Symmetric diameter two graphs arising from the classical groups

Carmen Amarra (University of Western Australia, Australia)

Let $V$ be a vector space of dimension $d$ over a field of $q$ elements, and let $H \leq \mathrm{GL}{V}$ where $H$ is irreducible on $V$. Let $S$ be an orbit of $H$ on $V \backslash \{\mathbf{0}\}$ with $-S := \{ -v \;|\; v \in S \} = S$, and define $\Gamma$ to be the graph with vertex set $V$ and whose edges are the pairs $\{v,w\}$ where $v-w \in S$. Then
$\Gamma$ is $G$-arc-transitive, where G = V \rtimes H. It has diameter 2 if $V \backslash S \subseteq S + S$. In this talk we present the symmetric diameter 2 graphs that arise for some of the groups in Aschbacher's class $C_8$, that is, for the case when $H$ is a classical group.