Resistors in dual networks

Abstract

Let G be a finite plane multigraph and G ' its dual. Each edge e of G is interpreted as a resistor of resistance R e , and the dual edge e' is assigned the dual resistance R e ' :=1/ R e . Then the equivalent resistance r e over e and the equivalent resistance r e' over e ' satisfy r e / R e + r e ' / R e ' =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 ' respectively.