1 Triangle-free Distance-regular Graphs with Pentagons Speaker : Yeh-jong Pan Advisor : Chih-wen Weng

2 Outline Introduction Preliminaries A combinatorial characterization An upper bound of c 2 A constant bound of c 2 Summary

3 Introduction Distance-regular graph: Biggs introduced as a combinatorial generalization of distance-transitive graphs Desarte studied P-polynomial schemes motivated by problems of coding theory in his thesis Leonard derived recurrsive formulae of the intersection numbers of Q-polynomial DRG Eiichi Bannai and Tatsuro Ito classified Q-polynomial DRG

4 Introduction Distance-regular graph: Brouwer, Cohen, and Neumaier invented the term classical parameters (D, b, α, β) 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 ? » a 1 ≠0 : by C. Weng » a 1 = 0 and a 2 ≠0 : our object

5 Introduction Let Γ be a distance-regular graph with Q-polynomial property. Assume the diameter and the intersection numbers a 1 = 0 and a 2 ≠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 c 2 is either 1 or 2, and if c 2 =1 then (b, α, β) = (-2, -2, ((-2) D+1 -1)/3).

6 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 : » a 1 ≠0 : by C. Weng and H. Suzuki » a 1 = 0 and a 2 ≠0 : our object b>1 : ??

7 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.

8 Strongly Regular Graph A strongly regular graph is a distance-regular graph with diameter 2.

9 Intersection Numbers b i, c i, a i Let Γ=(X, R) be a distance-regular graph. For two vertices with. Set

10 Intersection Numbers (cont.) Set Note that k := b 0 is the valency of Γ and

11 Examples Example : A pentagon. Diameter D=2.

12 Examples (cont.) Example : The Petersen graph. Diameter D=2.

13 Q-polynomial Property Let Γ= (X, R) be a distance-regular graph. M : commutative algebra spanned by distance matrices M has second basis If, then are called the dual eigenvalues associated with E. p14

14 Q-polynomial Property Theorem (Terwilliger 1995): Let Γ= (X, R) be Q-polynomial with respect to a primitive idempotent E, and are the corresponding dual eigenvalues. Then for all integers and for all such that p14

15 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

16 Classical Parameters (cont.)

17 Examples Example : Petersen graph. Diameter D=2. a 1 = 0, a 2 = 2, c 1 = c 2 = 1, b 0 = 3, and b 1 = 2. Classical parameters (D, b, α,β) D=2, b= -2, α= -2 and β = -3.

18 Examples (cont.) Example : Hermitian forms graph Her 2 (D). Let U be a D-dimension vector space over GF(4). Let H be the D 2 -dimensional vector space over GF(2) of the Hermitian forms on U. Set X=H, and if and only if rk(x-y)=1 for all Then Γ= ( X, R ) is a distance-regular graph with diameter D.

19 Examples (cont.) Example : Hermitian forms graph Her 2 (D). Classical parameters (D, b, α,β) with b=-2, α=-3 and β=-((-2) D +1). a 1 =0, a 2 =3, and c 2 = 2.

20 Examples (cont.) Example : Gewirtz graph. Delete a point in 3-(22, 6, 1) design → 2-(21, 6, 4) design. Two blocks are adjcent if they are disjoint. Diameter D = 2. a 1 =0, a 2 =8, c 1 =1, c 2 =2, b 0 =10, and b 1 =9. (Unique) Classical parameters (D, b, α,β) with D=2, b=-3, α=- 2 and β=-5.

21 Examples (cont.) Example : Witt graph M 23. Delete a point in 5-(24, 8, 1) design → 4-(23, 8, 4) design. Two blocks are adjcent if they are disjoint. Diameter D = 3. a 1 =0, a 2 =2, a 3 =6, c 1 = c 2 = 1, c 3 = 9, b 0 =15, b 1 =14, and b 2 = 12. (Unique) Classical parameters (D, b, α,β) with D=3, b=-2, α=- 2 and β=5.

22 Examples (cont.) Example : Gewirtz graph. a 1 =0, a 2 =8, c 1 =1, c 2 =2, b 0 =10, and b 1 =9. (Unique) Classical parameters (D, b, α,β) with D=2, b=-3, α=- 2 and β=-5. Example : Witt graph M 23. a 1 =0, a 2 =2, a 3 =6, c 1 = c 2 = 1, c 3 = 9, b 0 =15, b 1 =14, and b 2 = 12. (Unique) Classical parameters (D, b, α,β) with D=3, b=-2, α=- 2 and β=5.

23 Classical Parameters (cont.) Lemma : Let Γ denote a distance-regular graph with classical parameters (D, b, α,β). Suppose intersection numbers a 1 = 0, a 2 ≠0. Then α<0 and b< - 1.

24 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

25 Classical Parameters (Combinatorial) Theorem : Let Γ be a distance-regular graph with diameter and intersection numbers a 1 = 0, a 2 ≠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

26 Weak-geodetic Closed Definition : Let Γ= (X, R) be a distance-regular graph. A subset is weak-geodetically closed whenever for any and,

27 Weak-geodetically Closed (cont.) Lemma : (Weng 1998) Ω is weak-geodetically closed if and only if and for any

28 Weak-geodetically Closed Subgraph Theorem: (Weng 1998): Let Γ be a distance-regular graph. Supose Ω is a regular weak-geodetically closed subgraph with diameter d of Γ. Then Ω is distance-regular with intersection numbers

29 i-bounded Definition : A graph Γ=(X, R) is said to be i- bounded whenever for all with there is a regular weak-geodetically closed subgraph of diameter which contains x, y.

30 2-bounded Theorem : (Weng 1998) (Suzuki 1996): Let Γ be a distance-regular graph with diameter Suppose a 1 = 0 and a 2 ≠0 and Γ contains no parallelograms of length 3. Then Γ is 2-bounded.

31 An Upper Bound of c 2 Theorem : Let Γ be a distance-regular graph with diameter and intersection numbers a 1 = 0, a 2 ≠0. Suppose Γ has classical parameters (D, b,α,β). Then the following (i), (ii) hold. (i) Each of is an integer. (ii)

32 3-bounded Property Theorem : Let Γ be a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbers a 1 = 0, a 2 ≠0. Then Γ is

33 A Constant Bound of c 2 Theorem : Let Γ denote a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbers a 1 = 0, a 2 ≠0. Then c 2 is either 1 or 2.

34 The Case c 2 =1 Theorem : Let Γ denote a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbers a 1 = 0, a 2 ≠0, and c 2 =1. Then

35 Summary namea1a1 a2a2 c2c2 Dbαβ Petersen graph Witt graph M ??021 D≥4 -2 Hermitian forms graph Her 2 (D). 032D-2-3-((-2) D +1) Gewirtz graph ??082 D≥3 -3-2

36 Future Work Determine (b, α, β) when c 2 = 2. Hiraki : b = -2 or -3 ? Determine graphs for kwown b when c 2 = 1, 2. The case b>1.

37 Thank you very much !