The Number of Digraphs with Cycles of Length K

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 [1] 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 [2].


Introduction
, n  is a directed graph whose edges are oriented from vertex i to vertex j, 1 , i j n ≤ ≤ .In this note, the digraphs to be considered are with loops or cycles, but parallel edges are forbidden.An acyclic digraph is a digraph that has no cycles of any length.Let , n k D be a digraph of n vertices with cycles of length k plus, possibly, an acyclic digraph with n mk − vertices, where m is the number of cycles, and where k is fixed, 0 k n ≤ ≤ .Then it is seen that the cycles in , n k D are disjoined (therefore all the cycles that it has are simple) and if k n < the digraph is not strongly connected.And, in particular, , n n D is a Hamiltonian directed cycle of size n and ,0 n D is an acyclic digraph of n vertices, respectively.The acyclic directed graphs have been considered by several authors in the past decades.A first related result appeared in the literature seems to be the one described in [2].It says that a digraph G contains no cycle if and only if all eigenvalues of its adjacency matrix are 0. Subsequently, to the best of our knowledge, Robinson [3,4] and Stanley [5] counted the acyclic digraphs independently and showed that if n R stands for the number of acyclic digraphs of n vertices then  , and 0.474 M = .Later, in [6,7], Bender et al. considered the asymptotic number of acyclic digraphs with q edges, and subsequently, Gessel counted the acyclic digraphs by their sources and sinks in [8] 7 because the numbers were observed to coincide with the first five values of the sequence of the number of acyclic digraphs with n vertices that is obtained by Sloane in [9].Weisstein conjectured that the two sequences are identical.The conjecture has recently been proven in [1].Motivated by the above literature, we extend the acyclic digraphs to consider the digraphs , n k

D
with cycles of length k in this note, where 0 k n ≤ ≤ .Our theorem established in the next section indicates that similar counting theorem holds for more general graphs.

The Main Results
Let us first prove the following lemma.
Lemma 1 Given a positive integer n , and a nonnegative integer k with 0 k n ≤ ≤ , the eigenvalues of the adjacency matrix ( ) are copies of the entire kth unit roots plus, possibly, 0's.Conversely, if B is an n n × ( ) 0,1 -matrix whose eigenvalues are copies of the entire kth unit roots plus, possibly, 0's then its digraph ( ) has m cycles of length k.We show that the eigenvalues of A are m copies of the entire kth unit roots plus n mk − 0's.Since relabeling the vertices of a graph does not change the eigenvalues of its adjacency matrix, and since the m cycles of length k are disjoined, we may number the vertices consistently with the partial order so that A has the upper block-triangular as follows:  , is the adjacency matrix of a (directed simple) cycles of length k, and where '*' is either a block matrix of 0, 1, or 1, or 0's.From linear algebra, it can be easily proven that, for any i ( m ≤ ), the characteristic polynomial of i A is 1 B are either 1 or 0, the diagonal elements of k B must be 1 or 0. In fact, from Perron-Frobenius theory (e. g., [10], p. 28, (1,6) Corollary (a)), we have and k B has exactly mk 1's on its diagonal.Thus, counting all the closed walks in the kth power graph Note that when 0 k = , because of the one to one corresponding, this leads to an alternative proof of the above conjecture (the main theorem of [1]).That is, for each 1, 2, 3, n =  , the number of acyclic digraphs 0,1 -matrices whose eigenvalues are 0's ---from linear algebra, which is equivalent to saying that, for each 1, 2, 3, n =  , the number of acyclic digraphs ,0 n D is equal to the number of n n × ( ) 0,1 -matrices whose n eigenvalues are equal to 1 ---and from [1], which is also equivalent to saying that, for each 1, 2, 3, n =  , the number of acyclic digraphs ,0 n D is equal to the number of n n × ( ) 0,1 -matrices whose eigenvalues are positive real numbers.
Corollary 1 If a digraph D has cycles of lengths i k , 1, 2, , i t =  , and the cycles are piecewise disjoined then the eigenvalues of its adjacency matrix A are collection of the entire k i th unit roots, 1, 2, , i t =  , plus 0's.And vice versa.
So the eigenvalues of i A are the kth unit roots.Since the eigenvalues of A are collection of the eigenvalues of these i A and n mk − 0's, its eigenvalues are m copies of the entire kth unit roots plus n mk − zeroes.Conversely, if B is an n n × ( ) 0,1 -matrix whose eigenvalues are m copies of the entire kth unit roots plus n mk − 0's, then its graph ( ) D B is a digraph and, for any ( ) i n ≤ , the ith eigenvalues of k B , 0. We now consider the power digraphs of ( ) D B with adjacency matrix B. Since for all 1closed walks of length l in the kth power graph mk .Since the eigenvalues of k


is a digraph with m disjoined cycles of length k plus an acyclic graph with n mk − vertices.Putting the thing back to B implies that B is the adjacency matrix of a digraph with m cycles of length k plus, possibly, an acyclic digraph with n mk − vertices, counting the number of the digraphs and the ( ) 0,1 -matrices in the above lemma, we immediately have the following Theorem 1 For each 1, 2, 3, n =  , and nonnegative integer k, 0 k n ≤ ≤ , the number of digraphs , n k D with cycles of length k 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.