{
  "id": "1905.02118",
  "version": "v1",
  "published": "2019-05-06T16:06:21.000Z",
  "updated": "2019-05-06T16:06:21.000Z",
  "title": "The average simplex cardinality of a finite abstract simplicial complex",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "19 page, 8 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We study the average simplex cardinality Dim^+(G) = sum_x |x|/(|G|+1) of a finite abstract simplicial complex G. The functional is a homomorphism from the monoid of simplicial complexes to the rationals: the formula Dim^+(G + H) = Dim^+(G) + Dim^+(H) holds for the join + similarly as for the augmented inductive dimension dim^+(G) = dim(G)+1 where dim is the inductive dimension dim(G) = 1+ sum_x dim(S(x))/|G| with unit sphere S(x) (a recent theorem of Betre and Salinger). In terms of the generating function f(t) = 1+v_0 t + v_1 t^2 + ... +v_d t^(d+1) defined by the f-vector (v_0,v_1, \\dots) of G for which f(-1) is the genus 1-X(G) with Euler characteristic X and f(1)=|G|+1 is the augmented number of simplices, the average cardinality is the logarithmic derivative Dim^+(f) = f'(1)/f(1) of f at 1. Beside introducing the average cardinality and establishing its compatibility with arithmetic, we prove two results: 1) the inequality dim^+(G)/2 <= Dim^+(G) with equality for complete complexes. 2) the limit C_d of Dim^+(G_n) for n to infinity is the same for any initial complex G_0 of maximal dimension d and the constant c_d is explicitly given in terms of the Perron-Frobenius eigenfunction of the universal Barycentric refinement operator and is for positive d always a rational number in the open interval ((d+1)/2,d+1).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-06T16:06:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05Cxx",
      "05Exx",
      "68Rxx",
      "54F45",
      "55U10"
    ],
    "keywords": [
      "finite abstract simplicial complex",
      "average simplex cardinality",
      "average cardinality",
      "universal barycentric refinement operator",
      "inductive dimension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}