| Summary: | <p>Let <em>G</em> be a finite plane multigraph and <em>G</em>' its dual. Each edge <em>e</em> of <em>G</em> is interpreted as a resistor of resistance <em>R<sub>e</sub></em>, and the dual edge <em>e'</em> is assigned the dual resistance <em>R</em><sub><em>e</em>'</sub>:=1/<em>R<sub>e</sub></em>. Then the equivalent resistance <em>r</em><sub>e</sub> over <em>e</em> and the equivalent resistance <em>r</em><sub>e'</sub> over <em>e</em>' satisfy r<sub>e</sub>/<em>R<sub>e</sub></em>+<em>r</em><sub><em>e</em>'</sub>/<em>R</em><sub><em>e</em>'</sub>=1. We provide a graph theoretic proof of this relation by expressing the resistances in terms of sums of weights of spanning trees in <em>G</em> and <em>G</em>' respectively.</p>
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