The Critical Group of a Directed Graph

For a finite, directed graph G = (V,E) we define the critical groupK(G) to be the cokernel of the transpose of the Laplacian matrix of G acting on Z , and K(G) to be its torsion subgroup. This generalizes the case of undirected graphs studied by Bacher, de la Harpe and Nagnibeda, and by Biggs. We prove a variety of results about these critical groups, among which are: that K(G/π) is a subgroup of K(G) when π is an equitable partition and G is strongly connected; that K(G) depends only on the graphic matroid of G when G is undirected; that there is no ‘natural’ bijection between spanning trees of G and K(G) when G is undirected, even though these sets are equicardinal; and that the ‘dollar game’ of Biggs can be generalized slightly to provide a combinatorial interpretation for the elements of K(G) when G is strongly connected.