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.