Line graphs and $2$-geodesic transitivity

Abstract

For a graph $Γ$, a positive integer $s$ and a subgroup $G\leq \Aut(Γ)$, we prove that $G$ is transitive on the set of $s$-arcs of $Γ$ if and only if $Γ$ 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.