Affine flag graphs and classification of a family of symmetric graphs with complete quotients

Abstract

A graph $\Gamma$ is $G$-symmetric if $G$ is a group of automorphisms of $\Gamma$ which is transitive on the set of ordered pairs of adjacent vertices of $\Gamma$. If $V\left(\Gamma \right)$ admits a nontrivial $G$-invariant partition $\mathcal{B}$ such that for blocks $B,C\in \mathcal{B}$ adjacent in the quotient graph ${\Gamma }_{\mathcal{B}}$ of $\Gamma$ relative to $\mathcal{B}$, exactly one vertex of $B$ has no neighbour in $C$, then $\Gamma$ is called an almost multicover of ${\Gamma }_{\mathcal{B}}$. In this case an incidence structure with point set $\mathcal{B}$ arises naturally, and it is a $(G,2)$-point-transitive and $G$-block-transitive 2-design if in addition ${\Gamma }_{\mathcal{B}}$ is a complete graph. In this paper we classify all $G$-symmetric graphs $\Gamma$ such that (i) $\mathcal{B}$ has block size $\left|B\right|\ge 3$; (ii) ${\Gamma }_{\mathcal{B}}$ is complete and almost multi-covered by $\Gamma$; (iii) the incidence structure involved is a linear space; and (iv) $G$ contains a regular normal subgroup which is elementary abelian. This classification together with earlier results in Gardiner and Praeger (2018), Giulietti et al. (2013) and Fang et al. (2016) completes the classification of symmetric graphs satisfying (i) and (ii).