{
  "id": "math/0510246",
  "version": "v1",
  "published": "2005-10-12T11:39:29.000Z",
  "updated": "2005-10-12T11:39:29.000Z",
  "title": "Edge-local equivalence of graphs",
  "authors": [
    "Maarten Van den Nest",
    "Bart De Moor"
  ],
  "comment": "25 pages, 2 figures",
  "categories": [
    "math.CO",
    "quant-ph"
  ],
  "abstract": "The local complement G*i of a simple graph G at one of its vertices i is obtained by complementing the subgraph induced by the neighborhood of i and leaving the rest of the graph unchanged. If e={i,j} is an edge of G then G*e=((G*i)*j)*i is called the edge-local complement of G along the edge e. We call two graphs edge-locally equivalent if they are related by a sequence of edge-local complementations. The main result of this paper is an algebraic description of edge-local equivalence of graphs in terms of linear fractional transformations of adjacency matrices. Applications of this result include (i) a polynomial algorithm to recognize whether two graphs are edge-locally equivalent, (ii) a formula to count the number of graphs in a class of edge-local equivalence, and (iii) a result concerning the coefficients of the interlace polynomial, where we show that these coefficients are all even for a class of graphs; this class contains, as a subset, all strongly regular graphs with parameters (n, k, a, c), where k is odd and a and c are even.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-12T11:39:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "edge-local equivalence",
      "linear fractional transformations",
      "regular graphs",
      "edge-local complementations",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10246V"
    }
  }
}