Let \(F_ n\) be the family of digraphs with n vertices consisting of n- cycles with circulant adjacency matrices. (1) We characterize the isomorphic digraphs in \(F_ n.\) (2) Let \(n=p_ 1^{k_ 1}p_ 2^{k_ 2}...p_ t^{k_ t}\) be the prime-power decomposition of the positive integer n. We show that the vector space, \(A(F_ n)\), of adjacency matrices of \(F_ n\) over the integers modulo 2 is \[ A(F_{p_ 1^{k_ 1}})\otimes A(F_{p_ 2^{k_ 2}})\otimes...\otimes A(F_{p_ t^{k_ t}}). \] (3) We use Pólya's theorem to enumerate \(F_ n\). We also use an algorithm to determine the digraphs in each of the equivalence classes in \(F_ n.\) (4) We present an algorithm to obtain the group of automorphisms for each digraph in \(F_ n\).