{
  "id": "1601.04431",
  "version": "v1",
  "published": "2016-01-18T09:05:06.000Z",
  "updated": "2016-01-18T09:05:06.000Z",
  "title": "Normal Subgroup Based Power Graph of a finite Group",
  "authors": [
    "A. K. Bhuniya",
    "Sudip Bera"
  ],
  "comment": "14 pages, 4 figures",
  "categories": [
    "math.CO",
    "math.GR"
  ],
  "abstract": "For a finite group $G$ with a normal subgroup $H$, the normal subgroup based power graph of $G$, denoted by $\\Gamma_H(G)$ whose vertex set $V(\\Gamma_H(G))=(G\\setminus H)\\bigcup \\{e\\}$ and two vertices $a$ and $b$ are edge connected if $aH=b^mH$ or $bH=a^nH$ for some $m, n \\in \\mathbb{N}$. In this paper we obtain some fundamental characterizations of the normal subgroup based power graph. We show some relation between the graph $\\Gamma_H(G)$ and the power graph $\\Gamma(\\frac{G}{H})$. We show that $\\Gamma_H(G)$ is complete if and only of $\\frac{G}{H}$ is cyclic group of order $1$ or $p^m$, where $p$ is prime number and $m\\in \\mathbb{N}$. $\\Gamma_H(G)$ is planar if and only if $|H|=2$ or $3$ and $\\frac{G}{H}\\cong \\mathbb{Z}_2\\times \\mathbb{Z}_2 \\times \\cdots \\times \\mathbb{Z}_2$. Also $\\Gamma_H(G)$ is Eulerian if and only if $|G|\\equiv |H|$ mod$ 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-18T09:05:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25",
      "05C50"
    ],
    "keywords": [
      "power graph",
      "normal subgroup",
      "finite group",
      "vertex set",
      "fundamental characterizations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}