Line graphs and geodesic transitivity - Devillers, Alice (Ars mathematica contemporanea, 2013)
For a graph ?$Γ$?, a positive integer ?$s$? and a subgroup ?$G \le \text{Aut}(Γ)$?, 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 classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive. Secondly we prove that the only non-complete locally cyclic 2-geodesic transitive graphs are the octahedron and the icosahedron.
Keywords: line graphs; s-arc transitive graphs; s-geodesic transitive graphs