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

Abstract

Let Γ be an antipodal distance-regular graph of diameter 4, with eigenvalues $$\theta_0>\theta_1>\theta_2>\theta_3>\theta_4$$ . Then its Krein parameter $$q_{11}^4$$ 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 $$p:=\theta_2$$ and $$-q:=\theta_3$$ . 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.