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Line graphs and geodesic transitivity
Devillers, Alice ...
For a graph ▫$\Gamma$▫, a positive integer ▫$s$▫ and a subgroup ▫$G \le \text{Aut}(\Gamma)$▫, 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 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.
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