The paper extends the results given by M. Křížek and L. Somer, {\it On a connection of number theory with graph theory}, Czech. Math. J. 54 (129) (2004), 465--485 (see [5]). For each positive integer $n$ define a digraph $\Gamma (n)$ whose set of vertices is the set $H=\{0,1,\dots ,n - 1\}$ and for which there is a directed edge from $a\in H$ to $b\in H$ if $a^3\equiv b\pmod n.$ The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph $\Gamma (n)$ is proved. The formula for the number of fixed points of $\Gamma (n)$ is established. Moreover, some connection of the length of cycles with the Carmichael $\lambda $-function is presented.