Enumeration of cyclic components of the power digraph Γ(n, 13)

Source abstract

For each positive integer $n$, we assign a digraph $\Gamma(n,13)$ whose set of vertices is $\mathbb{Z}_n=\lbrace 0,1,2, \ldots ,n-1\rbrace$ and attach exactly one directed edge from a vertex $a$ to the vertex $b$ if and only if $a^{13}\equiv b \pmod n$. In this work, the number of cycles and the number of components in the digraph $\Gamma(n,13)$ are presented for $n=p^k$ with the help of Carmichael’s lambda function. It is proved that for $k\geq 1$ and for the prime $p=3, 5, 7$ and $11$, the respective numbers of components in the digraph $\Gamma(p^k,13)$ are $(4k-1)$, $(4k+1)$, $(18k-11)$ and $(6k-1)$. The structure of tree associated with power digraph $\Gamma(p^k, 13)$ for $p=3, 5, 7$ and $11$ are also studied. We have also studied the structure of power digraph $\Gamma(p, 13)$, where $p=2^{2^m}+1, m\geq 0$ is a Fermat prime.