{
  "id": "2407.19771",
  "version": "v1",
  "published": "2024-07-29T08:10:25.000Z",
  "updated": "2024-07-29T08:10:25.000Z",
  "title": "Characteristic Polynomial of Power Graphs on Direct Product of Any Two Finite Cyclic Groups",
  "authors": [
    "Komal Kumari",
    "Pratima Panigrahi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The power graph $\\mathscr{P}(G)$ of a group $G$ is defined as the simple graph with vertex set $G$, and where two distinct vertices $x$ and $y$ are joined by an edge if and only if either $x= y^k$ or $y= x^k$, $k \\in \\mathbb{N}$. Here we determine the characteristic polynomial of $\\mathscr{P}(\\mathbb{Z}_m \\times \\mathbb{Z}_{n})$ for any positive integers $m$ and $n$. Additionally, for some particular values of $m$ and $n$, we simplify the above characteristic polynomials and provide the full spectrum in a few cases.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-29T08:10:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C75",
      "05C50",
      "05C25"
    ],
    "keywords": [
      "characteristic polynomial",
      "finite cyclic groups",
      "power graph",
      "direct product",
      "simple graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}