A note on triangle-free distance-regular graphs with a(2) not equal 0

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Journal of Combinatorial Theory,

Series B

www.elsevier.com/locate/jctb

Note

A note on triangle-free distance-regular graphs with a

₂

=

0 ✩

Yeh-jong Pan, Chih-wen Weng

Department of Applied Mathematics, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC

a r t i c l e i n f o a b s t r a c t

Article history:

Received 12 September 2007 Available online 9 August 2008

Keywords:

Distance-regular graphs Classical parameters

Let Γ denote a distance-regular graph with classical parameters

(D,b,

α

, β)and D3. Assume the intersection numbers a1=0

and a2=0. We show that the intersection number c2 is either 1

or 2, and if c2=1, then(b,

α

1. Introduction

Brouwer, Cohen, and Neumaier invented the term classical parameters

(

D

,

b

,

α

, β)

to describe the intersection numbers of most known families of distance-regular graphs [3, p. ix, p. 193]. All classical parameters

(

D

,

b

,

α

, β)

of distance-regular graphs with b

=

1 are classiﬁed by Y. Egawa, A. Neumaier and P. Terwilliger in a sequence of papers (see [3, p. 195] for a detailed description). For b

<

−

1, the classiﬁcation is done in the case c2

=

1, a2

>

a1

>

1, D

4 [14], and in the case a1

=

0

,

c2

>

1

,

D

4 [15]. For the case a1

=

0, Miklaviˇc shows the graph is 1-homogeneous [6]. A. Juriši ´c, J. Koolen, and Š. Miklaviˇc study distance-regular graphs in the cases a1

=

0, a2

=

0

,

without the assumption of classical parameters, but instead with an additional assumption that the graphs have an eigen-value multiplicity equal to the valency, and they almost classify such graphs [5]. In this note, we study distance-regular graphs with classical parameters

(

D

,

b

,

α

, β)

, D

3, a1

=

0, and a2

=

0 (hence b

<

−

1 by [7, Lemma 3.3]). We prove c2

2, and if c2

=

1 then

(

b

,

α

, β)

= (−

2

,

−

2

, ((

−

2

)

D+1

−

1

)/

3

)

. Note that Witt graph M23is the only known example of such graph with c2

=

1 and the class of Her-mitian forms graphs Her2

(

D

)

is the only known family satisfying the conditions with c2

=

2.

First we review some deﬁnitions and basic concepts concerning distance-regular graphs. See Ban-nai and Ito [1] or Terwilliger [10,11] for more background information.

Let

Γ

= (

X

,

R

)

denote a ﬁnite undirected, connected graph without loops or multiple edges with vertex set X , edge set R, distance function

∂

, and diameter D

:=

max

{∂(

x

,

y

)

|

x

,

y

∈

X

}

.

✩ _{Research partially supported by the NSC grant 96-2628-M-009-015 of Taiwan, ROC.}

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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

Γ

1

(

x

)

. The graph

Γ

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

A graph

Γ

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_{i j}

=

Γ

i

(

x

)

∩ Γ

j

(

y

)

is independent of x

,

y. The constants ph_{i j} are known as the intersection numbers of

Γ

.

For convenience, set ci

:=

pi_{1 i}₋₁for 1

i

D, ai

:=

pi_{1 i} for 0

i

D, bi

:=

pi_{1 i}₊₁ for 0

i

D

−

1,

ki

:=

p0i i for 0

i

D, and set bD

:=

0, c0

:=

0, k

:=

b0. Note that k is the valency of

Γ

. It follows immediately from the deﬁnition of ph_{i j} that bi

=

0 for 0

i

D

−

1 and ci

=

0 for 1

i

D. Moreover

k

=

ai

+

bi

+

ci for 0

i

D

,

(1.1) and ki

=

b0

· · ·

bi−1 c1

· · ·

ci for 1

i

D

.

(1.2)

A strongly regular graph is a distance-regular graph with diameter 2. We quote a lemma about strongly regular graphs which will be used in the next section.

Lemma 1.1. (See [2, Theorem 19, p. 276].) Let

Ω

be a strongly regular graph with valency k which has k2

+

1 vertices. Then k

∈ {

2

,

3

,

7

,

57

}

.

Let

Γ

be a distance-regular graph with diameter D.

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

,

(1.3) bi

=

D 1

−

i 1

β

−

α

i 1

for 0

i

D

,

(1.4) where

i 1

:=

1

+

b

+

b2

+ · · · +

bi−1

.

(1.5) Suppose

Γ

has classical parameters

(

D

,

b

,

α

, β)

. Combining (1.3)–(1.5) with (1.1), we have

ai

=

i 1

β

−

1

+

α

D 1

−

i 1

−

i

−

1 1

for 0

i

D

.

