Resistors in dual networks - Martina Furrer; Norbert Hungerbühler; Simon Jantschgi (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2020)
Let G be a finite plane multigraph and G' its dual. Each edge e of G is interpreted as a resistor of resistance Re, and the dual edge e' is assigned the dual resistance Re':=1/Re. Then the equivalent resistance re over e and the equivalent resistance re' over e' satisfy re/Re+re'/Re'=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.