Distance-regular graphs with complete multipartite 渭 -graphs and AT4 family

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• 作者：Aleksandar Juri拧i膰 and Jack Koolen
• 关键词：Distance ; regular graphs ; Antipodal ; Tight ; Locally strongly regular ; μ ; graphs ; AT4 family
• 刊名：Journal of Algebraic Combinatorics
• 出版时间：June, 2007
• 出版年：2007
• 期刊代码：75_09259899
• 类别：bio
• 卷：25
• 期：4
• 页码：459-471
• 数据来源：sp

摘要

Let Γ be an antipodal distance-regular graph of diameter 4, with eigenvalues $\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$\theta_0>\theta_1>\theta_2>\theta_3>\theta_4. Then its Krein parameter q₁₁4q_{11}^4 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:=q₂p:=\theta_2 and -q:=q₃-q:=\theta_3. When this is the case, the intersection parameters of Γ can be parametrized by p, q and the size of the antipodal classes r of Γ. Let Γ be an antipodal tight graph of diameter 4, denoted by AT4 (p, q, r), and let the μ-graph be a graph that is induced by the common neighbours of two vertices at distance 2. Then we show that all the μ-graphs of Γ are complete multipartite if and only if Γ is AT4(sq,q,q) for some natural number s. As a consequence, we derive new existence conditions for graphs of the AT4 family whose μ-graphs are not complete multipartite. Another interesting application of our results is also that we were able to show that the μ-graphs of a distance-regular graph with the same intersection array as the Patterson graph are the complete bipartite graph K _4,4.