Q-polynomial Graph With an Intersections Array {60, 45, 8; 1, 12, 50} Does not Exist

Abstract

When studying amply regular graphs Γ of diameter d, in which for some vertex a the pair (Γd(a), Γd-1(a)) is a 2-cheme, it is proved that the subgraph induced by the set of points is a clique, coclique, or strongly regular graph. For a graph of diameter 3, it is established that this construction is a 2-cheme for any vertex a if and only if the graph is distance-regular and for any vertex a the subgraph Γ3(a) is a clique, coclique, or strongly regular graph (A.L. Gavrilyuk, A.A. Makhnev). An interesting question is whether there is a distance-regular graph with intersection array {60,45,8;1,12,50} such that Γ3(a) can be a 6×6-lattice and the pair (Γ3(a), Γ2(a)) will be a 2-scheme. In the paper of I.N. Belousov and A.A. Makhnev (2018) published a proof of the non-existence of the abovementioned graph, that contained errors. In this paper, we give a correct proof of this result.