Two classes of asymmetric, circulant, and r-regular digraphs were defined in . These digraphs are denoted by ~Crn and d~C n. The former is an orientation of the rth power of the cycle Cn. Another pair of asymmetric, circulant, and r-regular digraphs were introduced in . One belongs to the class of tournaments, and the other is an orientation of a class of complete bipartite graphs. The former is denoted by ~Tn, and the latter is denoted by ~K m;m. In  and , the singularity and nonsingularity of these classes of digraphs were investigated. In , the spectra of the aforementioned special classes of digraphs and their complements were determined.
A binary operation on digraphs is the cartesian product of digraphs. This paper gives the spectra and establishes some properties of the resulting digraph when the cartesian product of the digraphs given above are obtained.