Directed Power Graphs Of Non-Coprime Cyclic Group Products

Abstract

In this paper, we investigate the directed power graph of the direct product of cyclic groups, Z_m and Z_n where m and n are not relatively prime. We conduct a detailed structural analysis of the graphs G(Z_p×p), G(Z_p×2p)and  G(Z_p×p²). Here we are utilizing the algebraic properties of Z_n and Z_m, where m and n are coprime. The study focuses on how the lack of coprimality influences the connectivity and hierarchical structure of these directed power graphs.