(1.6)

A subgraph

Δ

of

Γ

is called weak-geodetically closed whenever for any vertices x

,

y

∈ Δ

with

∂(

x

,

y

)

=

i

,

Γ

1

(

x

)

∩

Γ

i

(

y

)

∪ Γ

i−1

(

y

)

⊆ Δ.

Weak-geodetically closed subgraphs are called strongly closed subgraphs in [9]. Let t be a positive in-teger.

Γ

is said t-bounded, whenever for any integer 0

i

t

,

and any two vertices x

,

y

∈

X with

∂(

x

,

y

)

=

i, x

,

y are contained in a regular weak-geodetically closed subgraph

Δ(

x

,

y

)

of diameter i. Furthermore by [13, Theorem 4.6],

Δ(

x

,

y

)

is a distance-regular graph with intersection numbers

aj

Δ(

x

,

y

)

=

aj

(Γ ),

(1.7) cj

Δ(

x

,

y

)

=

cj

(Γ ),

(1.8) bj

Δ(

x

,

y

)

=

bj

(Γ )

−

bi

(Γ )

(1.9)

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Lemma 1.2. (See [15, Lemma 4.10].) Let

Γ

denote a distance-regular graph with classical parameters

(

D

,

b

,

α

, β)

. Let

Δ

be a regular weak-geodetically closed subgraph of

Γ

. Then

Δ

is distance-regular with classical parameters

(

t

,

b

,

α

, β

)

, where t denotes the diameter of

Δ

, and

β

= β +

α

(

D₁

−

₁t

)

.

Lemma 1.3. (See [14, Lemma 4.1(iii)].) Let

Γ

(

D

,

b

,

α

, β)

and D

3. Assume

Γ

is D-bounded. Then

b3D−3i−4

(

b

+

1

−

c2

)

α

bD

+

1

+ β(

b

−

1

)

0 (1.10)

for 1

i

D

−

2.

Lemma 1.4. (See [7, Lemma 3.3] and [8, Theorem 1.3].) Let

Γ

denote a distance-regular graph with classical parameters

(

D

,

b

,

α

, β)

and D

3. Assume the intersection numbers a1

=

0 and a2

=

0. Then the following (i) and (ii) hold.

(i)

α

<

0 and b

<

−

1. (ii)

Γ

is 3-bounded.

2. Main result

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 1.4. Now we are ready to prove our main theorem.

Theorem 2.1. Let

Γ

(

D

,

b

,

α

, β)

and D

3. Assume the intersection numbers a1

=

0 and a2

=

0. Then c2

2.

Proof.

Γ

is 3-bounded by Lemma 1.4. Let

Δ

be a weak-geodetically closed subgraph of

Γ

with di-ameter 3. Then

Δ

has classical parameters

(

3

,

b

,

α

, β

)

, where

β

= β +

α

(

D₁

−

3₁

)

by Lemma 1.2. Applying a1

=

0 to (1.6) and by (1.5) we have

β

=

1

+

α

−

α

3 1

=

1

−

α

b

−

α

b2

.

(2.1)

Applying Lemma 1.3 to

Γ

= Δ

with D

=

3 and i

=

1, we have

b2

(

b

+

1

−

c2

)

α

b3

+

1

+ β

(

b

−

1

)

0

.

(2.2)

Note that b

+

1

−

c2

<

0 since b

<

−

1. Combining this with inequality (2.2) we ﬁnd

α

b3

+

1

+ β

(

b

−

1

)

0

.

(2.3)

Evaluating (2.3) using (2.1) we ﬁnd

α

b

+

α

+

b

−

1

0

.

(2.4)

Note that

α

b

+

α

+

b

−

1

=

c2

−

2 by (1.3) and hence c2

2.

2

For the case c2

=

1, we have the following result.

Theorem 2.2. Let

Γ

(

D

,

b

,

α

, β)

and D

3. As-sume the intersection numbers a1

=

0, a2

=

0, and c2

=

1. Then

(

b

,

α

, β)

= (−

2

,

−

2

, ((

−

2

)

D+1

−

1

)/

3

)

.

Proof. Combining a1

=

0 and c2

=

1 with (1.6), (1.3), and (1.5) we have

α

=

−

b

1

+

b

,

(2.5)

β

=

b

D+1

₋

₁

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Let

Ω

⊂ Δ

be two weak-geodetically closed subgraphs of

Γ

with diameter 2 and 3 respectively. Note that

Ω

is a strongly regular graph with a1

(Ω)

=

0, c2

(Ω)

=

1 by (1.7) and (1.8). Combining this with (1.1) and (1.2) we have

|Ω| =

1

+

k1

(Ω)

+

k2

(Ω)

=

1

+

b0

(Ω)

2

.

