a note on the power graph of a finite group

suppose $gamma$ is a graph with $v(gamma) = { 1,2, cdots, p}$and $ mathcal{f} = {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 $ mathcal{f}$.the power graph $mathcal{p}(g)$ 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 $mathcal{p}(mathbb{z}_{n}) = k_{phi(n)+1} +delta_n[k_{phi(d_1)},k_{phi(d_2)},cdots, k_{phi(d_{p})}]$,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