Critical Groups for Complete Multipartite Graphs and Cartesian Products of Complete Graphs

The critical group of a connected graph is a finite abelian group, whose order is the number of spanning trees in the graph, and which is closely related to the graph Laplacian. Its group structure has been determined for relatively few classes of graphs, e.g., complete graphs and complete bipartite graphs. For complete multipartite graphs K_n1,...,n_k, we describe the critical group structure completely. For Cartesian products of complete graphs K_n1x⋯x K_nk, we generalize results of H. Bai on the k-dimensional cube, by bounding the number of invariant factors in the critical group, and describing completely its p-primary structure for all primes p that divide none of n₁,..., n _k.