{
  "id": "1603.02711",
  "version": "v1",
  "published": "2016-03-08T21:29:24.000Z",
  "updated": "2016-03-08T21:29:24.000Z",
  "title": "Spectral radius and fractional matchings in graphs",
  "authors": [
    "Suil O"
  ],
  "journal": "European Journal of Combinatorics, Volume 55, 2016, Pages 144-148",
  "doi": "10.1016/j.ejc.2016.02.004",
  "categories": [
    "math.CO"
  ],
  "abstract": "A {\\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\\sum_{e \\in \\Gamma(v)} f(e) \\le 1$ for each $v\\in V(G)$, where $\\Gamma(v)$ is the set of edges incident to $v$. The {\\it fractional matching number} of $G$, written $\\alpha'_*(G)$, is the maximum of $\\sum_{e \\in E(G)} f(e)$ over all fractional matchings $f$. Let $G$ be an $n$-vertex connected graph with minimum degree $d$, let $\\lambda_1(G)$ be the largest eigenvalue of $G$, and let $k$ be a positive integer less than $n$. In this paper, we prove that if $\\lambda_1(G) < d\\sqrt{1+\\frac{2k}{n-k}}$, then $\\alpha'_*(G) > \\frac{n-k}{2}$. As a result, we prove $\\alpha'_*(G) \\ge \\frac{nd^2}{\\lambda_1(G)^2 + d^2}$, we characterize when equality holds in the bound.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-08T21:29:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "05C70"
    ],
    "keywords": [
      "spectral radius",
      "equality holds",
      "edges incident",
      "fractional matching number",
      "vertex connected graph"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160302711O"
    }
  }
}