{
  "id": "1403.1916",
  "version": "v1",
  "published": "2014-03-08T01:46:59.000Z",
  "updated": "2014-03-08T01:46:59.000Z",
  "title": "On Zero-free Intervals of Flow Polynomials",
  "authors": [
    "Fengming Dong"
  ],
  "comment": "26 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "This article studies real roots of the flow polynomial $F(G,\\lambda)$ of a bridgeless graph $G$. For any integer $k\\ge 0$, let $\\xi_k$ be the supremum in $(1,2]$ such that $F(G,\\lambda)$ has no real roots in $(1,\\xi_k)$ for all graphs $G$ with $|W(G)|\\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $\\xi_k$ can be determined by considering a finite set of graphs and show that $\\xi_k=2$ for $k\\le 2$, $\\xi_3=1.430\\cdots$, $\\xi_4=1.361\\cdots$ and $\\xi_5=1.317\\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,\\lambda)$ are real but some of these roots are not in the set $\\{1,2,3\\}$, then $|E|\\ge |V|+17$ and $F(G,\\lambda)$ has at least 9 real roots in $(1,2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-08T01:46:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15"
    ],
    "keywords": [
      "flow polynomial",
      "zero-free intervals",
      "article studies real roots",
      "bridgeless graph",
      "degrees larger"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.1916D"
    }
  }
}