Resistors in dual networks

Thursday, 3 May, 2018

Published in: 

arXiv:1805.01380

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.

Author(s): 

Martina Furrer
Norbert Hungerbühler
Simon Jantschgi