Enumeration of balanced finite group valued functions on directed graphs

TL;DR

This paper develops a method to count the number of balanced functions from the edges and vertices of a directed graph to a finite group, based on the property that the product along any cycle equals the identity.

Contribution

It provides a formula for enumerating balanced group-valued functions on directed graphs, extending previous work on graph labelings and group functions.

Findings

Derived a counting formula for balanced functions on directed graphs.

Connected cycle conditions to algebraic properties of finite groups.

Enhanced understanding of graph group labelings and their enumeration.

Abstract

A group valued function on a graph is called balanced if the product of its values along any cycle is equal to the identity element of the group. We compute the number of balanced functions from edges and vertices of a directed graph to a finite group.