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On semiregular digraphs of the congruence \(x^k\equiv y\pmod n\). (English) Zbl 1174.05058

Summary: We assign to each pair of positive integers \(n\) and \(k\geq 2\) a digraph \(G(n,k)\) whose set of vertices is \(H=\{0,1,\dots ,n-1\}\) and for which there is a directed edge from \(a\in H\) to \(b\in H\) if \(a^k\equiv b\pmod n\). The digraph \(G(n,k)\) is semiregular if there exists a positive integer \(d\) such that each vertex of the digraph has indegree \(d\) or 0. Generalizing earlier results of the authors for the case in which \(k=2\), we characterize all semiregular digraphs \(G(n,k)\) when \(k\geq 2\) is arbitrary.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11A07 Congruences; primitive roots; residue systems
11A15 Power residues, reciprocity
20K01 Finite abelian groups
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