{
  "id": "1705.05963",
  "version": "v1",
  "published": "2017-05-17T00:50:24.000Z",
  "updated": "2017-05-17T00:50:24.000Z",
  "title": "Sharp bounds for the Randic index of graphs with given minimum and maximum degree",
  "authors": [
    "Suil O",
    "Yongtang Shi"
  ],
  "comment": "7 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The Randi{\\' c} index of a graph $G$, written $R(G)$, is the sum of $\\frac 1{\\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\\sum_{uv \\in E(G)} \\frac 1{\\sqrt{d(u)d(v)}}$, which is called the Randi{\\' c} index of it. Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \\ge \\frac{\\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \\le \\frac n2- \\sum_{i=d}^{D-1}\\frac 12 \\left( \\frac 1{\\sqrt{i}} - \\frac 1{\\sqrt{i+1}}\\right)^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-17T00:50:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C07",
      "05C35"
    ],
    "keywords": [
      "maximum degree",
      "sharp bounds",
      "randic index",
      "minimum degree",
      "equality holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}