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

Abstract

Let Gamma 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 Gamma is tight in the sense of Jurisic, Koolen and Terwilliger, and furthermore, precisely when Gamma is locally strongly regular with nontrivial eigenvalues p :=theta(2) and - q :=theta(3). When this is the case, the intersection parameters of Gamma can be parametrized by p, q and the size of the antipodal classes r of Gamma. Let Gamma be an antipodal tight graph of diameter 4, denoted by AT4(p, q, r), and let the mu-graph be a graph that is induced by the common neighbours of two vertices at distance 2. Then we show that all the mu-graphs of Gamma are complete multipartite if and only if Gamma 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 mu-graphs are not complete multipartite. Another interesting application of our results is also that we were able to show that the mu-graphs of a distance-regular graph with the same intersection array as the Patterson graph are the complete bipartite graph K-4,K-4.