{
  "id": "2401.11873",
  "version": "v1",
  "published": "2024-01-22T12:00:25.000Z",
  "updated": "2024-01-22T12:00:25.000Z",
  "title": "Some Properties of Proper Power Graphs in Finite Abelian Groups",
  "authors": [
    "Dhawlath. G",
    "Raja. V"
  ],
  "comment": "12 pages and 2 figures",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "abstract": "The power graph of a group $G$, denoted as $P(G)$, constitutes a simple undirected graph characterized by its vertex set $G$. Specifically, vertices $a,b$ exhibit adjacency exclusively if $a$ belongs to the cyclic subgroup generated by $b$ or vice versa. The corresponding proper power graph of $G$ is obtained by taking $P(G)$ and removing a vertex corresponding to the identity element, which is denoted as $P^*(G)$. In the context of finite abelian groups, this article establishes the sufficient and necessary conditions for the proper power graph's connectedness. Moreover, a precise upper bound for the diameter of $P^*(G)$ in finite abelian groups is provided with sharpness. This article also explores the study of vertex connectivity, center, and planarity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-22T12:00:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "G.2"
    ],
    "keywords": [
      "finite abelian groups",
      "proper power graphs connectedness",
      "properties",
      "precise upper bound",
      "corresponding proper power graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}