Some digraphs arising from number theory and remarks on the zero-divisor graph of the ring Zn

In the first part of the paper we investigate a digraph Γ(n) whose set of vertices is the set H={0,1,…,n−1} and for which there is a directed edge from a∈H to b∈ H if . We specify two subdigraphs Γ1(n) and Γ2(n) of Γ(n). Let Γ1(n) be induced by the vertices which are coprime to n and Γ2(n) be induced by the set of vertices which are not coprime with n. The conditions for regularity and semiregularity of these subdigraphs are presented. The digraph Γ(n) has an interesting structure for some special n. It is shown that every component of the digraph Γ(n) is a cycle if and only if 3 does not divide the Euler totient function φ(n) and n is square-free. It is proved that Γ1(k2) contains only cycles and Γ2(k2) is a tree with the root in 0. Besides Γ1(k3) contains two ternary trees with roots in 1 and k3−1 and Γ2(k3) is a tree with the root in 0. All digraphs with 3 components are described.In the second part we consider the zero-divisor graph G(Zn) of the ring Zn. Its maximum degree and the clique number are calculated.