{
  "id": "1311.7238",
  "version": "v2",
  "published": "2013-11-28T08:27:42.000Z",
  "updated": "2014-11-15T09:05:21.000Z",
  "title": "A Combinatorial Formula for Principal Minors of a Matrix with Tree-metric Exponents and Its Applications",
  "authors": [
    "Hiroshi Hirai",
    "Akihiro Yabe"
  ],
  "comment": "16 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $T$ be a tree with a vertex set $\\{ 1,2,\\dots, N \\}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-11-28T08:27:42.000Z",
      "abstract": "Let $T$ be a tree with a vertex set $\\{ 1,\\dots, N \\}$. Denote by $d_{ij}$ the distance between vertices $i$ and $j$. In this paper, we present an explicit combinatorial formula of principal minors of the matrix $(t^{d_{ij}})$, and its applications to tropical geometry, study of multivariate stable polynomials, and representation of valuated matroids. We also give an analogous formula for a skew-symmetric matrix associated with $T$.",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-15T09:05:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "principal minors",
      "tree-metric exponents",
      "applications",
      "explicit combinatorial formula",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.7238H"
    }
  }
}