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 $d\ge 7$ 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 $K[G]$ is non-normal and satisfies $(S_2)$-condition.