TY - JOUR AU - Goswami , Pinkimani AU - Sanjay Kumar Thakur , AU - Ray, Gautam Chandra PY - 2023/03/27 Y2 - 2024/03/29 TI - The structure of power digraph connected with the congruence a¹¹ ≡ b(mod n) JF - Proyecciones (Antofagasta, On line) JA - Proyecciones (Antofagasta, On line) VL - 42 IS - 2 SE - DO - 10.22199/issn.0717-6279-5600 UR - https://www.revistaproyecciones.cl/index.php/proyecciones/article/view/5600 SP - 457-477 AB - <p>We assign to each positive integer n a digraph Γ(n, 11) whose set of vertices is Z<sub>n</sub> = {0, 1, 2, ..., n − 1} and there exists exactly one directed edge from a to b if and only if a<sup>11</sup> ≡ b(mod n), where a, b ∈ Z<sub>n</sub>. Let Γ<sub>1</sub>(n, 11) be the subdigraph induced by the vertices which are coprime to n. We discuss when the subdigraph Γ<sub>1</sub>(n, 11) is regular or semi-regular. 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.</p> ER -