{
  "id": "2208.10897",
  "version": "v1",
  "published": "2022-08-23T12:03:31.000Z",
  "updated": "2022-08-23T12:03:31.000Z",
  "title": "The Moore-Penrose Inverse of the Distance Matrix of a Helm Graph",
  "authors": [
    "I. Jeyaraman",
    "T. Divyadevi",
    "R. Azhagendran"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we give necessary and sufficient conditions for a real symmetric matrix, and in particular, for the distance matrix $D(H_n)$ of a helm graph $H_n$ to have their Moore-Penrose inverses as the sum of a symmetric Laplacian-like matrix and a rank one matrix. As a consequence, we present a short proof of the inverse formula, given by Goel (Linear Algebra Appl. 621:86--104, 2021), for $D(H_n)$ when $n$ is even. Further, we derive a formula for the Moore-Penrose inverse of singular $D(H_n)$ that is analogous to the formula for $D(H_n)^{-1}$. Precisely, if $n$ is odd, we find a symmetric positive semidefinite Laplacian-like matrix $L$ of order $2n-1$ and a vector $\\mathbf{w}\\in \\mathbb{R}^{2n-1}$ such that \\begin{eqnarray*} D(H_n)\\ssymbol{2} = -\\frac{1}{2}L + \\frac{4}{3(n-1)}\\mathbf{w}\\mathbf{w^{\\prime}}, \\end{eqnarray*} where the rank of $L$ is $2n-3$. We also investigate the inertia of $D(H_n)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-23T12:03:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C50",
      "15A09"
    ],
    "keywords": [
      "moore-penrose inverse",
      "distance matrix",
      "helm graph",
      "linear algebra appl",
      "real symmetric matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}