{
  "id": "1201.4297",
  "version": "v1",
  "published": "2012-01-20T14:04:53.000Z",
  "updated": "2012-01-20T14:04:53.000Z",
  "title": "Line graphs and $2$-geodesic transitivity",
  "authors": [
    "Alice Devillers",
    "Wei Jin",
    "Cai Heng Li",
    "Cheryl E. Praeger"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-01-20T14:04:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "line graph",
      "geodesic transitivity",
      "complete multipartite graph",
      "non-complete locally cyclic",
      "geodesic transitive graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.4297D"
    }
  }
}