Classification of the family AT4(qs,q,q) of antipodal tight graphs

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• 作者：Aleksandar Juriš ; ić ; ^a ; ^{aj@fri.uni-lj.si} ; Jack Koolen^b ; ^{koolen@postech.ac.kr}
• 关键词：Distance-regular graphs ; Antipodal graphs ; Tight graphs ; Locally strongly regular ; μ ; -graphs ; AT4 family
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2011
• 出版时间：April 2011
• 年：2011
• 卷：118
• 期：3
• 页码：842-852
• 全文大小：225 K

文摘

Let Γ be an antipodal distance-regular graph with diameter 4 and eigenvalues θ₀>θ₁>θ₂>θ₃>θ₄. Then its Krein parameter vanishes precisely when Γ is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when Γ is locally strongly regular with nontrivial eigenvalues p:=θ₂ and −q:=θ₃. When this is the case, the intersection parameters of Γ can be parameterized by p, q and the size of the antipodal classes r of Γ, hence we denote Γ by AT4(p,q,r).

Jurišić conjectured that the AT4(p,q,r) family is finite and that, aside from the Conway–Smith graph, the Soicher2 graph and the graph, all graphs in this family have parameters belonging to one of the following four subfamilies: In this paper we settle the first subfamily. Specifically, we show that for a graph AT4(qs,q,q) there are exactly five possibilities for the pair (s,q), with an example for each: the Johnson graph J(8,4) for (1,2), the halved 8-cube for (2,2), the graph for (1,3), the Meixner2 graph for (2,4) and the 3.O₇(3) graph for (3,3). The fact that the μ-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper.