Non-Abelian Flows in Networks

TL;DR

This paper generalizes graph flows by labeling edges with group elements instead of real numbers, revealing that only planar graphs maintain a specific conservation property in this non-abelian setting.

Contribution

It introduces a novel non-abelian flow framework and characterizes planar graphs as the unique class preserving certain flow conservation properties.

Findings

Graphs with conservation in all but one vertex are exactly the planar graphs.

Non-abelian group labelings do not guarantee flow conservation across all vertices.

The work extends classical flow theory to non-abelian group contexts.

Abstract

In this work we consider a generalization of graph flows. A graph flow is, in its simplest formulation, a labeling of the directed edges with real numbers subject to various constraints. A common constraint is conservation in a vertex, meaning that the sum of the labels on the incoming edges of this vertex equals the sum of those on the outgoing edges. One easy fact is that if a flow is conserving in all but one vertex, then it is also conserving in the remaining one. In our generalization we do not label the edges with real numbers, but with elements from an arbitrary group, where this fact becomes false in general. As we will show, graphs with the property that conservation of a flow in all but one vertex implies conservation in all vertices are precisely the planar graphs.