1 triangle-free distance-regular graphs with pentagons speaker : yeh-jong pan advisor : chih-wen...
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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 !