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

Abstract

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