Balanced Abelian group valued functions on directed graphs

TL;DR

This paper investigates balanced functions from graph edges and vertices to Abelian groups, analyzing their structure under different traversal constraints, and characterizes the resulting groups.

Contribution

It provides a detailed study of the structure of balanced Abelian group valued functions on directed graphs under different conditions.

Findings

Characterization of the group of balanced functions when walking against edges is allowed.

Analysis of the group structure when walking against edges is not permitted.

Insights into how traversal rules affect the algebraic properties of these functions.

Abstract

We discuss functions from the edges and vertices of a directed graph to an Abelian group. Such functions, when the sum of their values along any cycle is zero, are called balanced and form an Abelian group. We study this group in two cases: when we allowed to walk against the direction of an edge taking the opposite value of the function and when we are not allowed to walk against the direction.