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

Let @C be an antipodal distance-regular graph with diameter 4 and eigenvalues @q"0>@q"1>@q"2>@q"3>@q"4. Then its Krein parameter q"1"1^4 vanishes precisely when @C is tight in the sense of Jurisic, Koolen and Terwilliger, and furthermore, precisely when @C is locally strongly regular with nontrivial eigenvalues p:[email protected]"2 and -q:[email protected]"3. When this is the case, the intersection parameters of @C can be parameterized by p, q and the size of the antipodal classes r of @C, hence we denote @C 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"2"4^- graph, all graphs in this family have parameters belonging to one of the following four subfamilies:(i)q|p,r=q,(ii)q|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 @m-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper.