On the uniqueness of the factorization of power digraphs modulo \(n\) (Q2414666)

Summary: For each pair of integers \(n=\prod_{i=1}^{r}p_i^{e_i}\) and \(k \geq 2\), a digraph \(G(n,k)\) is one with vertex set \(\{0,1,\ldots ,n-1\}\) and for which there exists a directed edge from \(x\) to \(y\) if \(x^k \equiv y \pmod n\). Using the Chinese remainder theorem, the digraph \(G(n,k)\) can be written as a direct product of digraphs \(G(p_i^{e_i},k)\) for all \(i\) such that \(1 \leq i \leq r\). A fundamental constituent \(G_{P}^\ast(n,k)\), where \(P \subseteq Q=\{p_1,p_2,\ldots,p_r\}\), is a subdigraph of \(G(n,k)\) induced on the set of vertices which are multiples of \(\prod_{{p_i} \in P}p_i\) and are relatively prime to all primes \(p_j \in Q \smallsetminus P\). \par 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 of \(G(n,k)\). Moreover, we provide a necessary and sufficient condition for the isomorphism of the fundamental constituents \(G_{P}^\ast (n,k_1)\) and \(G_{P}^\ast(n,k_2)\) of \(G(n,k_1)\) and \(G(n,k_2)\) respectively for \(k_1 \neq k_2\).