## On the heights of power digraphs modulo $$n$$.(English)Zbl 1265.05274

To each pair of positive integers $$k\geq 2$$ and $$n$$ a digraph $$G(n,k)$$ is assigned whose set of vertices is $$\{0,1,\dots ,n-1\}$$ and $$\{(a,b)\: a^k\equiv b\pmod n\}$$ is the set of its directed edges. The authors define the height of each vertex $$a$$ as the shortest distance to the unique cycle of the component of $$G(n,k)$$ that contains $$a$$. They establish necessary and sufficient conditions on $$n$$ such that each vertex of indegree 0 of a certain subdigraph of $$G(n,k)$$ is at height $$q\geq 1$$.
Reviewer: Michal Krizek

### MSC:

 05C20 Directed graphs (digraphs), tournaments 11A07 Congruences; primitive roots; residue systems 11A15 Power residues, reciprocity 20K01 Finite abelian groups

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