Modulus on graphs as a generalization of standard graph theoretic quantities

TL;DR

This paper explores the modulus of walk families on graphs, revealing it as a versatile measure that unifies shortest path, effective resistance, and max-flow/min-cut concepts, with new insights into parameter effects.

Contribution

It introduces new results on graph modulus, demonstrating its role as a unifying framework for key graph theoretic quantities and analyzing parameter dependencies.

Findings

Modulus generalizes shortest path, effective resistance, and max-flow/min-cut.

Shows how modulus interpolates among these quantities.

Provides insights into the dependence of modulus on its parameters.

Abstract

This paper presents new results for the modulus of families of walks on a graph---a discrete analog of the modulus of curve families due to Beurling and Ahlfors. Particular attention is paid to the dependence of the modulus on its parameters. Modulus is shown to generalize (and interpolate among) three important quantities in graph theory: shortest path, effective resistance, and max-flow or min-cut.