### 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*. Then the equivalent resistance

_{e}*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.

#### Keywords

#### Full Text:

PDFDOI: http://dx.doi.org/10.5614/ejgta.2020.8.2.6

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