Contractors for flows

TL;DR

This paper constructs contractors for counting B-flows in graphs, using duality and Fourier analysis, contributing to the understanding of flow-related graph parameters and their conjectures.

Contribution

It provides the first explicit contractors for B-flows, advancing the algebraic and analytical tools in graph flow theory.

Findings

Constructed explicit contractors for B-flows.

Linked contractors to B-flow conjectures.

Demonstrated applications of Fourier analysis in graph parameters.

Abstract

We answer a question raised by Lov\'asz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions and finite Fourier analysis. We exhibit several examples of contractors for B-flows, which are of interest in relation to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.