{
  "id": "2404.06410",
  "version": "v1",
  "published": "2024-04-09T15:57:16.000Z",
  "updated": "2024-04-09T15:57:16.000Z",
  "title": "The maximum degree of the $r$th power of a sparse random graph",
  "authors": [
    "Alan Frieze",
    "Aditya Raut"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G^r_{n,p}$ denote the $r$th power of the random graph $G_{n,p}$, where $p=c/n$ for a positive constant $c$. We prove that w.h.p. the maximum degree $\\Delta\\left(G^r_{n,p}\\right)\\sim \\frac{\\log n}{\\log_{(r+1)}n}$. Here $\\log_{(k)}n$ indicates the repeated application of the log-function $k$ times. So, for example, $\\log_{(3)}n=\\log\\log\\log n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-09T15:57:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sparse random graph",
      "th power",
      "maximum degree",
      "positive constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}