The Cubic Mapping Graph of the Residue Classes of Integers

Abstract

In this paper, we study the connection of number theory with graph theory via investigating some uncharted properties of the directed graph ${\mathrm{\Gamma }}^{\prime }(n)$ whose vertex set is ${\mathbb{Z}}_n=\{0,1,\dots ,n-1\}$, and for which there is a directed edge from $a\in {\mathbb{Z}}_n$ to $b\in {\mathbb{Z}}_n$ if and only if $a^3\equiv b(\mathrm{mod}n)$. For an arbitrary prime $p$, the formula for the decomposition of the graph $\mathrm{\Gamma }(p)$ is established. We specify two subgraphs ${\mathrm{\Gamma }}_1(n)$ and ${\mathrm{\Gamma }}_2(n)$ of $\mathrm{\Gamma }(n)$. Let ${\mathrm{\Gamma }}_1(n)$ be induced by the vertices which are coprime to $n$ and ${\mathrm{\Gamma }}_2(n)$ by induced by the set of vertices which are not coprime to $n$. We determine the level of every component of ${\mathrm{\Gamma }}_1(n)$, and establish necessary and sufficient conditions when ${\mathrm{\Gamma }}_1(n)$ or ${\mathrm{\Gamma }}_2(n)$ has no cycles with length greater than $1$, respectively. Moreover, the conditions for the semiregularity of ${\mathrm{\Gamma }}_2(n)$ are presented.