A note on the power graph of a finite group

‎Suppose $Gamma$ is a graph with $V(Gamma) = { 1‎, ‎2,dots‎, ‎p}$‎ ‎and $ mathcal{F} = {Gamma_1,dots‎, ‎Gamma_p} $ is a family of‎ ‎graphs such that $n_j = |V(Gamma_j)|$‎, ‎$1 leq j leq p$‎. ‎Define‎ ‎$Lambda = Gamma[Gamma_1,dots‎, ‎Gamma_p]$ to be a graph with‎ ‎vertex set $ V(Lambda)=bigcup_{j=1}^pV(Gamma_j)$ and edge set‎ ‎$E(Lambda)=big(bigcup_{j=1}^pE(Gamma_j)big)cupbig(bigcup_{ijin‎ ‎E(Gamma)}{uv;uin V(Gamma_i),vin V(Gamma_j)}big) $‎. ‎The‎ ‎graph $ Lambda$ is called the $Gamma$-join of $ mathcal{F}$‎. ‎The power graph $mathcal{P}(G)$ of a group $G$ 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 is‎ ‎to prove that $mathcal{P}(mathbb{Z}_{n}) = K_{phi(n)+1}‎ + ‎Delta_n[K_{phi(d_1)}‎, ‎K_{phi(d_2)},dots‎, ‎K_{phi(d_{p})}]$‎, ‎where $Delta_n$ is a graph with vertex and edge sets‎ ‎$V(Delta_n)={d_i | 1,nnot = d_i | n‎, ‎1leq ileq p}$ and‎ ‎$ E(Delta_n)={ d_id_j | d_i|d_j‎, ‎1leq i<jleq p}$‎, ‎respectively‎. ‎As a consequence it is proved that‎ ‎$Aut(mathcal{P}(mathbb{Z}_{n}))cong‎ ‎S_{phi(n)+1}timesprod_{1,nnot=d|n}S_{phi(d)}.$ This proves a‎ ‎recent conjecture by Doostabadi et al‎. ‎[A‎. ‎Doostabadi‎, ‎A‎. ‎Erfanian and A‎. ‎Jafarzadeh‎, ‎Some results on the power graph of groups, ‎The Extended Abstracts of the 44th Annual Iranian Mathematics Conference‎, ‎27-30 August 2013‎, ‎Ferdowsi University of Mashhad‎, ‎Iran]‎. ‎Finally‎, ‎we‎ ‎apply our results to obtain complete descriptions of the power‎ ‎graphs of some finite groups‎.