{
  "id": "math/0201211",
  "version": "v1",
  "published": "2002-01-22T18:51:42.000Z",
  "updated": "2002-01-22T18:51:42.000Z",
  "title": "The kernel of the adjacency matrix of a rectangular mesh",
  "authors": [
    "Carlos Tomei",
    "Tania Vieira"
  ],
  "comment": "15 pages, 17 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given an m x n rectangular mesh, its adjacency matrix A, having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K. We describe the kernel of A: it is a direct sum of two natural subspaces whose dimensions are equal to $\\lceil c/2 \\rceil$ and $\\lfloor c/2 \\rfloor$, where c = gcd (m+1,n+1) - 1. We show that there are bases to both vector spaces, with entries equal to 0, 1 and -1. When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-01-22T18:51:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05B45",
      "05C50"
    ],
    "keywords": [
      "adjacency matrix",
      "rectangular mesh",
      "vector spaces",
      "integer entries",
      "integer sides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......1211T"
    }
  }
}