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