{
  "id": "1709.00706",
  "version": "v1",
  "published": "2017-09-03T12:01:10.000Z",
  "updated": "2017-09-03T12:01:10.000Z",
  "title": "The Automorphism Group of the Reduced Complete-Empty $X-$Join of Graphs",
  "authors": [
    "Adel Tadayyonfar",
    "Ali Reza Ashrafi"
  ],
  "comment": "19 pages, 0 figure",
  "categories": [
    "math.GR"
  ],
  "abstract": "Suppose $X$ is a simple graph. The $X-$join $\\Gamma$ of a set of complete or empty graphs $\\{X_x \\}_{x \\in V(X)}$ is a simple graph with the following vertex and edge sets: \\begin{eqnarray*} V(\\Gamma) &=& \\{(x,y) \\ | \\ x \\in V(X) \\ \\& \\ y \\in V(X_x) \\},\\\\ E(\\Gamma) &=& \\{(x,y)(x^\\prime,y^\\prime) \\ | \\ xx^\\prime \\in E(X) \\ or \\ else \\ x = x^\\prime \\ \\& \\ yy^\\prime \\in E(X_x)\\}. \\end{eqnarray*} The $X-$join graph $\\Gamma$ is called reduced if for vertices $x, y \\in V(X)$, $x \\ne y$, $N_X(x) \\setminus \\{ y\\} = N_X(y) \\setminus \\{ x\\}$ implies that $(i)$ if $xy \\not\\in E(X)$ then the graphs $X_x$ or $X_y$ are non-empty; $(ii)$ if $xy \\in E(X)$ then $X_x$ or $X_y$ are not complete graphs. In this paper, we want to explore how the graph theoretical properties of $X-$join of graphs effect on its automorphism group. Among other results we compute the automorphism group of reduced complete-empty $X-$join of graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-03T12:01:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20B25",
      "05C50"
    ],
    "keywords": [
      "automorphism group",
      "reduced complete-empty",
      "simple graph",
      "graphs effect",
      "graph theoretical properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}