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Triangle-free Distance-regular Graphs with Pentagons

This presentation by Yeh-jong Pan explores the properties and classification of distance-regular graphs with pentagons and classical parameters, including results on intersection numbers and combinatorial characterizations. The talk covers key theorems and achievements in the field from various researchers since the 1970s.

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Triangle-free Distance-regular Graphs with Pentagons

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  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 c2 • A constant bound of c2 • Summary

  3. 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

  4. 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 anda2≠0 : our object

  5. Introduction Let Γ be a distance-regular graph with Q-polynomial property. Assume the diameter and the intersection numbersa1= 0anda2≠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 numberc2is either1or2, and ifc2=1then (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 :byY. Egawa,A.NeumaierandP. Terwilliger • b<-1 : • a1≠0 : by C. WengandH. Suzuki • a1= 0 anda2≠0 : our object • b>1 : ??

  7. Distance-regular Graph • A graphΓ=(X, R)is said to bedistance-regular whenever for all integers , and all vertices with , the number is independent of x, y. • The constant is called theintersection numberof Γ.

  8. Strongly Regular Graph • A strongly regular graph is adistance-regular graphwithdiameter 2.

  9. Intersection Numbers bi, ci,ai Let Γ=(X, R) be a distance-regular graph. For twovertices with . • Set

  10. Intersection Numbers(cont.) • Set • Note that k :=b0 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. 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

  14. 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= -2andβ = -3.

  15. Examples (cont.) • Example : Hermitian forms graphHer2(D). • Classical parameters(D, b, α,β) withb=-2, α=-3andβ=-((-2)D+1). • a1 =0, a2 =3, andc2 = 2.

  16. 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) withD=2, b=-3, α=-2andβ=-5. • Example : Witt graphM23. • 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) withD=3, b=-2, α=-2andβ=5.

  17. Classical Parameters(cont.) • Lemma 3.1.3 : Let Γ denote a distance-regular graph with classical parameters (D, b, α,β) . Suppose intersection numbersa1= 0,a2≠0. Then α<0 and b<-1.

  18. Parallelogram of Length i • Definition : LetΓbe a distance-regular graph. By a parallelogram of length i, we mean a 4-tuple xyzwconsisting of vertices of X such that

  19. p21 Classical Parameters(Combinatorial) • Theorem 3.2.1 : Let Γ be a distance-regular graph with diameter and intersection numbersa1= 0,a2≠0. Then the following (i)-(iii) are equivalent. (i) Γ isQ-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.

  20. An Upper Bound of c2 • Theorem : Let Γ be a distance-regular graph with diameter and intersection numbersa1= 0,a2≠0. SupposeΓ has classical parameters (D, b,α,β). Then the following (i), (ii) hold. (i) Each of is an integer. (ii)

  21. 3-bounded Property • Theorem : Let Γ be a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbersa1= 0,a2≠0. ThenΓ is

  22. A Constant Bound ofc2 • Theorem 6.2.1 : Let Γ denote a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbersa1= 0,a2≠0. Then c2 is either 1 or 2.

  23. The Casec2=1 • Theorem 6.2.2 : Let Γ denote a distance-regular graph with classical parameters (D, b,α,β) and Assume intersection numbersa1= 0,a2≠0, andc2=1.Then

  24. Summary

  25. Future Work • Determine (b, α, β) when c2 = 2. Hiraki : b = -2 or -3 ? • Determine graphs for kwown b when c2 = 1, 2. • The caseb>1.

  26. Thank you very much !

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