{
  "id": "2007.04479",
  "version": "v1",
  "published": "2020-07-08T23:57:39.000Z",
  "updated": "2020-07-08T23:57:39.000Z",
  "title": "Signless Laplacian spectral radius and matching in graphs",
  "authors": [
    "Chang Liu",
    "Yingui Pan",
    "Jianping Li"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral radius, denoted by $q_1=q_1(G)$. In this paper, some properties between the signless Laplacian spectral radius and perfect matching in graphs are establish. Let $r(n)$ be the largest root of equation $x^3-(3n-7)x^2+n(2n-7)x-2(n^2-7n+12)=0$. We show that $G$ has a perfect matching for $n=4$ or $n\\geq10$, if $q_1(G)>r(n)$, and for $n=6$ or $n=8$, if $q_1(G)>4+2\\sqrt{3}$ or $q_1(G)>6+2\\sqrt{6}$ respectively, where $n$ is a positive even integer number. Moreover, there exists graphs $K_{n-3}\\vee K_1 \\vee \\overline{K_2}$ such that $q_1(K_{n-3}\\vee K_1 \\vee \\overline{K_2})=r(n)$ if $n\\geq4$, a graph $K_2\\vee\\overline{K_4}$ such that $q_1(K_2\\vee\\overline{K_4})=4+2\\sqrt{3}$ and a graph $K_3\\vee\\overline{K_5}$ such that $q_1(K_3\\vee\\overline{K_5})=6+2\\sqrt{6}$. These graphs all have no prefect matching.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-08T23:57:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C70"
    ],
    "keywords": [
      "signless laplacian spectral radius",
      "vertex degrees",
      "signless laplacian matrix",
      "integer number",
      "diagonal matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}