{
  "id": "1710.00252",
  "version": "v1",
  "published": "2017-09-30T19:35:06.000Z",
  "updated": "2017-09-30T19:35:06.000Z",
  "title": "Laplacian Simplices Associated to Digraphs",
  "authors": [
    "Gabriele Balletti",
    "Takayuki Hibi",
    "Marie Meyer",
    "Akiyoshi Tsuchiya"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We associate to a finite digraph $D$ a lattice polytope $P_D$ whose vertices are the rows of the Laplacian matrix of $D$. This generalizes a construction introduced by Braun and the third author. As a consequence of the Matrix-Tree Theorem, we show that the normalized volume of $P_D$ equals the complexity of $D$, and $P_D$ contains the origin in its relative interior if and only if $D$ is strongly connected. Interesting connections with other families of simplices are established and then used to describe reflexivity, $h^*$-polynomial, and integer decomposition property of $P_D$ in these cases. We extend Braun and Meyer's study of cycles by considering cycle digraphs. In this setting we characterize reflexivity and show there are only four non-trivial reflexive Laplacian simplices having the integer decomposition property.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-30T19:35:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "05C20"
    ],
    "keywords": [
      "integer decomposition property",
      "non-trivial reflexive laplacian simplices",
      "lattice polytope",
      "considering cycle digraphs",
      "meyers study"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}