Theory of resistor networks: The two-point resistance

TL;DR

This paper derives explicit formulas for calculating the resistance between any two nodes in various finite resistor networks, using eigenvalues of the Laplacian matrix, with applications to regular lattices and complex boundary conditions.

Contribution

It provides new explicit formulas and identities for two-point resistances in finite resistor networks, including regular lattices with complex boundary conditions.

Findings

Explicit resistance formulas for 1D, 2D, 3D lattices

Summation and product identities for large lattices

Analysis of finite-size effects in resistor networks

Abstract

The resistance between arbitrary two nodes in a resistor network is obtained in terms of the eigenvalues and eigenfunctions of the Laplacian matrix associated with the network. Explicit formulas for two-point resistances are deduced for regular lattices in one, two, and three dimensions under various boundary conditions including that of a Moebius strip and a Klein bottle. The emphasis is on lattices of finite sizes. We also deduce summation and product identities which can be used to analyze large-size expansions of two-and-higher dimensional lattices.