On Graphs in Which Neighborhoods of Vertices Are Strongly Regular with Parameters (85,14,3,2) or (325,54,3,10)

Abstract

J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal eigenvalue at most t for a given positive integer t. This problem was solved earlier for t = 3. In the case t = 4, the problem was reduced to studying graphs in which neighborhoods of vertices have parameters (352,26,0,2), (352,36,0,4), (243,22,1,2), (729,112,1,20), (204,28,2,4), (232,33,2,5), (676,108,2,20), (85,14,3,2), or (325,54,3,10). In the present paper, we prove that a distance-regular graph in which neighborhoods of vertices are strongly regular with parameters (85, 14, 3, 2) or (325, 54, 3, 10) has intersection array {85, 70, 1; 1, 14, 85} or {325, 270, 1; 1, 54, 325}. In addition, we find possible automorphisms of a graph with intersection array {85, 70, 1; 1, 14, 85}.