{
  "id": "1012.0287",
  "version": "v2",
  "published": "2010-12-01T20:26:47.000Z",
  "updated": "2011-09-23T06:26:25.000Z",
  "title": "Chip-Firing and Riemann-Roch Theory for Directed Graphs",
  "authors": [
    "Arash Asadi",
    "Spencer Backman"
  ],
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We investigate Riemann-Roch theory for directed graphs. The Riemann-Roch criteria of Amini and Manjunath is generalized to all integer lattices orthogonal to some positive vector. Using generalized notions of a $v_0$-reduced divisor and Dhar's algorithm we investigate two chip-firing games coming from the rows and columns of the Laplacian of a strongly connected directed graph. We discuss how the \"column\" chip-firing game is related to directed $\\vec{G}$-parking functions and the \"row\" chip-firing game is related to the sandpile model. We conclude with a discussion of arithmetical graphs, which after a simple transformation may be viewed as a special class of directed graphs which will always have the Riemann-Roch property for the column chip-firing game. Examples of arithmetical graphs are provided which demonstrate that either, both, or neither of the two Riemann-Roch conditions may be satisfied for the row chip-firing game.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-09-23T06:26:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "directed graph",
      "riemann-roch theory",
      "integer lattices orthogonal",
      "arithmetical graphs",
      "special class"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.0287A"
    }
  }
}