A Note on the Critical Group of a Line Graph

TL;DR

This paper investigates the relationship between the critical groups of directed graphs and their line graphs, providing conditions under which the critical group of a graph is a quotient of that of its line graph.

Contribution

It establishes a connection between the critical groups of directed graphs and their line graphs, answering a question posed by Levine.

Findings

Identifies conditions where the critical group of a directed graph is a quotient of its line graph's critical group.

Provides a theoretical result linking critical groups of graphs and their line graphs.

Addresses an open question posed by Levine in the context of critical groups.

Abstract

This note answers a question posed by Levine in arkiv:math.CO/0906.2809. The main result is Theorem 1 which shows that under certain circumstances a critical group of a directed graph is the quotient of a critical group of its directed line graph.