On the power graphs of finite groups and Hamilton cycle

The power graph $\mathcal{P}(G)$ of a finite group $G$ is a graph whose vertex set is the group $G$ and distinct elements $x,y\in G$ are adjacent if one is a power of the other, that is, $x$ and $y$ are adjacent if $x\in \langle y\rangle$ or $y \in \langle x\rangle$. In this paper, we study existence of the Hamilton cycle in the power graph of some finite nilpotent groups $G$ with a cyclic subgroup as direct factor when $G$ is written as direct product Sylow $p$-subgroups. For this purpose we use of cartesian product a spanning tree and a cycle. Finally, we determined values of $n$ such that $\mathcal{P}(U_n)$ is Hamiltonian, where $U_n$ is a group consist of all positive integers less than $n$ and relatively prime to $n$ under multiplication modulo $n$.