{
  "id": "1004.1778",
  "version": "v1",
  "published": "2010-04-11T09:16:38.000Z",
  "updated": "2010-04-11T09:16:38.000Z",
  "title": "The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degree",
  "authors": [
    "Xueliang Li",
    "Yiyang Li"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "Let $\\mathcal {T}^{\\Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\\Delta$. Suppose that every tree in $\\mathcal {T}^{\\Delta}_n$ is equally likely. We show that the number of vertices of degree $j$ in $\\mathcal {T}^{\\Delta}_n$ is asymptotically normal with mean $(\\mu_j+o(1))n$ and variance $(\\sigma_j+o(1))n$, where $\\mu_j$, $\\sigma_j$ are some constants. As a consequence, we give estimate to the value of the general Zagreb index for almost all trees in $\\mathcal {T}^{\\Delta}_n$. Moreover, we obtain that the number of edges of type $(i,j)$ in $\\mathcal {T}^{\\Delta}_n$ also has mean $(\\mu_{ij}+o(1))n$ and variance $(\\sigma_{ij}+o(1))n$, where an edge of type $(i,j)$ means that the edge has one end of degree $i$ and the other of degree $j$, and $\\mu_{ij}$, $\\sigma_{ij}$ are some constants. Then, we give estimate to the value of the general Randi\\'{c} index for almost all trees in $\\mathcal {T}^{\\Delta}_n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-04-11T09:16:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C12",
      "05C30",
      "05D40",
      "05A15",
      "05A16",
      "92E10"
    ],
    "keywords": [
      "bounded maximum degree",
      "asymptotic values",
      "general zagreb index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1778L"
    }
  }
}