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.

کلیدواژه ها:

Cartesian Product ، Hamilton cycle ، Power graph ، Spanning Tree