Finite symmetric graphs with two-arc transitive quotients

This paper forms part of a study of 2-arc transitivity for finite imprimitive symmetric graphs, namely for graphs Gamma admitting an automorphism group G that is transitive on ordered pairs of adjacent vertices, and leaves invariant a nontrivial vertex partition B. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph Gamma(B) corresponding to B. If in addition G is transitive on the 2-arcs of Gamma (that is, on vertex triples (alpha, beta, gamma) such that alpha, beta and beta, gamma are adjacent and alpha not equal gamma), then G is not in general transitive on the 2-arcs of Gamma(B), although it does have this property in the special case where B is the orbit set of a vertex-intransitive normal subgroup of G. On the other hand, G is sometimes transitive on the 2-arcs of Gamma(B) even if it is not transitive on the 2-arcs of Gamma. We study conditions under which G is transitive on the 2-arcs of Gamma(B). Our conditions relate to the structure of the bipartite graph induced on B boolean OR C for adjacent blocks B, C of B, and a graph structure induced on B. We obtain necessary and sufficient conditions for G to be transitive on the 2-arcs of one or both of Gamma, Gamma(B) for certain families of imprimitive symmetric graphs. (c) 2004 Elsevier Inc. All rights reserved.