on the power graphs of finite groups and hamilton cycle

The power graph p(g) of a finite group g is a graph whose vertex set is the group g and distinct elements x, y ∈ g are adjacent if one is a power of the other, that is, x and y are adjacent if x ∈ ⟨y⟩ or y ∈ ⟨x⟩. 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 p(un) is hamiltonian, where un is a group consist of all positive integers less than n and relatively prime to n under multiplication modulo n.