Graphs with circulant adjacency matrices

Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied. Examples are given showing that the connection set determined by the first row of such a matrix need not be multiplicatively related to the connection set of an isomorphic graph. Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related. The first criterion is that the graphs have a prime number of vertices. The second criterion is that the adjacency matrices have non-repeated eigenvalues. The final section gives a partial characterization of graphs with n vertices whose automorphism group is the cyclic group C_n.