A graph related to Euler $\varphi$ function

Euler function $\varphi (n)$ is the number of positive integers less than $n$ and relatively prime to $n$. Suppose that ${\varphi }^1(n)=\varphi (n)$ and ${\varphi }^i(n)=\varphi ({\varphi }^{i-1}(n))$. Let $A\subseteq \mathbb{N}$, and $A_{\phi}={ \phi^k(n)| n\in A , k\in $\mathbb{N}$ \cup {0}}.$Weconsideragraph$G_{\phi}(A)=(V,E)$,where$V=A_{\phi}$and$E={{r,s}| r,s\in V, \phi®=s }$\mathrm{.}Wesayagraph$H$isa$G_\phi$-graph, if there exists a set of natural numbers $A$, such that $H=G_{\varphi }(A)$. In this paper we study the graph $G_{\varphi }(A)$ and investigate some specific graphs and some chemical trees as $G_{\varphi }$-graph.