{
  "id": "1709.04189",
  "version": "v1",
  "published": "2017-09-13T08:43:31.000Z",
  "updated": "2017-09-13T08:43:31.000Z",
  "title": "The Graovac-Pisanski index of a connected bipartite graph is an integer number",
  "authors": [
    "Matevž Črepnjak",
    "Martin Knor",
    "Niko Tratnik",
    "Petra Žigert Pleteršek"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, \\v{C}repnjak, Tratnik, and \\v{Z}igert Pleter\\v{s}ek recently found out that the Graovac-Pisanski index is correlated with the melting points of some hydrocarbon molecules. The examples showed that for all the considered molecular graphs the Graovac-Pisanski index is an integer number. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an even number of vertices is an integer number. By using a computer programme, the graphs with a non-integer Graovac-Pisanski index on at most nine vertices are counted. Finally, an infinite class of unicyclic graphs with a non-integer Graovac-Pisanski index is described.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-13T08:43:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "92E10",
      "05C12",
      "05C60"
    ],
    "keywords": [
      "connected bipartite graph",
      "integer number",
      "non-integer graovac-pisanski index",
      "original wiener index",
      "unicyclic graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}