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 admitting an automorphism groupG 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 B corresponding toB. If in additionG is transitive on the 2-arcs of (that is, on vertex triples ( , , ) such that , and , are adjacent and = ), then G is not in general transitive on the 2-arcs of 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 B even if it is not transitive on the 2-arcs of . We study conditions under which G is transitive on the 2-arcs of B. Our conditions relate to the structure of the bipartite graph induced on B ∪ 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 , B for certain families of imprimitive symmetric graphs. © 2004 Elsevier Inc. All rights reserved.