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