Line graphs and $2$-geodesic transitivity

Details Line graphs and $2$-geodesic transitivity Line graphs and $2$-geodesic transitivity

2012-01-20

arxiv.org/abs/1201.4297v1

Mathematics

Combinatorics

Scientific paper

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.

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