Affine primitive symmetric graphs of diameter two

Abstract

Let $n$ be a positive integer, $q$ be a prime power, and $V$ be a vector space of dimension $n$ over $\mathbb{F}_q$. Let $G := V \rtimes G_0$, where $G_0$ is an irreducible subgroup of ${\rm GL}(V)$ which is maximal by inclusion with respect to being intransitive on the set of nonzero vectors. We are interested in the class of all diameter two graphs $\Gamma$ that admit such a group $G$ as an arc-transitive, vertex-quasiprimitive subgroup of automorphisms. In particular, we consider those graphs for which $G_0$ is a subgroup of either ${\rm \Gamma L}(n,q)$ or ${\rm \Gamma Sp}(n,q)$ and is maximal in one of the Aschbacher classes $\mathcal{C}_i$, where $i \in \{2,4,5,6,7,8\}$. We are able to determine all graphs $\Gamma$ which arise from $G_0 \leq {\rm \Gamma L}(n,q)$ with $i \in \{2,4,8\}$, and from $G_0 \leq {\rm \Gamma Sp}(n,q)$ with $i \in \{2,8\}$. For the remaining classes we give necessary conditions in order for $\Gamma$ to have diameter two, and in some special subcases determine all $G$-symmetric diameter two graphs.