Sum rules for effective resistances in infinite graphs

TL;DR

This paper extends Foster Theorems to infinite graphs, deriving sum rules for effective resistances, and demonstrates their application on common planar grids like square, triangular, and hexagonal grids.

Contribution

It introduces a method to extend finite graph resistance results to infinite graphs, enabling easier calculation of resistance sums over paths.

Findings

Sum rules for effective resistances in infinite graphs derived.

Application demonstrated on square, triangular, and hexagonal grids.

Facilitates resistance calculations in infinite network models.

Abstract

Extending work of Foster, Doyle, and others, we show how the Foster Theorems, a family of results concerning effective resistances on finite graphs, can in certain cases be extended to infinite graphs. A family of sum rules is then obtained, which allows one to easily calculate the sum of the resistances over all paths of a given length. The results are illustrated with some of the most common grids in the plane, including the square, triangular, and hexagonal grids.