Symmetries of regular $q$-graphs

Abstract

Given a finite vector space $V=\mathbb{F}_q^n$, the $q$-analogue of a graph, called a $q$-graph, is a pair $\Gamma=(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ is the set of $1$-dimensional subspaces of $V$ and $\mathcal{E}$ is a subset of the $2$-dimensional subspaces of $V$. Elements of $\mathcal{V}$ and $\mathcal{E}$ are called vertices and edges, respectively. If the edges through a vertex $X$ consist of all $2$-spaces of a $(k+1)$-dimensional space which contain $X$, regardless of the choice of vertex, then $\Gamma$ is $k$-regular. Moreover, $\Gamma$ is flag-transitive if there is a subgroup of $\Gamma {\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all incident vertex-edge pairs; and symmetric if there is a subgroup of $\Gamma {\rm L}_n(q)$ preserving $\mathcal{E}$ and acting transitively on the set of all ordered pairs of adjacent vertices. This paper classifies all $k$-regular $q$-graphs that are either flag-transitive or symmetric. The $q$-graphs in the classification are constructed from familiar objects in finite geometry, including spreads, symplectic polar spaces, and generalised hexagons. The classification depends essentially on the classification of transitive linear groups, and thus ultimately on the classification of finite simple groups.