# Undirected power graphs of semigroups

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## Abstract

The undirected power graph *G*(*S*) of a semigroup *S* is an undirected graph whose vertex set is *S* and two vertices *a*,*b*∈*S* are adjacent if and only if *a*≠*b* and *a* ^{ m }=*b* or *b* ^{ m }=*a* for some positive integer *m*. In this paper we characterize the class of semigroups *S* for which *G*(*S*) is connected or complete. As a consequence we prove that *G*(*G*) is connected for any finite group *G* and *G*(*G*) is complete if and only if *G* is a cyclic group of order 1 or *p* ^{ m }. Particular attention is given to the multiplicative semigroup ℤ_{ n } and its subgroup *U* _{ n }, where *G*(*U* _{ n }) is a major component of *G*(ℤ_{ n }). It is proved that *G*(*U* _{ n }) is complete if and only if *n*=1,2,4,*p* or 2*p*, where *p* is a Fermat prime. In general, we compute the number of edges of *G*(*G*) for a finite group *G* and apply this result to determine the values of *n* for which *G*(*U* _{ n }) is planar. Finally we show that for any cyclic group of order greater than or equal to 3, *G*(*G*) is Hamiltonian and list some values of *n* for which *G*(*U* _{ n }) has no Hamiltonian cycle.

## Keywords

Semigroup Group Divisibility graph Power graph Connected graph Complete graph Fermat prime Planar graph Eulerian graph Hamiltonian graph## Preview

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