Fulkerson duality for modulus of spanning trees and partitions

TL;DR

This paper explores Fulkerson duality in the context of spanning tree modulus, introducing Beurling partitions and hierarchical deflation processes to deepen understanding of graph structure.

Contribution

It introduces Beurling partitions, links duality to graph properties, and proposes a hierarchical deflation approach for analyzing graphs.

Findings

Identification of Beurling partitions related to graph strength and denseness

Development of deflation processes revealing hierarchical graph structure

Introduction of weighted spanning tree modulus

Abstract

One of the main properties of modulus on graphs is Fulkerson duality. In this paper, we study Fulkerson duality for spanning tree modulus. We introduce a new notion of Beurling partition, and we identify two important ones, which correspond to the notion of strength and maximum denseness of an arbitrary graph. These special partitions, also give rise to two deflation processes that reveal a hierarchical structure for general graphs. While Fulkerson duality for spanning tree families can be deduced from a well-known result in combinatorics due to Chopra, we give an alternative approach based on a result of Nash-Williams and Tutte. Finally, we introduce the weighted variant of spanning tree modulus.