Resistors in dual networks

Abstract

Let $G$ be a finite plane multigraph and $G^{\prime }$ its dual. Each edge $e$ of $G$ is interpreted as a resistor of resistance $R_e$, and the dual edge $e^{\prime }$ is assigned the dual resistance $R_{e^{\prime }}:=1/R_e$. Then the equivalent resistance $r_e$ over $e$ and the equivalent resistance $r_{e^{\prime }}$ over $e^{\prime }$ satisfy $r_e/R_e+r_{e^{\prime }}/R_{e^{\prime }}=1$. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in $G$ and $G^{\prime }$ respectively.

7 pages, 2 figures