Resistors in dual networks

TL;DR

This paper proves a relation between resistances in a planar graph and its dual using graph theory, expressing resistances via spanning trees, and provides a new proof of a classical electrical network duality.

Contribution

It offers a novel graph-theoretic proof of the resistance relation in dual networks, connecting electrical properties with combinatorial spanning tree sums.

Findings

Resistances in dual networks satisfy a specific reciprocal relation.

Resistances can be expressed as sums over spanning trees.

The proof links electrical network theory with graph combinatorics.

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.