{
  "id": "1610.08855",
  "version": "v1",
  "published": "2016-10-27T15:54:47.000Z",
  "updated": "2016-10-27T15:54:47.000Z",
  "title": "The signless Laplacian spectral radius of subgraphs of regular graphs",
  "authors": [
    "Qi Kong",
    "Ligong Wang"
  ],
  "comment": "9 pages, 2 figures, 1 table",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this paper, we prove that \\\\1. Let $H$ be a proper subgraph of a $\\Delta$-regular graph $G$ with $n$ vertices and diameter $D$. Then $$2\\Delta - q(H)>\\frac{1}{n(D-\\frac{1}{4})}.$$ \\\\2. Let $H$ be a proper subgraph of a $k$-connected $\\Delta$-regular graph $G$ with $n$ vertices, where $k\\geq 2$. Then $$2\\Delta-q(H)>\\frac{2(k-1)^{2}}{2(n-\\Delta)(n-\\Delta+2k-4)+(n+1)(k-1)^{2}}.$$ Finally, we compare the two bounds. We obtain that when $k>2\\sqrt{\\frac{(n-\\Delta)(n+\\Delta-4)}{n(4D-3)-2}}+1$, the second bound is always better than the first. On the other hand, when $k<\\frac{2(n-\\Delta)}{\\sqrt{n(4D-3)-2}}+1$, the first bound is always better than the second.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-27T15:54:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "15A18"
    ],
    "keywords": [
      "signless laplacian spectral radius",
      "regular graph",
      "proper subgraph",
      "second bound",
      "first bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}