{
  "id": "2311.15144",
  "version": "v1",
  "published": "2023-11-26T00:04:10.000Z",
  "updated": "2023-11-26T00:04:10.000Z",
  "title": "Some results on the Wiener index related to the Šoltés problem of graphs",
  "authors": [
    "Andrey A. Dobrynin",
    "Konstantin V. Vorob'ev"
  ],
  "comment": "9 pages, 3 tables, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Wiener index, $W(G)$, of a connected graph $G$ is the sum of distances between its vertices. In 2021, Akhmejanova et al. posed the problem of finding graphs $G$ with large $R_m(G)= |\\{v\\in V(G)\\,|\\,W(G)-W(G-v)=m \\in \\mathbb{Z} \\}|/ |V(G)|$. It is shown that there is a graph $G$ with $R_m(G) > 1/2$ for any integer $m \\ge 0$. In particular, there is a regular graph of even degree with this property for any odd $m \\ge 1$. The proposed approach allows to construct new families of graphs $G$ with $R_0(G) \\rightarrow 1/2$ when the order of $G$ increases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-26T00:04:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C09",
      "05C12",
      "05C92"
    ],
    "keywords": [
      "wiener index",
      "regular graph",
      "akhmejanova",
      "finding graphs",
      "connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}