{
  "id": "2008.08621",
  "version": "v1",
  "published": "2020-08-19T18:24:10.000Z",
  "updated": "2020-08-19T18:24:10.000Z",
  "title": "Symmetric edge polytopes and matching generating polynomials",
  "authors": [
    "Hidefumi Ohsugi",
    "Akiyoshi Tsuchiya"
  ],
  "comment": "17 pages",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "Symmetric edge polytopes $\\mathcal{A}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between $\\mathcal{A}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of $\\mathcal {A}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^*$-polynomial of $\\mathcal{A}_{\\widehat{G}}$ by using matching generating polynomials, where $\\widehat{G}$ is the suspension of $G$. This gives also a formula for the normalized volume of $\\mathcal{A}_{\\widehat{G}}$. Moreover, via the chemical graph theory, we show that for any cactus graph $G$, the $h^*$-polynomial of $\\mathcal{A}_{\\widehat{G}}$ is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type $B$, which are lattice polytopes arising from the root system $B_n$ and finite simple graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-19T18:24:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05C31",
      "13P10",
      "52B12",
      "52B20"
    ],
    "keywords": [
      "symmetric edge polytopes",
      "matching generating polynomials",
      "finite simple graphs",
      "lattice polytopes arising",
      "cactus graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}