Automorphisms of a Distance-Regular Graph with Intersection Array {176, 135, 32, 1; 1, 16, 135, 176}

Abstract

A distance-regular graph Γ with intersection array {176, 135, 32, 1; 1, 16, 135, 176} is an AT4-graph. Its antipodal quotient \(\overline {\rm{\Gamma }} \) is a strongly regular graph with parameters (672, 176, 40, 48). In both graphs the neighborhoods of vertices are strongly regular with parameters (176, 40, 12, 8). We study the automorphisms of these graphs. In particular, the graph Γ is not arc-transitive. If G = Aut (Γ) contains an element of order 11, acts transitively on the vertex set of Γ, and S(G) fixes each antipodal class, then the full preimage of the group (G/S(G))′ is an extension of a group of order 3 by M₂₂ or U₆ (2). We describe automorphism groups of strongly regular graphs with parameters (176, 40, 12, 8) and (672, 176, 40, 48) in the vertex-symmetric case.