Symmetries of regular $q$-graphs

TL;DR

This paper classifies all symmetric and flag-transitive $q$-graphs derived from finite vector spaces, connecting their structure to well-known finite geometric objects and the classification of finite simple groups.

Contribution

It provides a complete classification of $k$-regular $q$-graphs with symmetry properties, linking them to classical finite geometries and group theory.

Findings

Classified all $k$-regular $q$-graphs that are flag-transitive or symmetric.

Connected $q$-graphs to finite geometries like spreads and polar spaces.

Relied on the classification of finite simple groups for the results.

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.