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Characterization of power digraphs modulo n

Uzma Ahmad, Syed Husnine (2011)

Commentationes Mathematicae Universitatis Carolinae

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.

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