{
  "id": "1804.11345",
  "version": "v1",
  "published": "2018-04-30T17:50:39.000Z",
  "updated": "2018-04-30T17:50:39.000Z",
  "title": "Linear maps on graphs preserving a given independence number",
  "authors": [
    "Yanan Hu",
    "Zejun Huang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Denote by $\\mathscr{G}_n$ the set of simple graphs with vertex set $\\{1,2,\\ldots,n\\}$. A complete linear map $\\phi: \\mathscr{G}_n \\rightarrow\\mathscr{G}_n$ is a map satisfying $\\phi(K_n)=K_n$ and $$\\phi(G_1\\cup G_2)=\\phi(G_1)\\cup \\phi(G_2)~~~~{\\rm for ~all~}G_1,G_2\\in \\mathscr{G}_n.$$ Given positive integers $n$ and $t$ such that $t\\le n-1$, we prove that a complete linear map $\\phi: \\mathscr{G}_n\\rightarrow \\mathscr{G}_n$ satisfies $$\\alpha(\\phi(G))=t~\\iff \\alpha(G)=t {~~~~\\rm for~ all~}G\\in \\mathscr{G}_n$$ iff $\\phi$ is an edge permutation when $t=1$ and $\\phi$ is a vertex permutation when $t\\ge 2$, where $\\alpha(G)$ is the independence number of a graph $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-30T17:50:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independence number",
      "complete linear map",
      "graphs preserving",
      "vertex set",
      "vertex permutation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}