{
  "id": "1310.4717",
  "version": "v1",
  "published": "2013-10-17T14:31:57.000Z",
  "updated": "2013-10-17T14:31:57.000Z",
  "title": "The domination number and the least $Q$-eigenvalue",
  "authors": [
    "Guanglong Yu",
    "Shu-Guang Guo",
    "Rong Zhang",
    "Yarong Wu"
  ],
  "comment": "13 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\\gamma(G)$ ($\\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\\gamma$, we show that $n\\geq 3\\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-10-17T14:31:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "domination number",
      "eigenvalue",
      "nonbipartite graph",
      "dominating set",
      "minimum cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1310.4717Y"
    }
  }
}