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Digraphs from powers modulo \(p\). (English) Zbl 0855.05067

Let \(G^k_p\) denote the digraph whose vertices are the nonzero residues modulo the prime \(p\) in which there is an edge directed from vertex \(a\) to vertex \(b\) if and only if \(a^k\equiv b\pmod p\); each component of such a graph consists of a collection of rooted trees whose roots lie on a cycle. The authors describe a number of graph-theoretical features of \(G^k_p\) that can be determined in terms of number-theoretical properties of \(p\) and \(k\).

MSC:

05C20 Directed graphs (digraphs), tournaments
11B50 Sequences (mod \(m\))
05C38 Paths and cycles
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