{
  "id": "1606.03751",
  "version": "v1",
  "published": "2016-06-12T18:37:23.000Z",
  "updated": "2016-06-12T18:37:23.000Z",
  "title": "Distinguishing number and distinguishing index of neighbourhood corona of two graphs",
  "authors": [
    "Saeid Alikhani",
    "Samaneh Soltani"
  ],
  "comment": "15 pages, 11 figures",
  "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 neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \\star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\\star G_2$. We obtain upper bounds for $D(G_1\\star G_2)$ and $D'(G_1\\star G_2)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-12T18:37:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "05E18"
    ],
    "keywords": [
      "distinguishing number",
      "neighbourhood corona",
      "distinguishing index",
      "trivial automorphism",
      "th vertex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}