{
  "id": "2009.12680",
  "version": "v1",
  "published": "2020-09-26T20:33:18.000Z",
  "updated": "2020-09-26T20:33:18.000Z",
  "title": "Generalizing Kirchhoff laws for Signed Graphs",
  "authors": [
    "Lucas J. Rusnak",
    "Josephine Reynes",
    "Skyler J. Johnson",
    "Peter Ye"
  ],
  "comment": "22 pages, 16 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Kirchhoff-type Laws for signed graphs are characterized by generalizing transpedances through the incidence-oriented structure of bidirected graphs. The classical $2$-arborescence interpretation of Tutte is shown to be equivalent to single-element Boolean classes of reduced incidence-based cycle covers, called contributors. A generalized contributor-transpedance is introduced using entire Boolean classes that naturally cancel in a graph; classical conservation is proven to be property of the trivial Boolean classes. The contributor-transpedances on signed graphs are shown to produce non-conservative Kirchhoff-type Laws, where every contributor possesses the unique source-sink path property. Finally, the maximum value of a contributor-transpedance is calculated through the signless Laplacian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-26T20:33:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C22",
      "05C05",
      "05C50",
      "05B20",
      "05B45"
    ],
    "keywords": [
      "signed graphs",
      "generalizing kirchhoff laws",
      "unique source-sink path property",
      "single-element boolean classes",
      "produce non-conservative kirchhoff-type laws"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}