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

Abstract

Let Gamma be an antipodal distance-regular graph with diameter 4 and 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 parameterized by p, q and the size of the antipodal classes r of Gamma, hence we denote Gamma by AT4(p, q, r). Jurisic conjectured that the AT4(p, q, r) family is finite and that, aside from the Conway-Smith graph, the Soicher2 graph and the 3.Fi(24)(-) graph, all graphs in this family have parameters belonging to one of the following four subfamilies: (i) q vertical bar p, r = q, (ii) q vertical bar p, r = 2, (iii) p = q - 2, r = q - 1, (iv) p = q - 2, r = 2. 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 3.O-6(-) (3) graph for (1,3), the Meixner2 graph for (2,4) and the 3.O-7 (3) graph for (3,3). The fact that the mu-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper. (C) 2010 Elsevier Inc. All rights reserved.