A graph theoretic characterization of the classical generalized hexagon on $364$ vertices

TL;DR

This paper characterizes a specific class of highly symmetric graphs related to the classical generalized hexagon, providing a graph-theoretic perspective on its structure and classification.

Contribution

It offers a new graph-theoretic characterization of the classical generalized hexagon on 364 vertices, connecting it to 2-arc-transitive graphs and their covers.

Findings

Identifies the classical generalized hexagon as a unique 2-arc-transitive graph of order 728.

Classifies certain 2-arc-transitive graphs as either the known hexagon or covers of smaller transitive graphs.

Provides a characterization linking graph symmetry properties to geometric structures.

Abstract

A tetravalent $2$-arc-transitive graph of order $728$ is either the known $7$-arc-transitive incidence graph of the classical generalized hexagon $GH(3,3)$ or a normal cover of a $2$-transitive graph of order $182$ denoted $A[182,1]$ or $A[182,2]$ in the $2009$ list of Poto\v{c}nik.