁(x₃)∩ Δ| = a₃+c₃. Hence (5.2.1) follows from (5.2.4).

Remark 5.2.2. The 3-bounded property is enough to obtain the main result of this

Chapter 6

A Constant Bound of c

₂

Let Γ = (X, R) be a distance-regular graph which has classical parameters (D, b, α, β) with D ≥ 3. Assume the intersection numbers a₁ = 0 and a₂ = 0. We shall show that

c2 ≤ 2, and if c2 = 1 then (b, α, β) = (−2, −2, ((−2)D+1− 1)/3).

6.1 Preliminary Lemmas

Let Γ = (X, R) be a distance-regular graph with diameter D ≥ 3 and intersection numbers a_i, c_i, b_i for 0≤ i ≤ D. Assume that Γ is D-bounded. By Theorem 2.2.9, for any x, y ∈ X with ∂(x, y) = t, there exists a unique weak-geodetically closed subgraph Δ(x, y) containing x, y of diameter t, and Δ(x, y) is a distance-regular graph with the intersection numbers

ai(Δ(x, y)) = ai, (6.1.1)

ci(Δ(x, y)) = ci, (6.1.2)

bi(Δ(x, y)) = bi− bt (6.1.3)

for 0 ≤ i ≤ t by Theorem 2.2.4 and (2.1.1). In particular, Δ(x, y) is a clique of size 1 +b₀− b₁ =a₁+ 2 when t = 1.

Lemma 6.1.1. [27, Lemma 4.10] Let Γ denote a distance-regular graph which has

of Γ. Then Δ is a distance-regular graph which has classical parameters (t, b, α, β), where t denotes the diameter of Δ, and β =β + α(D₁−t₁). 

Proof. By Theorem 2.2.4, Δ is distance-regular with intersection numbers ci(Δ) = ci = i 1  1 +α i − 1 1  , ai(Δ) = ai = i 1  a1+α  1− i 1  − i − 1 1  , and bi(Δ) =bi− bt =  D 1  − i 1   β − α i 1  −  D 1  − t 1   β − α t 1  =  t 1  − i 1   β + α D 1  − α t 1  − α i 1 

for 0≤ i ≤ t. Hence Δ has classical parameters (t, b, α, β), whereβ =β +αD₁−α₁t.

Lemma 6.1.2. Let Γ = (X, R) denote a D-bounded distance-regular graph with D ≥ 3.

Let Λ be a weak-geodetically closed subgraph of Γ with diameters, where 0 ≤ s ≤ D−1. Suppose x, y ∈ Λ with ∂(x, y) = s. Then the following (i)-(iii) hold.

(i) For any w ∈ X, let M(w) = {m − {w} | m ⊆ X is a clique of size a₁ + 2

containing w}. Then M(w) is a partition of Γ₁(w) with |M(w)| = b0

a1+ 1

. (ii) If z ∈ B(y, x), then Δ(x, z) ⊇ Λ and Δ(x, z) has diameter s + 1.

(iii) If Δ is a weak-geodetically closed subgraph of Γ with diameter s + 1 and contains

Λ, then Δ = Δ(x, z) for some z ∈ B(y, x).

Proof. Note that Λ = Δ(x, y) by Theorem 2.2.9.

(i) The 1-bounded property implies each edge is contained in a clique of sizea₁+ 2. Since there are b₀ edges in Γ containing a fixed vertex w, we have (i).

(ii) Note that Δ(x, z) ∩ Λ is a weak-geodetically closed subgraph of Γ and y ∈ Δ(x, z) ∩ Λ since y ∈ C(z, x). This implies the diameter of Δ(x, z) ∩ Λ is s and we have Δ(x, z) ∩ Λ = Λ by Theorem 2.2.9. Hence Δ(x, z) ⊇ Λ. The diameter of Δ(x, z) is s + 1 since ∂(x, z) = s + 1.

(iii) Suppose that Δ is a weak-geodetically closed subgraph of Γ with diameters+1 and contains Λ. Note thatx, y ∈ Δ. Choose z ∈ Δ and z ∈ B(y, x). Then Δ = Δ(x, z) by (ii).

