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

Let Γ   be an antipodal distance-regular graph with diameter 4 and eigenvalues θ0>θ1>θ2>θ3>θ4θ0>θ1>θ2>θ3>θ4. Then its Krein parameter q114 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:=θ2p:=θ2 and −q:=θ3−q:=θ3. When this is the case, the intersection parameters of Γ can be parameterized by p, q and the size of the antipodal classes r of Γ, hence we denote Γ   by AT4(p,q,r)AT4(p,q,r).Jurišić conjectured that the AT4(p,q,r)AT4(p,q,r) family is finite and that, aside from the Conway–Smith graph, the Soicher2 graph and the 3.Fi24− 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)AT4(qs,q,q) there are exactly five possibilities for the pair (s,q)(s,q), with an example for each: the Johnson graph J(8,4)J(8,4) for (1,2)(1,2), the halved 8-cube for (2,2)(2,2), the 3.O6−(3) graph for (1,3)(1,3), the Meixner2 graph for (2,4)(2,4) and the 3.O7(3)3.O7(3) graph for (3,3)(3,3). The fact that the μ-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper.