On Automorphisms of a Distance-Regular Graph with Intersection Array {125, 96, 1; 1, 48, 125}

Abstract

J. Koolen posed the problem of studying distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs whose second eigenvalue is at most t for a given positive integer t. This problem is reduced to the description of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with a nonprincipal eigenvalue t for t = 1, 2,.... In the paper “Distance regular graphs in which local subgraphs are strongly regular graphs with the second eigenvalue at most 3”, Makhnev and Paduchikh found intersection arrays of distance-regular graphs in which the neighborhoods of vertices are strongly regular graphs with second eigenvalue t, where 2 < t ≤ 3. Graphs with intersection arrays {125, 96, 1; 1, 48, 125}, {176, 150, 1; 1, 25, 176}, and {256, 204, 1; 1, 51, 256} remained unexplored. In this paper, possible orders and fixed-point subgraphs of automorphisms are found for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125}. It is proved that the neighborhoods of the vertices of this graph are pseudogeometric graphs for GQ(4, 6). Composition factors of the automorphism group for a distance-regular graph with intersection array {125, 96, 1; 1, 48, 125} are determined.