On the uniqueness of the factorization of power digraphs

For each pair of integers n = ∏r i=1 p ei i and k ≥ 2, a digraph G(n, k) is one with vertex set {0, 1, . . . , n − 1} and for which there exists a directed edge from x to y if x ≡ y (mod n). Using the Chinese Remainder Theorem, the digraph G(n, k) can be written as a direct product of digraphs G(pi i , k) for all i such that 1 ≤ i ≤ r. A fundamental constituent GP (n, k), where P ⊆ Q = {p1, p2, . . . , pr}, is a subdigraph of G(n, k) induced on the set of vertices which are multiples of ∏ pi∈P pi and are relatively prime to all primes pj ∈ Q r P . In this paper, we investigate the uniqueness of the factorization of trees attached to cycle vertices of the type 0, 1, and (1, 0), and in general, the uniqueness ofG(n, k). Moreover, we provide a necessary and sufficient condition for the isomorphism of the fundamental constituents GP (n, k1) and GP (n, k2) of G(n, k1) and G(n, k2) respectively for k1 6= k2. Mathematics Subject Classification (2010). 05C20, 05C05, 11A15, 11A07.