... partially forconvenience, some assumptions are made about the connectivity of G which are not madein this note. For related work on the critical group of a line graph for an undirected graph,see ... for a careful reading and helpful com-ments.References[1] Andrew Berget, Andrew Manion, Molly Maxwell, Aaron Potechin, and Victor Reiner. The critical group of a line graph. arxiv:math.CO/0904.1246.[2] ... f−. As with G, we have the Laplacian ∆LGand the critical group K( LG, e∗) := cok φLG,e∗for each e∗∈ E.If every vertex of G has a directed path to w∗then K(G, w∗) is called the sandpilegroup...