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

TL;DR

This paper constructs specific finite simple connected graphs with prescribed numbers of vertices and edges such that their edge rings are non-normal yet satisfy the $(S_2)$-condition, expanding understanding of algebraic properties of graph-based rings.

Contribution

It demonstrates the existence of graphs with non-normal edge rings that still satisfy the $(S_2)$-condition for certain vertex and edge counts.

Findings

Existence of graphs with non-normal edge rings satisfying $(S_2)$ for given parameters.

Construction method for such graphs based on vertex and edge counts.

Provides new examples linking graph structure to algebraic properties of edge rings.

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.