{
  "id": "2307.04740",
  "version": "v1",
  "published": "2023-07-10T17:52:19.000Z",
  "updated": "2023-07-10T17:52:19.000Z",
  "title": "On the image of graph distance matrices",
  "authors": [
    "William Dudarov",
    "Noah Feinberg",
    "Raymond Guo",
    "Ansel Goh",
    "Andrea Ottolini",
    "Alicia Stepin",
    "Raghavenda Tripathi",
    "Joia Zhang"
  ],
  "comment": "4 figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "Let $G=(V,E)$ be a finite, simple, connected, combinatorial graph on $n$ vertices and let $D \\in \\mathbb{R}^{n \\times n}$ be its graph distance matrix $D_{ij} = d(v_i, v_j)$. Steinerberger (J. Graph Theory, 2023) empirically observed that the linear system of equations $Dx =\\mathbf{1}$, where $\\mathbf{1} = (1,1,\\dots, 1)^{T}$, very frequently has a solution (even in cases where $D$ is not invertible). The smallest nontrivial example of a graph where the linear system is not solvable are two graphs on 7 vertices. We prove that, in fact, counterexamples exists for all $n\\geq 7$. The construction is somewhat delicate and further suggests that such examples are perhaps rare. We also prove that for Erd\\H{o}s-R\\'enyi random graphs the graph distance matrix $D$ is invertible with high probability. We conclude with some structural results on the Perron-Frobenius eigenvector for a distance matrix.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-10T17:52:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C50"
    ],
    "keywords": [
      "graph distance matrix",
      "linear system",
      "smallest nontrivial example",
      "graph theory",
      "combinatorial graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}