Let G be a digraph and A( G) be its adjacency matrix. The spectrum of G, denoted by Spec G is where Ao, AI, ... , Ap-l are the eigenvalues of A( G) and mo, m1, ... , m - 1 are their corresponding multiplicities. This paper discusses some properties of the spectrum of four different classes of asymmetric, circulant, and r-regular digraphs and their complements. The digraphs considered in this paper are orientations of the rth power of a cycle, a complete graph, a complete bipartite graph, and a digraph whose adjacency matrix is
circulant with first row entries all zeros exc~t the ( d + 1) st and nth column entries which are' both 1's.