On finite nilpotent groups with the same enhanced power graph

Abstract

The enhanced power graph of a group $G$ is the graph $P_e(G)$ whose vertex set is $G$, such that two distinct vertices $x$ and $y$, are adjacent if $\langle x, y\rangle$ is cyclic. In this paper, we analyze the structure of the enhanced power graph of a finite nilpotent group in terms of the enhanced power graphs of its Sylow subgroups. We establish that for two nilpotent groups, their enhanced power graphs are isomorphic if and only if the enhanced power graphs of their Sylow subgroups are isomorphic. Additionally, we identify specific nilpotent groups for which the enhanced power graphs uniquely characterize the group structure, meaning that if $P_e(G)\cong P_e(H)$ then $G \cong H$. Finally, we extend these results to power graphs and cyclic graphs.