Smith normal form and Laplacians

Let M denote the Laplacian matrix of a graph G. Associated with G is a finite group Φ(G), obtained from the Smith normal form of M, and whose order is the number of spanning trees of G. We provide some general results on the relationship between the eigenvalues of M and the structure of Φ(G), and address the question of how often the group Φ(G) is cyclic.