Hence

b0

(Ω)

=

2

,

3

,

7

,

57 (2.7)

by Lemma 1.1. Note that

b0

(Ω)

=

b0

−

b2

=

1

+

b

+

b2 (2.8)

by (1.9), (1.4), (2.5), and (2.6). Solving (2.7) with (2.8) for integer b

<

−

1 we have b

= −

2,

−

3, or

−

8. Note that

k3

(Δ)

=

(

b0

−

b3

)(

b1

−

b3

)(

b2

−

b3

)

c1c2c3 (2.9)

by (1.2), (1.8), and (1.9). Evaluating (2.9) using (1.3)–(1.5), (2.5), and (2.6) we have k3

(Δ)

=

b3

₍

_b2

₊

₁

₎₍

_b2

₊

_b

₊

₁

₎₍

_b3

₊

_b2

₊

_2b

₊

₁

₎

1

−

b

.

(2.10)

The number k3

(Δ)

is not an integer when b

= −

3 or

−

8. Hence b

= −

2 and

α

= −

2,

β

=

((

−

2

)

D+1

−

1

)/

3 by (2.5) and (2.6) respectively.

2

Example 2.3. [4] Hermitian forms graph Her2

(

D

)

is a distance-regular graph with classical param-eters (D

,

b

,

α

, β

) with b

= −

2,

α

= −

3, and

β

= −(−

2

)

D

−

1, which has a1

=

0, a2

=

0, and c2

=

(

1

+

α

)(

b

+

1

)

=

2. This is the only known class of examples that satisﬁes the assumptions of Theo-rem 2.1 with c2

=

2.

Example 2.4. (See [12, p. 237].) Gewirtz graph is a distance-regular graph with diameter 2 and inter-section numbers a1

=

0, a2

=

8, k

=

10, which can be written as classical parameters (D

,

b

,

α

, β

) with D

=

2, b

= −

3,

α

= −

2, and

β

= −

5, so we have c2

= (

b

+

1

)

2

/

2

=

2. It is still open if there exists a class of distance-regular graphs with classical parameters

(

D

,

−

3

,

−

2

, (

−

1

− (−

3

)

D

_)/

₂

₎

_{for D}

_3.

Example 2.5. (See [3, Table 6.1].) Witt graph M23is a distance-regular graph with classical parameters (D

,

b

,

α

, β

) with D

=

3, b

= −

2,

α

= −

2, and

β

=

5, which has a1

=

0, a2

=

2, and c2

=

1. This is the only known example that satisﬁes the assumptions of Theorem 2.1 with c2

=

1.

Acknowledgment

Theorem 2.2 is an anonymous referee’s idea. References

[1] E. Bannai, T. Ito, Algebraic Combinatorics I: Association Schemes, Benjamin/Cummings, Menlo Park, 1984. [2] B. Bollobás, Modern Graph Theory, Springer-Verlag, New York, 1998.

[3] A.E. Brouwer, A.M. Cohen, A. Neumaier, Distance-Regular Graphs, Springer-Verlag, Berlin, 1989.

[4] A.A. Ivanov, S.V. Shpectorov, Characterization of the association schemes of Hermitian forms over G F(22₎_{, Geom.}

Dedi-cata 30 (1989) 23–33.

[5] A. Juriši ´c, J. Koolen, Š. Miklaviˇc, Triangle- and pentagon-free distance-regular graphs with an eigenvalue multiplicity equal to the valency, J. Combin. Theory Ser. B 94 (2) (2005) 245–258.

[6] Š. Miklaviˇc, Q -polynomial distance-regular graphs with a1=0, European J. Combin. 25 (7) (2004) 911–920.

[7] Y. Pan, M. Lu, C. Weng, Triangle-free distance-regular graphs, J. Algebraic Combin. 27 (2008) 23–34.

[8] Y. Pan, C. Weng, 3-bounded property in a triangle-free distance-regular graph, European J. Combin. 29 (2008) 1634–1642. [9] H. Suzuki, On strongly closed subgraphs of highly regular graphs, European J. Combin. 16 (1995) 197–220.

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[11] P. Terwilliger, A new inequality for distance-regular graphs, Discrete Math. 137 (1995) 319–332. [12] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, Cambridge, 1992. [13] C. Weng, Weak-geodetically closed subgraphs in distance-regular graphs, Graphs Combin. 14 (1998) 275–304. [14] C. Weng, D-bounded distance-regular graphs, European J. Combin. 18 (1997) 211–229.