Geodesic
Line graphs and geodesic transitivity
Ars Mathematica Contemporanea, Tome 6 (2013) no. 1, pp. 13-20.

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For a graph Γ, a positive integer s and a subgroup G ≤ 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.
DOI : 10.26493/1855-3974.248.aae
Keywords: Line graphs, s-geodesic transitive graphs, s-arc transitive graphs. 
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Alice Devillers; Wei Jin; Cai Heng Li; Cheryl E. Praeger. Line graphs and geodesic transitivity. Ars Mathematica Contemporanea, Tome 6 (2013) no. 1, pp. 13-20. doi : 10.26493/1855-3974.248.aae. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.248.aae/

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