{
  "id": "1602.03698",
  "version": "v1",
  "published": "2016-02-11T12:21:50.000Z",
  "updated": "2016-02-11T12:21:50.000Z",
  "title": "The variation of the Randic index with regard to minimum and maximum degree",
  "authors": [
    "Milica Milivojevic",
    "Ljiljana Pavlovic"
  ],
  "comment": "12 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "The variation of the Randi\\'c index $ R'(G) $ of a graph $G$ is defined by\\ $R(G) = \\sum_{uv \\in E(G)}\\frac 1{\\max \\{d(u) d(v)\\}}$, where $d(u)$ is the degree of vertex $u$ and the summation extends over all edges $uv$ of $G$. Let $G(k,n)$ be the set of connected simple $n$-vertex graphs with minimum vertex degree $k$. In this paper we found in $G(k,n)$ graphs for which the variation of the Randi\\'c index attains its minimum value. When $k \\leq \\frac n2$ the extremal graphs are complete split graphs $K_{k,n-k}^*$, which only vertices of two degrees, i.e. degree $k$ and degree $n-1$, and the number of vertices of degree $k$ is $n-k$, while the number of vertices of degree $n-1$ is $k$. For $k \\geq \\frac n2$ the extremal graphs have also vertices of two degrees $k$ and $n-1$, and the number of vertices of degree $k$ is $\\frac n2$. Further, we generalized results for graphs with given maximum degree.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-11T12:21:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "maximum degree",
      "extremal graphs",
      "complete split graphs",
      "minimum vertex degree",
      "randic index attains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203698M"
    }
  }
}