Metric graphs, cross ratios, and Rayleigh's laws

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.