A family of symmetric graphs in relation to 2-point-transitive linear spaces

Abstract

A graph $\Gamma$ is $G$-symmetric if it admits $G$ as a group of automorphisms acting transitively on the set of arcs of $\Gamma$, where an arc is an ordered pair of adjacent vertices. Let $\Gamma$ be a $G$-symmetric graph such that its vertex set admits a nontrivial $G$-invariant partition $\mathcal{B}$, and let $\mathcal{D}(\Gamma ,\mathcal{B})$ be the incidence structure with point set $\mathcal{B}$ and blocks $\{B\}\cup {\Gamma }_{\mathcal{B}}(\alpha )$, for $B\in \mathcal{B}$ and $\alpha \in B$, where ${\Gamma }_{\mathcal{B}}(\alpha )$ is the set of blocks of $\mathcal{B}$ containing at least one neighbour of $\alpha$ in $\Gamma$. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that ${\Gamma }_{\mathcal{B}}(\alpha )\ne {\Gamma }_{\mathcal{B}}(\beta )$ for distinct $\alpha ,\beta \in B$, the quotient graph of $\Gamma$ with respect to $\mathcal{B}$ is a complete graph, and $\mathcal{D}(\Gamma ,\mathcal{B})$ is isomorphic to the complement of a $(G,2)$-point-transitive linear space.

20 pages