Some coset actions in $G_2(q)$ and distance-transitive graphs

TL;DR

This paper investigates the existence of distance-transitive graphs derived from coset actions of the group G_2(q) and concludes that such graphs do not exist, even with group extensions by outer automorphisms.

Contribution

It provides a comprehensive analysis showing the non-existence of certain distance-transitive graphs from specific coset actions of G_2(q).

Findings

No distance-transitive graphs from G_2(q) coset actions.

Extensions by outer automorphisms do not produce such graphs.

Clarifies the limitations of coset actions in G_2(q).

Abstract

This paper studies whether there are distance-transitive graphs arising from the coset actions of $G_2(q)$ on the subfield subgroup $G_2(\sqrt{q})$ or $G_2(q)$ on the Ree subgroup $^2G_2(q)$. It is found that there are no such graphs, even if the groups are extended by outer automorphisms of $G_2(q)$.