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Displaying similar documents to “On the heights of power digraphs modulo n ”

The structure of digraphs associated with the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2011)

Czechoslovak Mathematical Journal

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We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod n ) . We investigate the structure of G ( n , k ) . In particular, upper bounds are given for the longest cycle in G ( n , k ) . We find subdigraphs of G ( n , k ) , called fundamental constituents of G ( n , k ) , for which all trees attached to cycle vertices are isomorphic.

Isomorphic digraphs from powers modulo p

Guixin Deng, Pingzhi Yuan (2011)

Czechoslovak Mathematical Journal

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Let p be a prime. We assign to each positive number k a digraph G p k whose set of vertices is { 1 , 2 , ... , p - 1 } and there exists a directed edge from a vertex a to a vertex b if a k ≡ b ( mod p ) . In this paper we obtain a necessary and sufficient condition for G p k 1 ≃ G p k 2 .

On semiregular digraphs of the congruence x k ≡ y ( mod n )

Lawrence Somer, Michal Křížek (2007)

Commentationes Mathematicae Universitatis Carolinae

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We assign to each pair of positive integers n and k ≥ 2 a digraph G ( n , k ) whose set of vertices is H = { 0 , 1 , ⋯ , n - 1 } and for which there is a directed edge from a ∈ H to b ∈ H if a k ≡ b ( mod 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 ≥ 2 is arbitrary.