A note on the power graph of a finite group - Zeinab Mehranian (International Journal of Group Theory, 2016)
Suppose $Gamma$ is a graph with $V(Gamma) = { 1,2, cdots, p}$and $ mathcalF = {Gamma_1,cdots, Gamma_p} $ is a family ofgraphs such that $n_j = |V(Gamma_j)|$, $1 leq j leq p$. Define$Lambda = Gamma[Gamma_1,cdots, Gamma_p]$ to be a graph withvertex set $ V(Lambda)=bigcup_{j=1}^pV(Gamma_j)$ and edge set$E(Lambda)=big(bigcup_{j=1}^pE(Gamma_j)big)cupbig(bigcup_{ijinE(Gamma)}{uv;uin V(Gamma_i),vin V(Gamma_j)}big) $. Thegraph $ Lambda$ is called the $Gamma-$join of $ mathcalF$.The power graph $mathcalP(G)$ of a group $G$ is the graphwhich has the group elements as vertex set and two elements areadjacent if one is a power of the other. The aim of this paper isto prove $mathcalP(mathbbZ_{n}) = K_{phi(n)+1} +Delta_n[K_{phi(d_1)},K_{phi(d_2)},cdots, K_{phi(d_{p})}]$,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