Power Digraphs Associated with the Congruence xn = y (mod m)

Abstract

For any positive integer m, we assign a digraph G(m) for which {0, 1, 2, 3, ..., m−1} is the set of vertices and there is an edge from a vertex u to a vertex v if m divides u 7 − v. We enumerate the self and isolated loops and study the structures of this digraph for the numbers 2 r and 7 r , for every positive integer r. Further, we characterize the existence of cycles by employing Carmichael’s Theorem. Also, we discuss the subdigraphs of proposed digraph induced by the vertices coprime to m and not coprime to m. Lastly, we characterize the regularity, semiregularity and results regarding components of these subdigraphs.