Elementary proof of Rayleigh formula for graphs

TL;DR

This paper provides a self-contained inductive proof of Rayleigh monotonicity, a principle linking electrical network theory and graph combinatorics, demonstrating negative correlation between edges in random spanning trees.

Contribution

It offers a new, self-contained inductive proof of Rayleigh monotonicity for graphs, clarifying its combinatorial interpretation.

Findings

Proves Rayleigh monotonicity using an inductive approach

Establishes negative correlation between edges in random spanning trees

Connects electrical network principles with graph combinatorics

Abstract

The Rayleigh monotonicity is a principle from the theory of electrical networks. Its combinatorial interpretation says for each two edges of a graph G, that the presence of one of them in a random spanning tree of G is negatively correlated with the presence of the other edge. In this paper we give a self-contained (inductive) proof of Rayleigh monotonicity for graphs.