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 .
Cite this paper
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