Properties of digraphs connected with some congruence relations |
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Authors: | J. Skowronek-Kaziów |
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Affiliation: | (1) Faculty of Mathematics, University of Zielona Góra, ul. prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland |
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Abstract: | The paper extends the results given by M. Křížek and L. Somer, On a connection of number theory with graph theory, Czech. Math. J. 54 (129) (2004), 465–485 (see [5]). For each positive integer n define a digraph Γ(n) whose set of vertices is the set H = {0, 1, ..., n − 1} and for which there is a directed edge from a ∈ H to b ∈ H if a 3 ≡ b (mod n). The properties of such digraphs are considered. The necessary and the sufficient condition for the symmetry of a digraph Γ(n) is proved. The formula for the number of fixed points of Γ(n) is established. Moreover, some connection of the length of cycles with the Carmichael λ-function is presented. |
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Keywords: | digraphs Chinese remainder theorem Carmichael λ -function group theory |
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