Quotient-complete arc-transitive graphs

Abstract

We examine the class of G-arc-transitive graphs with at least one nontrivial complete normal quotient, such that all other normal quotients are either complete or empty graphs. Such graphs arise naturally as basic graphs under normal quotient analysis on families of arc-transitive graphs. We construct all such graphs Γ, together with the corresponding group G≤Aut(Γ), which have 3 or more nontrivial complete G-normal quotients. In particular, we prove that these graphs have order the square of a prime power q, and arise from finite transitive linear groups H acting on a vector space of order q. The only disconnected graphs in this class are isomorphic to q copies of the complete graph on q vertices, and we classify all the connected graphs that arise from the transitive linear groups H where H is not a subgroup of a one-dimensional affine group ΓL(1,q). For the case where H≤ΓL(1,q) we construct some infinite families of examples. In most cases we determine which of the graphs constructed have diameter two.