{
  "id": "2407.01227",
  "version": "v1",
  "published": "2024-07-01T12:18:50.000Z",
  "updated": "2024-07-01T12:18:50.000Z",
  "title": "Non-intersecting Paths and the Determinant of the Distance Matrix of a Tree",
  "authors": [
    "Emmanuel Briand",
    "Luis Esquivias-Quintero",
    "Álvaro Gutiérrez",
    "Adrián Lillo",
    "Mercedes Rosas"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We present the first combinatorial proof of the Graham-Pollak Formula for the determinant of the distance matrix of a tree, via sign-reversing involutions and the Lindstr\\\"om-Gessel-Viennot Lemma. Our approach provides a cohesive and unified framework for the understanding of the existing generalizations and $q$-analogues of the Graham-Pollak Formula, and facilitates the derivation of a natural simultaneous generalizations for them.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-01T12:18:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05A19",
      "05A15"
    ],
    "keywords": [
      "distance matrix",
      "non-intersecting paths",
      "determinant",
      "graham-pollak formula",
      "first combinatorial proof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}