Finite 2-distance transitive graphs

TL;DR

This paper classifies finite graphs with specific symmetry properties related to vertex distances, focusing on those with valency up to 5 that are transitive but not arc transitive.

Contribution

It provides a complete classification of non-arc transitive, (G,2)-distance transitive graphs with valency at most 5.

Findings

Classified all such graphs with valency ≤ 5

Identified structural properties of these graphs

Extended understanding of symmetry in finite graphs

Abstract

A non-complete graph $\Gamma$ is said to be $(G,2)$-distance transitive if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set of $\Gamma$, and for any vertex $u$ of $\Gamma$, the stabilizer $G_u$ is transitive on the sets of vertices at distance 1 and 2 from $u$. This paper investigates the family of $(G,2)$-distance transitive graphs that are not $(G,2)$-arc transitive. Our main result is the classification of such graphs of valency not greater than 5.