Let $G$ be a finite simple connected graph on the vertex set $V(G)=[d]=\{1,\dots ,d\}$ with edge set $E(G)=\{e_{1},\dots , e_{n}\}$. Let $\mathbb {K}[\textbf{t}]=\mathbb {K}[t_{1},\dots ,t_{d}]$ be the polynomial ring in d variables over a field $\mathbb {K}$. The edge ring of G is the affine semigroup ring $\mathbb {K}[G]$ generated by monomials $\textbf{t}^{e}:=t_{i}t_{j}$, for $e=\{i,j\} \in E(G)$. In this paper, we will prove that, given integers $d$ and $n$, where $d\geq 7$ and $d+1\le n\le \frac{d^{2}-7d+24}{2}$, there exists a finite simple connected graph $G$ with $|V(G)|=d$ and $|E(G)|=n$, such that $\mathbb {K}[G]$ is non-normal and satisfies $(S_{2})$-condition.