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