{
  "id": "1810.12258",
  "version": "v1",
  "published": "2018-10-29T17:11:15.000Z",
  "updated": "2018-10-29T17:11:15.000Z",
  "title": "Reflexive polytopes arising from bipartite graphs with $γ$-positivity associated to interior polynomials",
  "authors": [
    "Hidefumi Ohsugi",
    "Akiyoshi Tsuchiya"
  ],
  "comment": "17 pages, 2 figures",
  "categories": [
    "math.CO",
    "math.AC"
  ],
  "abstract": "In this paper, we introduce polytopes ${\\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\\mathcal B}_G$ is a reflexive polytope with a regular unimodular triangulation if and only if $G$ is bipartite. This implies that the $h^*$-polynomial of ${\\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, the $\\gamma$-positivity and the real-rootedness of the $h^*$-polynomials. In fact, if $G$ is bipartite, then the $h^*$-polynomial of ${\\mathcal B}_G$ is $\\gamma$-positive and its $\\gamma$-polynomial is given by an interior polynomial (a version of Tutte polynomial of a hypergraph). Moreover, the $h^*$-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-10-29T17:11:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A15",
      "05C31",
      "13P10",
      "52B12",
      "52B20"
    ],
    "keywords": [
      "reflexive polytopes arising",
      "bipartite graphs",
      "positivity",
      "regular unimodular triangulation",
      "finite graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}