{
  "id": "2207.04641",
  "version": "v1",
  "published": "2022-07-11T05:50:35.000Z",
  "updated": "2022-07-11T05:50:35.000Z",
  "title": "The complement of enhanced power graph of a finite group",
  "authors": [
    "Parveen",
    "Jitender Kumar"
  ],
  "comment": "7 figures",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "The enhanced power graph $\\mathcal{P}_E(G)$ of a finite group $G$ is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \\in \\langle z \\rangle$ for some $z \\in G$. In this article, we give an affirmative answer of the question posed by Cameron [6] which states that: Is it true that the complement of the enhanced power graph $\\bar{\\mathcal{P}_E(G)}$ of a non-cyclic group $G$ has only one connected component apart from isolated vertices? We classify all finite groups $G$ such that the graph $\\bar{\\mathcal{P}_E(G)}$ is bipartite. We show that the graph $\\bar{\\mathcal{P}_E(G)}$ is weakly perfect. Further, we study the subgraph $\\bar{\\mathcal{P}_E(G^*)}$ of $\\bar{\\mathcal{P}_E(G)}$ induced by all the non-isolated vertices of $\\bar{\\mathcal{P}_E(G)}$. We classify all finite groups $G$ such that the graph is $\\bar{\\mathcal{P}_E(G^*)}$ is unicyclic and pentacyclic. We prove the non-existence of finite groups $G$ such that the graph $\\bar{\\mathcal{P}_E(G^*)}$ is bicyclic, tricyclic or tetracyclic. Finally, we characterize all finite groups $G$ such that the graph $\\bar{\\mathcal{P}_E(G^*)}$ is outerplanar, planar, projective-planar and toroidal, respectively.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-11T05:50:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C25"
    ],
    "keywords": [
      "finite group",
      "enhanced power graph",
      "complement",
      "vertex set",
      "distinct vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}