Graph Connectivity and Binomial Edge Ideals

TL;DR

This paper explores the relationship between algebraic properties of binomial edge ideals and graph connectivity measures, revealing new connections between homological invariants and graph toughness, connectivity, and multiplicities.

Contribution

It establishes a link between Cohen-Macaulayness of binomial edge ideals and graph toughness, and provides inequalities relating depth and vertex-connectivity, along with studying multiplicities.

Findings

If $R/\mathcal{J}_G$ is Cohen-Macaulay, then graph toughness is 1/2.

An inequality relates the depth of $R/\mathcal{J}_G$ to the vertex-connectivity of $G$.

The paper analyzes Hilbert-Samuel and Hilbert-Kunz multiplicities of $R/\mathcal{J}_G$.

Abstract

We relate homological properties of a binomial edge ideal $\mathcal{J}_G$ to invariants that measure the connectivity of a simple graph $G$. Specifically, we show if $R/\mathcal{J}_G$ is a Cohen-Macaulay ring, then graph toughness of $G$ is exactly $\frac{1}{2}$. We also give an inequality between the depth of $R/\mathcal{J}_G$ and the vertex-connectivity of $G$. In addition, we study the Hilbert-Samuel multiplicity, and the Hilbert-Kunz multiplicity of $R/\mathcal{J}_G$.