{
  "id": "2006.05511",
  "version": "v1",
  "published": "2020-06-09T21:08:23.000Z",
  "updated": "2020-06-09T21:08:23.000Z",
  "title": "On the largest real root of the independence polynomial of a unicyclic graph",
  "authors": [
    "Iain Beaton",
    "Ben Cameron"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \\textit{independence roots} of $G$. It is known that for every graph $G$, the independence root of smallest modulus, denoted $\\xi(G)$, is real. The relation $\\preceq$ on the set of all graphs is defined as follows, $H\\preceq G$ if and only if $I(H,x)\\ge I(G,x)\\text{ for all }x\\in [\\xi(G),0].$ We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to $\\preceq$. This extends recent work by Oboudi where the maximum and minimum trees of a given order were determined and also answers an open question posed in the same work. Corollaries of our results give the graphs that minimize and maximize $\\xi(G)$ among all connected (well-covered) unicyclic graphs. We also answer more open questions posed by Oboudi and disprove a related conjecture due to Levit and Mandrescu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-09T21:08:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unicyclic graph",
      "largest real root",
      "independence polynomial",
      "open question",
      "minimum connected unicyclic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}