Non-normal edge rings satisfying $(S_2)$-condition

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