{
  "id": "2206.10278",
  "version": "v1",
  "published": "2022-06-21T11:57:34.000Z",
  "updated": "2022-06-21T11:57:34.000Z",
  "title": "On Eccentricity Matrices of Wheel Graphs",
  "authors": [
    "I. Jeyaraman",
    "T. Divyadevi"
  ],
  "comment": "32 pages, one figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "The eccentricity matrix $E(G)$ of a simple connected graph $G$ is obtained from the distance matrix $D(G)$ of $G$ by retaining the largest distance in each row and column, and by defining the remaining entries to be zero. This paper focuses on the eccentricity matrix $E(W_n)$ of the wheel graph $W_n$ with $n$ vertices. By establishing a formula for the determinant of $E(W_n)$, we show that $E(W_n)$ is invertible if and only if $n \\not\\equiv 1\\Mod3$. We derive a formula for the inverse of $E(W_n)$ by finding a vector $\\mathbf{w}\\in \\mathbb{R}^n$ and an $n \\times n$ symmetric Laplacian-like matrix $\\widetilde{L}$ of rank $n-1$ such that \\begin{eqnarray*} E(W_n)^{-1} = -\\frac{1}{2}\\widetilde{L} + \\frac{6}{n-1}\\mathbf{w}\\mathbf{w^{\\prime}}. \\end{eqnarray*} Further, we prove an analogous result for the Moore-Penrose inverse of $E(W_n)$ for the singular case. We also determine the inertia of $E(W_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-21T11:57:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "eccentricity matrix",
      "wheel graph",
      "distance matrix",
      "largest distance",
      "simple connected graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}