Affine primitive symmetric graphs of diameter two

TL;DR

This paper classifies affine primitive symmetric graphs of diameter two based on the maximal subgroup structure of their automorphism groups, providing complete classifications for certain subgroup classes and necessary conditions for others.

Contribution

It determines all such graphs arising from specific subgroup classes within the Aschbacher classification, advancing understanding of diameter two symmetric graphs with affine automorphism groups.

Findings

Complete classification for G_0 in ext{ΓL}(n,q) with i ∈ {2,4,8}.

Complete classification for G_0 in ext{ΓSp}(n,q) with i ∈ {2,8}.

Necessary conditions for other subgroup classes to produce diameter two graphs.

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.