{
  "id": "1505.02029",
  "version": "v1",
  "published": "2015-05-08T13:22:37.000Z",
  "updated": "2015-05-08T13:22:37.000Z",
  "title": "Vertex-transitive graphs and their arc-types",
  "authors": [
    "Marston Conder",
    "Tomaž Pisanski",
    "Arjana Žitnik"
  ],
  "comment": "32 pages, 12 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $X$ be a finite vertex-transitive graph of valency $d$, and let $A$ be the full automorphism group of $X$. Then the arc-type of $X$ is defined in terms of the sizes of the orbits of the action of the stabiliser $A_v$ of a given vertex $v$ on the set of arcs incident with $v$. Specifically, the arc-type is the partition of $d$ as the sum $$n_1 + n_2 + \\dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \\dots + (m_s + m_s),$$ where $n_1, n_2, \\dots, n_t$ are the sizes of the self-paired orbits, and $m_1,m_1, m_2,m_2, \\dots, m_s,m_s$ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two `relatively prime' graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of $1+1$ and $(1+1)$, every partition as defined above is realisable, in the sense that there exists at least one graph with the given partition as its arc-type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-08T13:22:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "full automorphism group",
      "finite vertex-transitive graph",
      "arcs incident",
      "cartesian product",
      "natural sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}