{
  "id": "1307.3809",
  "version": "v1",
  "published": "2013-07-15T02:37:47.000Z",
  "updated": "2013-07-15T02:37:47.000Z",
  "title": "The Euler characteristic of an even-dimensional graph",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "16 pages, 4 figures",
  "categories": [
    "math.GT",
    "cs.DM"
  ],
  "abstract": "We write the Euler characteristic X(G) of a four dimensional finite simple geometric graph G=(V,E) in terms of the Euler characteristic X(G(w)) of two-dimensional geometric subgraphs G(w). The Euler curvature K(x) of a four dimensional graph satisfying the Gauss-Bonnet relation sum_x K(x) = X(G) can so be rewritten as an average 1-E[K(x,f)]/2 over a collection two dimensional \"sectional graph curvatures\" K(x,f) through x. Since scalar curvature, the average of all these two dimensional curvatures through a point, is the integrand of the Hilbert action, the integer 2-2 X(G) becomes an integral-geometrically defined Hilbert action functional. The result has an interpretation in the continuum for compact 4-manifolds M: the Euler curvature K(x), the integrand in the classical Gauss-Bonnet-Chern theorem, can be seen as an average over a probability space W of 1-K(x,w)/2 with curvatures K(x,w) of compact 2-manifolds M(w). Also here, the Euler characteristic has an interpretation of an exotic Hilbert action, in which sectional curvatures are replaced by surface curvatures of integral geometrically defined random two-dimensional sub-manifolds M(w) of M.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-07-15T02:37:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C50",
      "81Q10"
    ],
    "keywords": [
      "euler characteristic",
      "even-dimensional graph",
      "defined random two-dimensional sub-manifolds",
      "defined hilbert action functional",
      "geometrically defined random two-dimensional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.3809K"
    }
  }
}