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

Let be an antipodal distance-regular graph of diameter 4, with eigenvalues θ0 > θ1 > θ2 > θ3 > θ4. Then its Krein parameter q4 11 vanishes precisely when is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when is locally strongly regular with nontrivial eigenvalues p := θ2 and −q := θ3. When this is the case, the intersection parameters of can be parametrized by p, q and the size of the antipodal classes r of . Let 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 are complete multipartite if and only if 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 K4,4.