Graph Connectivity and Binomial Edge Ideals

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 $R/{\mathcal{J}}_G$.

12 pages, 1 figure