{
  "id": "1907.03369",
  "version": "v1",
  "published": "2019-07-07T23:47:46.000Z",
  "updated": "2019-07-07T23:47:46.000Z",
  "title": "The energy of a simplicial complex",
  "authors": [
    "Oliver Knill"
  ],
  "comment": "34 pages",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g of L has integer entries g(x,y). In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy summing all matrix elements g(x,y) is equal to the Euler characteristic X(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to X(G).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-07T23:47:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C10",
      "57M15",
      "68R10"
    ],
    "keywords": [
      "finite abstract simplicial complex",
      "green function entries define",
      "euclidean spaces",
      "integer entries",
      "potential energy"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}