{
  "id": "2105.02297",
  "version": "v1",
  "published": "2021-05-05T19:38:21.000Z",
  "updated": "2021-05-05T19:38:21.000Z",
  "title": "Remarks on the spectral radius of $K_{r+1}$-saturated graphs",
  "authors": [
    "V. Nikiforov"
  ],
  "comment": "6 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Write $\\rho\\left( G\\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\\vee\\overline{K}_{n-r}.$ Let $n>r\\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $\\rho\\left( G\\right) $ for $r=2$, and found an asymptotically tight bound on $\\rho\\left( G\\right) $ for $r\\geq3.$ They also conjectured that \\[ \\rho\\left( G\\right) >\\rho\\left( S_{n,r-1}\\right) , \\] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-05T19:38:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "spectral radius",
      "saturated graph",
      "minimum value",
      "asymptotically tight bound",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}