Lemma 6.1.3. Let Γ denote a D-bounded distance-regular graph with D ≥ 3. Let Λ,

Λ be two weak-geodetically closed subgraphs of Γ with diameter s, s + 3 respectively and Λ ⊆ Λ, where 0 ≤ s ≤ D − 3. Let P and B be the sets of weak-geodetically closed subgraphs of Λ which contain Λ, with diameter s + 1 and s + 2 respectively. Let

I = {(p, B) | p ∈ P, B ∈ B, and p ⊆ B}. Then (P, B, I) is a 2-(v, κ, 1) design, where

v = b_bs− bs+3

s− bs+1,

κ = b_bs− bs+2

s− bs+1,

and the replication number

r = b_bs+1− bs+3

s+1− bs+2.

Proof. Let x, y ∈ Λ with ∂(x, y) = s. Counting in two ways the number of pairs (, Ω),

where ⊆ Λ is a clique of sizea₁+ 2 containing y with   Λ, and Ω ∈ P with  ⊆ Ω. By Lemma 6.1.2, bs(Λ) (a₁+ 1) × 1 = |P| × bs(Ω) (a₁+ 1). (6.1.4) Simplifying (6.1.4) by (6.1.3) we have |P| = bs(Λ) bs(Ω) = bs− bs+3 bs− bs+1.

Fix Δ∈ B. Using the same technique as above, there are

bs− bs+2

bs− bs+1

distinct elements ofP incident with Δ. Note that the number is independent of choice of Δ.

Fix any distinct Ω, Ω∈ P. Pick z ∈ B(y, x) ∩ Ω. Then Ω = Δ(x, z) by Theorem 6.1.2. Pick w ∈ Ω₁(x) − Ω. Note that w ∈ B(x, z). Then Δ(w, z) ∈ B containing Ω and Ω. Suppose that Δ ∈ B is another block incident with Ω and Ω. Observe

that Ω, Ω⊆ Δ(w, z) ∩ Δ ⊆ Δ(w, z). This implies that the diameter of Δ(w, z) ∩ Δ is s + 1. We have Ω = Δ(w, z) ∩ Δ = Ω by Theorem 2.2.9, which contradicts to Ω = Ω.

The replication numberr = bs+1− bs+3

bs+1− bs+2 can be computed by the same argument of

counting of |P|.

6.2 An Application of 3-bounded Property

Let Γ = (X, R) be a distance-regular graph which has classical parameters (D, b, α, β) with D ≥ 3. Suppose the intersection numbers a₁ = 0 and a₂ = 0. Then α < 0 and

b < −1 by Lemma 3.1.2. Now we are ready to prove the main theorem of this chapter.

Theorem 6.2.1. Let Γ denote a distance-regular graph which has classical parameters

(D, b, α, β) and D ≥ 3. Assume the intersection numbers a₁ = 0 and a₂ = 0. Then

c2 ≤ 2.

Proof. It was shown in Theorem 5.2.1 that Γ is 3-bounded. Fix a vertex x ∈ X and

a weak-geodetically closed subgraph Δ containing x of diameter 3. By (6.1.1)-(6.1.3), and Lemma 6.1.1 we find a₁(Δ) = 0 and Δ has classical parameters (3, b, α, β) where

β _{=β + α(}D 1  −3 1  ) . Note that β _{= 1 +α − α(}3 1  ) = 1− αb − αb2 (6.2.1)

by applyinga₁(Δ) = 0 to (2.4.4). LetP denote the set of all maximal cliques containing

x in Δ, and B be the set of all weak-geodetically closed subgraphs of diameter 2

containing x in Δ. Let I = {(p, B) | p ∈ P, B ∈ B, and p ⊆ B}. Then (P,B, I) is a 2-(v, κ, 1) design by Lemma 6.1.3, where

κ = b0(Δ)− b2(Δ) b0(Δ)− b1(Δ)

= (1 +b)(1 − αb) (6.2.2)

and the replication number

r = _b b1(Δ)

1(Δ)− b2(Δ)

= b(1 + b)(1 − αb − αb

2_{− α)}