In this note, we show that the number of digraphs with n vertices and with cycles of length k, 0 ≤ k ≤ n, is equal to the number of n × n (0,1)-matrices whose eigenvalues are the collection of copies of the entire kth unit roots plus, possibly, 0’s. In particular, 1) when k = 0, since the digraphs reduce to be acyclic, our result reduces to the main theorem obtained recently in  stating that, for each n = 1, 2, 3, …, the number of acyclic digraphs is equal to the number of n × n (0,1)-matrices whose eigenvalues are positive real numbers; and 2) when k = n, the digraphs are the Hamiltonian directed cycles and it, therefore, generates another well-known (and trivial) result: the eigenvalues of a Hamiltonian directed cycle with n vertices are the nth unit roots .
 R. Robinson, “Enumeration of Acyclic Digraphs,” In: R. C. Bose, et al., Eds., Proceedings of 2nd Chapel Hill Conference on Combinatorial Mathematics and Its Applications, Chapel Hill, Orange County, 1970, pp. 391-399.
 R. Stanley, “Acyclic Orientations of Graphs,” Discrete Mathematics, Vol. 5, No. 2, 1973, pp. 171-178.
 E. Bender, L. Richmond, R. Robinson and N. Wormald, “The Asymptotic Number of Acyclic Digraphs (I),” Combinatorica, Vol. 6, No. 1, 1986, pp. 15-22. http://dx.doi.org/10.1007/BF02579404
 E. Bender and R. Robinson, “The Asymptotic Number of Acyclic Digraphs (II),” Journal of Combinatorial Theory, Series B, Vol. 44, No. 3, 1988, pp. 363-369. http://dx.doi.org/10.1137/1.9781611971262
 I. Gessel, “Counting the Acyclic Digraphs by Sources and Sinks,” Discrete Mathematics, Vol. 160, No. 1-3, 1996, pp. 253-258.