{
  "id": "2201.09835",
  "version": "v1",
  "published": "2022-01-24T17:52:25.000Z",
  "updated": "2022-01-24T17:52:25.000Z",
  "title": "On the gamma-vector of symmetric edge polytopes",
  "authors": [
    "Alessio D'Alì",
    "Martina Juhnke-Kubitzke",
    "Daniel Köhne",
    "Lorenzo Venturello"
  ],
  "comment": "29 pages, 4 figures. Comments are very welcome!",
  "categories": [
    "math.CO"
  ],
  "abstract": "We study $\\gamma$-vectors associated with $h^*$-vectors of symmetric edge polytopes both from a deterministic and a probabilistic point of view. On the deterministic side, we prove nonnegativity of $\\gamma_2$ for any graph and completely characterize the case when $\\gamma_2 = 0$. The latter also confirms a conjecture by Lutz and Nevo in the realm of symmetric edge polytopes. On the probabilistic side, we show that the $\\gamma$-vectors of symmetric edge polytopes of most Erd\\H{o}s-R\\'enyi random graphs are asymptotically almost surely nonnegative up to any fixed entry. This proves that Gal's conjecture holds asymptotically almost surely for arbitrary unimodular triangulations in this setting.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-24T17:52:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "05C80",
      "52B05",
      "52B12",
      "05E45"
    ],
    "keywords": [
      "symmetric edge polytopes",
      "gamma-vector",
      "arbitrary unimodular triangulations",
      "gals conjecture holds",
      "probabilistic point"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}