Resistors in dual networks

2018/05/03 by Furrer, Martina, Hungerbühler, Norbert, Jantschgi, Simon
#05C50 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1805.01380

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.

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