{
  "id": "1901.00053",
  "version": "v1",
  "published": "2018-12-31T21:54:20.000Z",
  "updated": "2018-12-31T21:54:20.000Z",
  "title": "Spanning 2-Forests and Resistance Distance in 2-Connected Graphs",
  "authors": [
    "Wayne Barrett",
    "Emily J. Evans",
    "Amanda E. Francis",
    "Mark Kempton",
    "John Sinkovic"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. It is well-known that the number of these spanning 2-forests separating $u$ and $v$ in a graph is equal to the determinant of the matrix obtained from the combinatorial Laplacian matrix of the graph by deleting the rows and columns corresponding to $u$ and $v$. For most interesting graphs, neither of these quantities can be easily found. For any connected graph $G$ with a 2-separator and its associated decomposition which separates $u$ and $v$, we show that the number of spanning trees and spanning 2-forests separating $u$ and $v$ can be expressed in terms of the number of spanning trees and 2-forests in the smaller graphs in the decomposition, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if $u$ and $v$ are on the \"same side\" of the decomposition. We apply these results to derive an exact formula for the number of spanning 2-forests separating $u$ and $v$ for any linear 2-tree with a single bend. We focus in particular on the case in which $u$ and $v$ are the end (degree 2) vertices of the linear 2-tree, interpret these as resistance distances, and analyze these values with respect to the location of the bend.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-31T21:54:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C05",
      "94C15"
    ],
    "keywords": [
      "resistance distance",
      "spanning trees",
      "decomposition",
      "connected graph",
      "combinatorial laplacian matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}