The structure of power digraph connected with the congruence a¹¹ ≡ b(mod n)
DOI:
https://doi.org/10.22199/issn.071762795600Keywords:
digraph, fixed point, power digraph, regular graph, Carmichael λfunctionAbstract
We assign to each positive integer n a digraph Γ(n, 11) whose set of vertices is Z_{n} = {0, 1, 2, ..., n − 1} and there exists exactly one directed edge from a to b if and only if a^{11} ≡ b(mod n), where a, b ∈ Z_{n}. Let Γ_{1}(n, 11) be the subdigraph induced by the vertices which are coprime to n. We discuss when the subdigraph Γ_{1}(n, 11) is regular or semiregular. A formula for the number of fixed points of Γ(n, 11) is established. A necessary and sufficient condition for the symmetry of the digraph Γ(n, 11) is proved. Moreover, using Carmichael ́s lambda function, the number of components and conditions for the existence of cycles in the digraph Γ(n, 11) is presented.
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Copyright (c) 2023 Pinkimani Goswami , Sanjay Kumar Thakur , Gautam Chandra Ray
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