A generalised formulation of the Laplacian approach to resistor networks

TL;DR

This paper introduces a generalized analytic method using Laplacian matrix minors to compute exact two-point resistances in arbitrary non-regular resistor networks, extending previous methods limited to regular geometries.

Contribution

It develops a new approach based on the second minor of the Laplacian matrix, enabling resistance calculations for more complex network structures.

Findings

Derived exact resistance expressions for non-regular networks.

Applied method to globe lattice, obtaining resistance as a single summation.

Extended the scope of resistor network analysis beyond regular geometries.

Abstract

An analytic approach is presented to developing exact expressions for the two-point resistance between arbitrary nodes on certain non-regular resistor networks. This generalises previous approaches, which only deliver results for networks of more regular geometry. The new approach exploits the second minor of the Laplacian matrix associated with the given network to obtain the resistance in terms its eigenvalues and eigenvectors. The method is illustrated by application to the resistor network on the globe lattice, for which the resistance between two arbitrary nodes is obtained in the form of single summation.