On the uniqueness of the factorization of power digraphs modulo $n$

Abstract

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 $G\_{P}^{}(n,k)$, 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 $p\_j \in Q \smallsetminus P$. 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 $G\_{P}^{}(n,k\_1)$ and $G\_{P}^{\*}(n,k\_2)$ of $G(n,k\_1)$ and $G(n,k\_2)$ respectively for $k\_1 \neq k\_2$.