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Triangle-free Distance-regular Graphs with Pentagons

Speaker : Yeh-jong Pan

Advisor : Chih-wen Weng

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Outline

Introduction

Preliminaries

A combinatorial characterization

An upper bound of c2

A constant bound of c2

Summary

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Introduction

Distance-regular graph: Biggs introduced as a combinatorial generalization of distance-transitive graphs. -----1970

Desarte studied P-polynomial schemes motivated by problems of coding theory in his thesis. -----1973 Leonard derived recurrsive formulae of the intersection numbers of Q-polynomial DRG. -----1982

Eiichi Bannai and Tatsuro Ito classified Q-polynomial DRG. -----1984

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Introduction

Distance-regular graph: Brouwer, Cohen, and Neumaier invented the term classical parameters (D, b, α, β). -----1989 The class of DRGs which have classical parameters is a special case of DRGs with the Q-polynomial property.

The converse is not true. Ex: n-gon The necessary and sufficient condition ?

» a1≠0 : by C. Weng

» a1= 0 and a2≠0 : our object

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Introduction

Let Γ be a distance-regular graph with Q-polynomial property. Assume the diameter and the intersection numbers a1= 0 and a2≠0.

We give a necessary and sufficient condition for Γ to have classical parameters (D, b, α, β).

When Γ satisfies this condition, we show 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).

3D

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Introduction

To classify distance-regular graphs with classical parameters (D, b, α, β).

b =1 : by Y. Egawa, A.Neumaier and P. Terwilliger

b<-1 :

» a1≠0 : by C. Weng and H. Suzuki

» a1= 0 and a2≠0 : our object

b>1 : ??

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Distance-regular Graph

A graph Γ=(X, R) is said to be distance-regular

whenever for all integers , and all

vertices with , the number

is independent of x, y.

The constant is called the intersection

number of Γ.

Xyx , hyx ),(

Djih ,,0

)()( yxp jih

ji h

jip

jih

jip

hx y

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Strongly Regular Graph

A strongly regular graph is a distance-regular

graph with diameter 2.

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Intersection Numbers bi, ci, ai

Let Γ=(X, R) be a distance-regular graph. For two

vertices with . Set

Xyx , iyx ),(

,)()(),( 11 yxyxB i

)()(),( 11 yxyxC i )()(),( 1 yxyxA i

y x

),( yxC1

),( yxB

),( yxA

11i 1i

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Intersection Numbers (cont.)

Set

Note that k := b0 is the valency of Γ and

|),(|: yxAai ).0( Di

|),(|: yxBbi ),10( Di

|),(|: yxCci ),1( Di

iii cbak ).0( Di

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Examples

Example : A pentagon. Diameter D=2.

,1,2 10 bb

,11 c

,01 a ,12 a

.12 c

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Examples (cont.)

Example : The Petersen graph. Diameter D=2.

,30 b

,11 c

,22 a,01 a

,21 b

.12 c

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Classical Parameters

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

where

1

i

1

1i,0for Di ic 1( )

11

iD ,0for Di ib

1

i( ))(

.1: 12 ibbb

1

i

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Examples

Example : Petersen graph. Diameter D=2. a1 = 0, a2 = 2, c1 = c2 = 1,

b0 = 3, and b1 = 2.

Classical parameters (D, b, α,β)

D=2, b= -2, α= -2 and β = -3.

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Examples (cont.)

Example : Hermitian forms graph Her2(D).

Classical parameters (D, b, α,β) with b=-2, α=-3 and

β=-((-2)D+1).

a1 =0, a2 =3, and c2 = 2.

ic ,3))1(2(2 1 iii

ib ,322 22 iD

ia .312)1(2 112 iii

(Unique)

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Examples (cont.)

Example : Gewirtz graph. a1 =0, a2 =8, c1 =1, c2 =2, b0 =10, and b1 =9. (Unique)

Classical parameters (D, b, α,β) with D=2, b=-3, α=-2 and β=-5.

Example : Witt graph M23. a1 =0, a2 =2, a3 =6, c1 = c2 = 1, c3 = 9, b0 =15, b1 =14, and

b2 = 12. (Unique)

Classical parameters (D, b, α,β) with D=3, b=-2, α=-2 and β=5.

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Classical Parameters (cont.)

Lemma 3.1.3 : Let Γ denote a distance-regular g

raph with classical parameters (D, b, α,β) . Suppose

intersection numbers a1= 0, a2≠0. Then α<0 and b

<-1.

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Parallelogram of Length i

Definition : Let Γbe a distance-regular graph. By a parallelogram of length i, we mean a 4-tuple xyzw consisting of vertices of X such that

,1),(),( wzyx ,1),(),(),( izywywx

.),( izx

11i

x

y z

w

1i

1i

1i

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Classical Parameters (Combinatorial)

Theorem 3.2.1 : Let Γ be a distance-regular graph with diameter 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

(iii) Γ has classical parameters (D, b, α,β) for some real constants b, α, β with b<-1.

p21

3D

.1 Di

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An Upper Bound of c2

Theorem : Let Γ be a distance-regular graph with diameter and intersection numbers a1= 0, a2≠0. Sup

pose Γ has classical parameters (D, b,α,β). Then the following (i), (ii) hold.

(i) Each of

is an integer.

(ii)

3D

22

2

22

)1()1)(2(,

)2()1(

cb

bbbb

c

bbb

).1(2 bbc

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3-bounded Property

Theorem : Let Γ be a distance-regular graph with classical parameters (D, b,α,β) and

Assume intersection numbers a1= 0, a2≠0.

Then Γ is

.3D

.bounded-3

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A Constant Bound of c2

Theorem 6.2.1 : Let Γ denote a distance-regular

graph with classical parameters (D, b,α,β) and

Assume intersection numbers a1= 0, a2≠0. T

hen c2 is either 1 or 2.

.3D

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The Case c2=1

Theorem 6.2.2 : Let Γ denote a distance-regular

graph with classical parameters (D, b,α,β) and

Assume intersection numbers a1= 0, a2≠0, a

nd c2=1. Then

.3D

).,2,2(),,( 31)2( 1

D

b

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Summary

name a1 a2 c2 D b α β

Petersen graph 0 2 1 2 -2 -2 -3

Witt graph M23 0 2 1 3 -2 -2 5

?? 0 2 1 D≥4 -2 -2

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

Gewirtz graph 0 8 2 2 -3 -2 -5

?? 0 8 2 D≥3 -3 -2

31)2( 1 D

2)3(1 D

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Future Work

Determine (b, α, β) when c2 = 2.

Hiraki : b = -2 or -3 ? Determine graphs for kwown b when c2 = 1, 2.

The case b>1.

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Thank you

very much !