Counting small cycles in generalized de Bruijn digraphs

In this paper, we count small cycles in generalized de Bruijn digraphs. Let n = pd h , where d? p, and g l = gcd(d l - 1, n). We show that if p d 3 and k ≤ h + 3, then the number of cycles of length k in a generalized de Bruijn digraph G B (n, d) is given by 1 /k Σ l\k μ(k/l)g l [d l / g l ], where μ is the Mobius function and [r] denotes the smallest integer not smaller than a real number r.