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On the tree structure of the power digraphs modulo \(n\). (English) Zbl 1363.11002

Summary: For any two positive integers \(n\) and \(k \geq 2\), let \(G(n,k)\) be a digraph whose set of vertices is \(\{0,1,\ldots ,n-1\}\) and such that there is a directed edge from a vertex \(a\) to a vertex \(b\) if \(a^k \equiv b \pmod n\). Let \(n=\prod \nolimits_{i=1}^r p_{i}^{e_{i}}\) be the prime factorization of \(n\). Let \(P\) be the set of all primes dividing \(n\) and let \(P_1,P_2 \subseteq P\) be such that \(P_1 \cup P_2=P\) and \(P_1 \cap P_2= \emptyset \). A fundamental constituent of \(G(n,k)\), denoted by \(G_{P_2}^{*}(n,k)\), is a subdigraph of \(G(n,k)\) induced on the set of vertices which are multiples of \(\prod \nolimits_{{p_i} \in P_2}p_i\) and are relatively prime to all primes \(q \in P_1\). L. Somer and M. Křížek [Czech. Math. J. 61, No. 2, 337–358 (2011; Zbl 1249.11006)] proved that the trees attached to all cycle vertices in the same fundamental constituent of \(G(n,k)\) are isomorphic. In this paper, we characterize all digraphs \(G(n,k)\) such that the trees attached to all cycle vertices in different fundamental constituents of \(G(n,k)\) are isomorphic. We also provide a necessary and sufficient condition on \(G(n,k)\) such that the trees attached to all cycle vertices in \(G(n,k)\) are isomorphic.

MSC:

11A07 Congruences; primitive roots; residue systems
05C05 Trees
05C20 Directed graphs (digraphs), tournaments
11A15 Power residues, reciprocity

Citations:

Zbl 1249.11006
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References:

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