Smith forms for adjacency matrices of circulant graphs

We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrasfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Mobius ladder. The proofs operate by finding the abelianization of a cyclically presented group whose relation matrix is column equivalent to the required adjacency matrix. (C) 2013 Elsevier Inc. All rights reserved.