A class of symmetric graphs with 2-arc-transitive quotients

Abstract

Let $\Gamma$ be a finite X-symmetric graph with a nontrivial X-invariant partition $\mathcal{B}$ on $V(\Gamma )$ such that ${\Gamma }_{\mathcal{B}}$ is a connected (X,2)-arc-transitive graph and $\Gamma$ is not a multicover of ${\Gamma }_{\mathcal{B}}$. This article aims to give a characterization of $(\Gamma ,X,\mathcal{B})$ for the case where $|\Gamma (C)\cap B|=3$ for $B\in \mathcal{B}$ and $C\in {\Gamma }_{\mathcal{B}}(B)$. This investigation requires a study on (X,2)-arc-transitive graphs of valency 4 or 7. We give a characterization of tetravalent (X,2)-arc-transitive graphs at first; and as a byproduct, we prove that every tetravalent (X,2)-transitive graph is either the complete graph on 5 vertices or a near n-gonal graph for some $n\ge 4$. Then we show that a heptavalent $(X,2)$-arc-transitive graph $\Sigma$ can occur as ${\Gamma }_{\mathcal{B}}$ if and only if $X_{\tau }^{\Sigma (\tau )}\cong PSL(3,2)$ for $\tau \in V(\Sigma )$.

22 pages