Heptavalent symmetric graphs with solvable stabilizers admitting vertex-transitive non-abelian simple groups

TL;DR

This paper classifies finite connected heptavalent symmetric graphs with solvable stabilizers that admit a vertex-transitive non-abelian simple automorphism group, identifying specific exception pairs and conditions for regular and arc-transitive cases.

Contribution

It provides a detailed classification of such graphs, explicitly listing exception pairs of non-abelian simple groups and their relation to the automorphism group structure.

Findings

If G is normal in Aut(Γ), then specific structural properties hold.

Identifies 11 exception pairs of non-abelian simple groups for the automorphism group.

Specifies 7 exception pairs when G is regular and particular pairs for arc-transitive cases.

Abstract

A graph $\Gamma$ is said to be symmetric if its automorphism group $\rm Aut(\Gamma)$ acts transitively on the arc set of $\Gamma$. In this paper, we show that if $\Gamma$ is a finite connected heptavalent symmetric graph with solvable stabilizer admitting a vertex-transitive non-abelian simple group $G$ of automorphisms, then either $G$ is normal in $\rm Aut(\Gamma)$, or $\rm Aut(\Gamma)$ contains a non-abelian simple normal subgroup $T$ such that $G\leq T$ and $(G,T)$ is explicitly given as one of $11$ possible exception pairs of non-abelian simple groups. Furthermore, if $G$ is regular on the vertex set of $\Gamma$ then the exception pair $(G,T)$ is one of $7$ possible pairs, and if $G$ is arc-transitive then the exception pair $(G,T)=(A_{17},A_{18})$ or $(A_{35},A_{36})$.