Classification of the family of antipodal tight graphs

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 11 4 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 AT 4 ( p , q , r ) . ć conjectured that the AT 4 ( 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 | 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 AT 4 ( 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.