{
  "id": "1707.06533",
  "version": "v1",
  "published": "2017-07-19T15:46:46.000Z",
  "updated": "2017-07-19T15:46:46.000Z",
  "title": "The distinguishing number and the distinguishing index of co-normal product of two graphs",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "8 pages. arXiv admin note: text overlap with arXiv:1703.01874",
  "categories": [
    "math.CO"
  ],
  "abstract": "The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The co-normal product $G\\star H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\\times V (H)$ and edge set $\\{\\{(x_1, x_2), (y_1, y_2)\\} | x_1y_1 \\in E(G) ~{\\rm or}~x_2y_2 \\in E(H)\\}$. In this paper we study the distinguishing number and the distinguishing index of the co-normal product of two graphs. We prove that for every $k \\geq 3$, the $k$-th co-normal power of a connected graph $G$ with no false twin vertex and no dominating vertex, has the distinguishing number and the distinguishing index equal two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-19T15:46:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05C60"
    ],
    "keywords": [
      "distinguishing number",
      "co-normal product",
      "false twin vertex",
      "th co-normal power",
      "distinguishing index equal"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}