A note on the power graph of a finite group

Suppose Gamma Gamma is a graph with V(Gamma)=1,2,cdots,p V(Gamma)=1,2,cdots,p and mathcalF=Gamma 1 ,cdots,Gamma p mathcalF=Gamma1,cdots,Gammap is a family of graphs such that n j =|V(Gamma j )| nj=|V(Gammaj)| , 1leqjleqp 1leqjleqp . Define Lambda=Gamma[Gamma 1 ,cdots,Gamma p ] Lambda=Gamma[Gamma1,cdots,Gammap] to be a graph with vertex set V(Lambda)=bigcup p j=1 V(Gamma j ) V(Lambda)=bigcupj=1pV(Gammaj) and edge set E(Lambda)=big(bigcup p j=1 E(Gamma j )big)cupbig(bigcup ijinE(Gamma) uv;uin (Gamma i ),vinV(Gamma j )big) E(Lambda)=big(bigcupj=1pE(Gammaj)big)cupbig(bigcupijinE(Gamma)uv;uin (Gammai),vinV(Gammaj)big) . The graph Lambda Lambda is called the Gamma− Gamma− join of mathcalF mathcalF .The power graph mathcalP(G) mathcalP(G) of a group G G is the graph which has the group elements as vertex set and two elements are adjacent if one is a power of the other. The aim of this paper isto prove mathcalP(mathbbZ n )=K phi(n) n [K phi(d 1 ) ,K phi(d 2 ) ,cdots,K phi(d p ) ] mathcalP(mathbbZn)=Kphi(n)ퟠ�횧[Kphi(d1),Kphi(d2),cdots,Kphi(dp)] , where Delta n Deltan is a graph with vertex and edge sets V(Delta n )=d i |1,nnot=d i |n,1leqileqp V(Deltan)=di|1,nnot=di|n,1leqileqp and $ E(Delta_n)={ d_id_j | d_i|d_j, 1leq i