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

Aleksandar Jurišić, Jack Koolen
2011 Journal of combinatorial theory. Series A  
Let Γ be an antipodal distance-regular graph with diameter 4 and eigenvalues θ 0 > θ 1 > θ 2 > θ 3 > θ 4 . Then its Krein parameter q 4 11 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 := θ 2 and −q := θ 3 . 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,
more » ... r). Jurišić 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: 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 μ-graphs of the graphs in this subfamily are completely multipartite is very crucial in this paper.
doi:10.1016/j.jcta.2010.10.001 fatcat:mzh7g2onifa7lcxbgf24kp3co4