Effective resistance is more than distance: Laplacians, Simplices and the Schur complement

TL;DR

This paper presents a geometric and algebraic approach to understanding effective resistance in graphs, using simplices and the Schur complement, unifying various concepts in graph theory and matrix analysis.

Contribution

It introduces a novel perspective combining geometric and algebraic methods to analyze effective resistance, extending classical proofs with new insights.

Findings

Effective resistance is a metric on graph nodes.

A matrix identity unifies geometric and algebraic perspectives.

The approach offers a deeper understanding of Laplacians and simplices in graph theory.

Abstract

This article discusses a geometric perspective on the well-known fact in graph theory that the effective resistance is a metric on the nodes of a graph. The classical proofs of this fact make use of ideas from electrical circuits or random walks; here we describe an alternative approach which combines geometric (using simplices) and algebraic (using the Schur complement) ideas. These perspectives are unified in a matrix identity of Miroslav Fiedler, which beautifully summarizes a number of related ideas at the intersection of graphs, Laplacian matrices and simplices, with the metric property of the effective resistance as a prominent consequence.