{
  "id": "0906.5230",
  "version": "v1",
  "published": "2009-06-29T09:17:24.000Z",
  "updated": "2009-06-29T09:17:24.000Z",
  "title": "Randić index, diameter and the average distance",
  "authors": [
    "Xueliang Li",
    "Yongtang Shi"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Randi\\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\\'c index $R(G)$ with relations to the diameter $D(G)$ and the average distance $\\mu(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $\\delta(G)$, if $\\delta(G)\\geq 5$, then $R(G)-D(G)\\geq \\sqrt 2-\\frac{n+1} 2$; if $\\delta(G)\\geq n/5$ and $n\\geq 15$, $\\frac{R(G)}{D(G)} \\geq \\frac{n-3+2\\sqrt 2}{2n-2}$ and $R(G)\\geq \\mu(G)$. Furthermore, for any arbitrary real number $\\varepsilon \\ (0<\\varepsilon<1)$, if $\\delta(G)\\geq \\varepsilon n$, then $\\frac{R(G)}{D(G)} \\geq \\frac{n-3+2\\sqrt 2}{2n-2}$ and $R(G)\\geq \\mu(G)$ hold for sufficiently large $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-29T09:17:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C35",
      "92E10"
    ],
    "keywords": [
      "average distance",
      "randic index",
      "arbitrary real number",
      "minimum degree",
      "connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.5230L"
    }
  }
}