Finite symmetric graphs with two-arc transitive quotients

Abstract

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 $\mathcal{B}$. Such a group G is also transitive on the ordered pairs of adjacent vertices of the quotient graph ${\Gamma }_{\mathcal{B}}$ corresponding to $\mathcal{B}$. 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 $\alpha \ne \gamma$), then G is not in general transitive on the 2-arcs of ${\Gamma }_{\mathcal{B}}$, although it does have this property in the special case where $\mathcal{B}$ 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 ${\Gamma }_{\mathcal{B}}$ 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 ${\Gamma }_{\mathcal{B}}$. Our conditions relate to the structure of the bipartite graph induced on $B\cup C$ for adjacent blocks $B,C$ of $\mathcal{B}$, 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 $\Gamma ,{\Gamma }_{\mathcal{B}}$ for certain families of imprimitive symmetric graphs.