On the power graphs of finite groups and Hamilton cycle

Document Type : Research Paper

Authors

¹ Faculty of Sciences, University of Zabol, Zabol, Iran.

² Department of Mathematics, Payame Noor University, Tehran, Iran.

³ University of Gonabad, Gonabad, Iran.

Abstract

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 \in G

are adjacent if one is a power of the other, that is,

x

and

y

are adjacent if

x \in ⟨ y ⟩

y \in ⟨ 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 (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

Keywords

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