Metric graphs, cross ratios, and Rayleigh's laws

TL;DR

This paper introduces a new framework using cross ratios on metric graphs and electrical networks to derive known results and generalize Rayleigh's laws, providing explicit formulas and efficient computations.

Contribution

It develops a novel approach with cross ratios to analyze metric graphs, leading to new expressions for Kirchhoff matrices and a generalized Rayleigh's law.

Findings

Projection matrices expressed via cross ratios

Quantitative Rayleigh's monotonicity law derived

Explicit behavior of the Laplacian's potential kernel under contractions

Abstract

We study a notion of cross ratios on metric graphs and electrical networks. We show that several known results immediately follow from the basic properties of cross ratios. We show that the projection matrices of Kirchhoff have nice (and efficiently computable) expressions in terms of cross ratios. Finally we prove a very general version of Rayleigh's law, relating energy pairings and cross ratios before and after contracting an edge segment. As a corollary, we obtain a quantitative version of Rayleigh's monotonicity law for effective resistances. Another consequence is an explicit description of the behavior of the potential kernel of the Laplacian operator under contractions.