Metric graphs, cross ratios, and Rayleigh’s laws

Abstract

We systematically study the notion of cross ratios and energy pairings on metric graphs and electrical networks. We show that several foundational results on electrical networks and metric graphs immediately follow from the basic properties of cross ratios. For example, the projection matrices of Kirchhoff have natural (and efficiently computable) expressions in terms of cross ratios. We prove a generalized version of Rayleigh’s law, relating energy pairings and cross ratios on metric graphs before and after contracting an edge segment. Quantitative versions of Rayleigh’s law for effective resistances, potential kernels, and cross ratios will follow as immediate corollaries.