{
  "id": "1212.5261",
  "version": "v1",
  "published": "2012-12-20T12:51:17.000Z",
  "updated": "2012-12-20T12:51:17.000Z",
  "title": "On signless Laplacian coefficients of bicyclic graphs",
  "authors": [
    "Jie Zhang",
    "Xiao-Dong Zhang"
  ],
  "comment": "22 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1212.5008",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph of order $n$ and $Q_G(x)= det(xI-Q(G))= \\sum_{i=1}^n (-1)^i \\varphi_i x^{n-i}$ be the characteristic polynomial of the signless Laplacian matrix of a graph $G$. We give some transformations of $G$ which decrease all signless Laplacian coefficients in the set $\\mathcal{B}(n)$ of all $n$-vertex bicyclic graphs. $\\mathcal{B}^1(n)$ denotes all n-vertex bicyclic graphs with at least one odd cycle. We show that $B_n^1$ (obtained from $C_4$ by adding one edge between two non-adjacent vertices and adding $n-4$ pendent vertices at the vertex of degree 3) minimizes all the signless Laplacian coefficients in the set $\\mathcal{B}^1(n)$. Moreover, we prove that $B_n^2$ (obtained from $K_{2,3}$ by adding $n-5$ pendent vertices at one vertex of degree 3) has minimum signless Laplacian coefficients in the set $\\mathcal{B}^2(n)$ of all $n$-vertex bicyclic graphs with two even cycles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-12-20T12:51:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "pendent vertices",
      "minimum signless laplacian coefficients",
      "n-vertex bicyclic graphs",
      "signless laplacian matrix",
      "characteristic polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1212.5261Z"
    }
  }
}