{
  "id": "2006.03289",
  "version": "v1",
  "published": "2020-06-05T08:17:45.000Z",
  "updated": "2020-06-05T08:17:45.000Z",
  "title": "On distance matrices of wheel graphs with odd number of vertices",
  "authors": [
    "R. Balaji",
    "R. B. Bapat",
    "Shivani Goel"
  ],
  "categories": [
    "math.CO",
    "math.FA"
  ],
  "abstract": "Let $W_n$ denote the wheel graph having $n$-vertices. If $i$ and $j$ are any two vertices of $W_n$, define \\[d_{ij}:= \\begin{cases} 0 & \\mbox{if}~i=j \\\\ 1 & \\mbox{if}~i~ \\mbox{and} ~j~ \\mbox{are adjacent} \\\\ 2 & \\mbox{else}. \\end{cases}\\] Let $D$ be the $n \\times n$ matrix with $(i,j)^{\\rm th}$ entry equal to $d_{ij}$. The matrix $D$ is called the distance matrix of $W_n$. Suppose $n \\geq 5$ is an odd integer. In this paper, we deduce a formula to compute the Moore-Penrose inverse of $D$. More precisely, we obtain an $n\\times n$ matrix $\\widetilde{L}$ and a rank one matrix $ww'$ such that \\[D^\\dagger = -\\frac{1}{2} \\widetilde{L}+\\frac{4}{n-1}ww'.\\] Here, $\\widetilde{L}$ is positive semidefinite, ${\\rm rank}(\\widetilde{L})=n-2$ and all row sums are equal to zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-05T08:17:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "distance matrix",
      "wheel graph",
      "odd number",
      "entry equal",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}