Line graphs and $2$-geodesic transitivity

TL;DR

This paper characterizes when a group acts transitively on s-arcs of a graph via girth and line graph geodesics, and classifies certain 2-geodesic transitive graphs with specific properties.

Contribution

It establishes a new equivalence relating s-arc transitivity to girth and line graph geodesic transitivity, and classifies specific 2-geodesic transitive graphs.

Findings

Only K_{3[2]} and the icosahedron are non-complete locally cyclic 2-geodesic transitive graphs.

Classified 2-geodesic transitive graphs of valency 4 and girth 3.

Identified which 2-geodesic transitive graphs are also geodesic transitive.

Abstract

For a graph $\Gamma$, a positive integer $s$ and a subgroup $G\leq \Aut(\Gamma)$, we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive.