{
  "id": "1709.01227",
  "version": "v1",
  "published": "2017-09-05T03:50:10.000Z",
  "updated": "2017-09-05T03:50:10.000Z",
  "title": "Electrical networks and hyperplane arrangements",
  "authors": [
    "Bob Lutz"
  ],
  "comment": "19 pages, 4 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "This paper connects the theory of hyperplane arrangements to the theory of linear resistor networks with fixed boundary voltages. Given a graph $G$, a set $\\partial V\\subsetneq V$, and a function $u:\\partial V\\to\\mathbb{R}$, our main object of study is the arrangement $\\mathcal{A}_{G,u}$ obtained from the real graphic arrangement $\\mathcal{A}_G$ by fixing the coordinate $x_j$ to $u(j)$ for all $j\\in\\partial V$. First, fixed-energy harmonic functions in the sense of Abrams and Kenyon are shown to be critical points of master functions in the sense of Varchenko. Second, the basic graph-theoretic descriptions of $\\mathcal{A}_G$ are generalized to $\\mathcal{A}_{G,u}$. It is also proven that the arrangements $\\mathcal{A}_{G,u}$ are equivalent to the $\\psi$-graphical arrangements introduced recently by Stanley.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-05T03:50:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C35",
      "34B45",
      "05C15"
    ],
    "keywords": [
      "hyperplane arrangements",
      "electrical networks",
      "real graphic arrangement",
      "basic graph-theoretic descriptions",
      "linear resistor networks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}