{
  "id": "2006.02841",
  "version": "v1",
  "published": "2020-06-04T13:20:19.000Z",
  "updated": "2020-06-04T13:20:19.000Z",
  "title": "An inverse formula for the distance matrix of a wheel graph with even number of vertices",
  "authors": [
    "R. Balaji",
    "R. B. Bapat",
    "Shivani Goel"
  ],
  "categories": [
    "math.CO",
    "math.FA"
  ],
  "abstract": "Let $n \\geq 4$ be an even integer and $W_n$ be the wheel graph with $n$ vertices. The distance $d_{ij}$ between any two distinct vertices $i$ and $j$ of $W_n$ is the length of the shortest path connecting $i$ and $j$. Let $D$ be the $n \\times n$ symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to $d_{ij}$. In this paper, we find a positive semidefinite matrix $\\widetilde{L}$ such that ${\\rm rank}(\\widetilde{L})=n-1$, all row sums of $\\widetilde{L}$ equal to zero and a rank one matrix $ww^T$ such that \\[D^{-1}=-\\frac{1}{2}\\widetilde{L} + \\frac{4}{n-1}ww^T. \\] An interlacing property between the eigenvalues of $D$ and $\\widetilde{L}$ is also proved.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-04T13:20:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "wheel graph",
      "distance matrix",
      "inverse formula",
      "off-diagonal entries equal",
      "shortest path"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}