{
  "id": "2012.12705",
  "version": "v1",
  "published": "2020-12-23T14:32:35.000Z",
  "updated": "2020-12-23T14:32:35.000Z",
  "title": "On distance matrices of helm graphs obtained from wheel graphs with an even number of vertices",
  "authors": [
    "Shivani Goel"
  ],
  "categories": [
    "math.CO",
    "math.FA"
  ],
  "abstract": "Let $n \\geq 4$. The helm graph $H_n$ on $2n-1$ vertices is obtained from the wheel graph $W_n$ by adjoining a pendant edge to each vertex of the outer cycle of $W_n$. Suppose $n$ is even. Let $D := [d_{ij}]$ be the distance matrix of $H_n$. In this paper, we first show that $\\det(D) = 3(n-1)2^{n-1}.$ Next, we find a matrix $\\L$ and a vector $u$ such that \\[D^{-1} = -\\frac{1}{2}\\L+\\frac{4}{3(n-1)}uu'.\\] We also prove an interlacing property between the eigenvalues of $\\L$ and $D$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-23T14:32:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50"
    ],
    "keywords": [
      "distance matrix",
      "helm graph",
      "wheel graph",
      "outer cycle",
      "pendant edge"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}