The critical group of a directed graph

TL;DR

This paper studies the critical group of directed graphs, extending known results from undirected graphs, and provides new algebraic and combinatorial insights into its structure and properties.

Contribution

It proves subgroup relations for critical groups under equitable partitions, links torsion subgroups to graphic matroids, and generalizes Biggs' dollar game for strongly connected graphs.

Findings

K(G/p) is a subgroup of K(G) under equitable partitions

Torsion subgroup depends only on the graphic matroid for undirected graphs

Generalization of Biggs' dollar game offers a combinatorial interpretation

Abstract

The critical group K(G) of a directed graph G=(V,E) is the cokernel of the transpose of the Laplacian matrix of G acting on the integer lattice Z^V. For undirected graphs G, this has been considered by Bacher, de la Harpe, and Nagnibeda, and by Biggs. We prove several things, among which are: K(G/p) is a subgroup of K(G) when p is an equitable partition and G is strongly connected; for undirected graphs, the torsion subgroup of K(G) depends only on the graphic matroid of G; and, the `dollar game' of Biggs can be generalized to give a combinatorial interpretation for the elements of K(G), when G is strongly connected.