{
  "id": "1612.03514",
  "version": "v1",
  "published": "2016-12-12T01:39:39.000Z",
  "updated": "2016-12-12T01:39:39.000Z",
  "title": "Upper bounds on the Q-spectral radius of book-free and/or $K_{s,t}$-free graphs",
  "authors": [
    "Qi Kong",
    "Ligong Wang"
  ],
  "comment": "8 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we prove two results about the signless Laplacian spectral radius $q(G)$ of a graph $G$ of order $n$ with maximum degree $\\Delta$. Let $B_{n}=K_{2}+\\overline{K_{n}}$ denote a book, i.e., the graph $B_{n}$ consists of $n$ triangles sharing an edge. (1) Let $1< k\\leq l< \\Delta < n$ and $G$ be a connected \\{$B_{k+1},K_{2,l+1}$\\}-free graph of order $n$ with maximum degree $\\Delta$. Then $$\\displaystyle q(G)\\leq \\frac{1}{4}[3\\Delta+k-2l+1+\\sqrt{(3\\Delta+k-2l+1)^{2}+16l(\\Delta+n-1)}.$$ with equality holds if and only if $G$ is a strongly regular graph with parameters ($\\Delta$, $k$, $l$). (2) Let $s\\geq t\\geq 3$, and let $G$ be a connected $K_{s,t}$-free graph of order $n$ $(n\\geq s+t)$. Then $$q(G)\\leq n+(s-t+1)^{1/t}n^{1-1/t}+(t-1)(n-1)^{1-3/t}+t-3.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-12T01:39:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A18"
    ],
    "keywords": [
      "free graph",
      "q-spectral radius",
      "upper bounds",
      "maximum degree",
      "signless laplacian spectral radius"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}