{
  "id": "math/0411239",
  "version": "v1",
  "published": "2004-11-10T18:13:58.000Z",
  "updated": "2004-11-10T18:13:58.000Z",
  "title": "Very well-covered graphs with log-concave independence polynomials",
  "authors": [
    "Vadim E. Levit",
    "Eugen Mandrescu"
  ],
  "comment": "8 pages, 4 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "If for any $k$ the $k$-th coefficient of a polynomial $I(G;x)$ is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Alavi, Malde, Schwenk and Erdos (1987) conjectured that $I(G;x)$ is unimodal, whenever $G$ is a forest, while Brown, Dilcher and Nowakowski (2000) conjectured that $I(G;x)$ is unimodal for any well-covered graph G. Michael and Traves (2003) showed that the assertion is false for well-covered graphs with $a(G)$ > 3 ($a(G)$ is the size of a maximum stable set of the graph $G$), while for very well-covered graphs the conjecture is still open. In this paper we give support to both conjectures by demonstrating that if $a(G)$ < 4, or $G$ belongs to ${K_{1,n}, P_{n}: n > 0}$, then $I(G*;x)$ is log-concave, and, hence, unimodal (where $G*$ is the very well-covered graph obtained from $G$ by appending a single pendant edge to each vertex).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-11-10T18:13:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05E99",
      "11B83",
      "05A20"
    ],
    "keywords": [
      "well-covered graph",
      "log-concave independence polynomials",
      "conjecture",
      "single pendant edge",
      "th coefficient"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....11239L"
    }
  }
}