Symmetric graphs with 2-arc transitive quotients

Abstract

A graph Γ is G-symmetric if Γ admits G as a group of automorphisms acting transitively on the set of vertices and the set of arcs of Γ, where an arc is an ordered pair of adjacent vertices. In the case when G is imprimitive on V(Γ), namely when V(Γ) admits a nontrivial G-invariant partition ℬ, the quotient graph Γℬ of Γ with respect to ℬ is always G-symmetric and sometimes even (G, 2)-arc transitive. (A G-symmetric graph is (G, 2)-arc transitive if G is transitive on the set of oriented paths of length two.) In this paper we obtain necessary conditions for Γℬ to be (G, 2)-arc transitive (regardless of whether Γ is (G, 2)-arc transitive) in the case when v-k is an odd prime p, where v is the ..