{
  "id": "1410.2941",
  "version": "v1",
  "published": "2014-10-11T02:53:01.000Z",
  "updated": "2014-10-11T02:53:01.000Z",
  "title": "New inequalities on the hyperbolicity constant of line graphs",
  "authors": [
    "Walter Carballosa",
    "José M. Rodríguez",
    "José M. Sigarreta"
  ],
  "comment": "Accepted for publication in Ars Combinatoria",
  "categories": [
    "math.CO"
  ],
  "abstract": "If X is a geodesic metric space and $x_1,x_2,x_3\\in X$, a {\\it geodesic triangle} $T=\\{x_1,x_2,x_3\\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\\delta$-\\emph{hyperbolic} $($in the Gromov sense$)$ if any side of $T$ is contained in a $\\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. We denote by $\\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\\delta(X):=\\inf\\{\\delta\\ge 0: \\, X \\, \\text{ is $\\delta$-hyperbolic}\\,\\}\\,. $ The main result of this paper is the inequality $\\delta(G) \\le \\delta(\\mathcal L(G))$ for the line graph $\\mathcal L(G)$ of every graph $G$. We prove also the upper bound $\\delta(\\mathcal L(G)) \\le 5 \\delta(G)+ 3 l_{max}$, where $l_{max}$ is the supremum of the lengths of the edges of $G$. Furthermore, if every edge of $G$ has length $k$, we obtain $\\delta(G) \\le \\delta(\\mathcal L(G)) \\le 5 \\delta(G)+ 5k/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-11T02:53:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "line graph",
      "inequality",
      "geodesic triangle",
      "geodesic metric space",
      "sharp hyperbolicity constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.2941C"
    }
  }
}