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 $V(\Gamma )$ 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 $|B|\ge 3$; (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 [A. Gardiner and C. E. Praeger, Australas. J. Combin. 71 (2018) 403--426], [M.~Giulietti et al., J. Algebraic Combin. 38 (2013) 745--765] and [T. Fang et al., Electronic J. Combin. 23 (2) (2016) P2.27] completes the classification of symmetric graphs satisfying (i) and (ii).

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