Given a graph G with a distinguished vertex s, the critical group of (G,s) is
the cokernel of their reduced Laplacian matrix L(G,s). In this article we
generalize the concept of the critical group to the cokernel of any matrix with
entries in a commutative ring with identity. In this article we find diagonal
matrices that are equivalent to some matrices that generalize the reduced
Laplacian matrix of the path, the cycle, and the complete graph over an
arbitrary commutative ring with identity. We are mainly interested in those
cases when the base ring is the ring of integers and some subrings of matrices.
Using these equivalent diagonal matrices we calculate the critical group of the
m-cones of the l-duplications of the path, the cycle, and the complete graph.
Also, as byproduct, we calculate the critical group of another matrices, as the
m-cones of the l-duplication of the bipartite complete graph with m vertices in
each partition, the bipartite complete graph with 2m vertices minus a matching.Comment: 18 pages, 5 figure