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On iteration digraph and zero-divisor graph of the ring \(\mathbb {Z}_n\). (English) Zbl 1349.05145

Summary: In the first part, we assign to each positive integer \(n\) a digraph \(\Gamma (n,5),\) whose set of vertices consists of elements of the ring \(\mathbb {Z}_n=\{0,1,\cdots ,n-1\}\) with the addition and the multiplication operations modulo \(n,\) and for which there is a directed edge from \(a\) to \(b\) if and only if \(a^5\equiv b\pmod n\). Associated with \(\Gamma (n,5)\) are two disjoint subdigraphs: \(\Gamma_1(n,5)\) and \(\Gamma_2(n,5)\) whose union is \(\Gamma (n,5).\) The vertices of \(\Gamma_1(n,5)\) are coprime to \(n,\) and the vertices of \(\Gamma_2(n,5)\) are not coprime to \(n.\) In this part, we study the structure of \(\Gamma (n,5)\) in detail.
In the second part, we investigate the zero-divisor graph \(G(\mathbb {Z}_n)\) of the ring \(\mathbb {Z}_n.\) Its vertex- and edge-connectivity are discussed.

MSC:

05C20 Directed graphs (digraphs), tournaments
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11A07 Congruences; primitive roots; residue systems
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References:

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