On the tree structure of the power digraphs modulo n

Abstract

For any two positive integers n and k ⩾ 2, let G(n, k) be a digraph whose set of vertices is {0, 1, …, n − 1} and such that there is a directed edge from a vertex a to a vertex b if ak ≡ b (mod n). Let \(n = \prod\nolimits_{i = 1}^r {p_i^{{e_i}}} \) be the prime factorization of n. Let P be the set of all primes dividing n and let P1, P2 ⊆ P be such that P1 ∪ P2 = P and P1 ∩ P2 = ∅. A fundamental constituent of G(n, k), denoted by \(G_{{P_2}}^*(n,k)\), is a subdigraph of G(n, k) induced on the set of vertices which are multiples of \(\prod\nolimits_{{p_i} \in {P_2}} {{p_i}} \) and are relatively prime to all primes q ∈ P1. L. Somer and M. Křižek proved that the trees attached to all cycle vertices in the same fundamental constituent of G(n, k) are isomorphic. In this paper, we characterize all digraphs G(n, k) such that the trees attached to all cycle vertices in different fundamental constituents of G(n, k) are isomorphic. We also provide a necessary and sufficient condition on G(n, k) such that the trees attached to all cycle vertices in G(n, k) are isomorphic.