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 ${ \cal B}$, and let ${ \cal D}(\Gamma, { \cal B})$ be the incidence structure with point set ${ \cal B}$ and blocks $\{ B\} \cup \Gamma_{ \cal B}(\alpha)$, for $B \in { \cal B}$ and $\alpha \in B$, where $\Gamma_{ \cal B}(\alpha)$ is the set of blocks of ${ \cal B}$ containing at least one neighbour of $\alpha$ in $\Gamma$. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that $\Gamma_{ \cal B}(\alpha) \ne \Gamma_{ \cal B}(\beta)$ for distinct $\alpha, \beta \in B$, the quotient graph of $\Gamma$ with respect to ${ \cal B}$ is a complete graph, and ${ \cal D}(\Gamma, { \cal B})$ is isomorphic to the complement of a $(G, 2)$-point-transitive linear space.
Categories: math.CO