Enumeration of cyclic vertices and components over the congruence $a^{11} \equiv b \pmod n$

Abstract

For each positive integer $n$, we assign a digraph $\Gamma(n,11)$ whose set of vertices is $Z_n=\lbrace 0,1,2, \ldots, n-1\rbrace$ and there exists exactly one directed edge from the vertex $a$ to the vertex $b$ iff $a^{11}\equiv b \pmod n$. Using the ideas of modular arithmetic, cyclic vertices are presented and established for $n=3^k$ in the digraph $\Gamma(n,11)$. Also, the number of cycles and the number of components in the digraph $\Gamma(n,11)$ is presented for $n=3^k,7^k$ with the help of Carmichael’s lambda function. It is proved that for $k\geq 1$, the number of components in the digraph $\Gamma(3^k,11)$ is $(2k+1)$ and for $k>2$ the digraph $\Gamma(3^k,11)$ has $(k-1)$ non-isomorphic cycles of length greater than $1$, whereas the number of components of the digraph $\Gamma(7^k,11)$ is $(8k-3)$.