Displaying similar documents to “The structure of digraphs associated with the congruence x k ≡ y ( mod n ) ”

On the heights of power digraphs modulo n

Uzma Ahmad, Husnine Syed (2012)

Czechoslovak Mathematical Journal

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A power digraph, denoted by G ( n , k ) , is a directed graph with ℤ n = { 0 , 1 , ⋯ , n - 1 } as the set of vertices and E = { ( a , b ) : a k ≡ b ( mod n ) } as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křížek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křížek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of G ( n , k ) for n ≥ 1 and...

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 .

Characterization of power digraphs modulo n

Uzma Ahmad, Syed Husnine (2011)

Commentationes Mathematicae Universitatis Carolinae

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A power digraph modulo n , denoted by G ( n , k ) , is a directed graph with Z n = { 0 , 1 , ⋯ , n - 1 } as the set of vertices and E = { ( a , b ) : a k ≡ b ( mod n ) } as the edge set, where n and k are any positive integers. In this paper we find necessary and sufficient conditions on n and k such that the digraph G ( n , k ) has at least one isolated fixed point. We also establish necessary and sufficient conditions on n and k such that the digraph G ( n , k ) contains exactly two components. The primality of Fermat number is also discussed.