Analytic spread of binomial edge ideals

TL;DR

This work investigates the analytic spread $\ell(J_G)$ of binomial edge ideals $J_G$ for finite simple graphs, providing tight general bounds $n-1 \le \ell(J_G) \le 2n-3$ for connected graphs and exact values for key families. The authors develop a framework based on the special fiber ring and transcendence degrees to compute $\ell(J_G)$, proving additivity over connected components and monotonicity under graph containment, and they connect closed graphs to the Newton-Okounkov region to obtain a convex-geometric perspective. They establish exact results for pseudo-forests, showing $\ell(J_G)=|E|$ for forests and unicyclic graphs, and they analyze how leaf additions affect $\ell(J_G)$. For closed graphs, they obtain a precise formula $\ell(J_G)=\mathrm{mcd}(\Delta(J_G))+1$ and characterize when equality with $|E|$ holds, linking to the absence of induced $K_4$. Overall, the work bridges combinatorial graph structure, commutative algebra, and convex geometry to advance understanding of analytic spread in binomial edge ideals.

Abstract

We investigate the analytic spread of binomial edge ideals of finite simple graphs. We provide tight bounds for this invariant in general. For special families of graphs (e.g., closed graphs, pseudo-forests), we compute the exact value for the analytic spread of the corresponding binomial edge ideals via combinatorial and convex geometric means.

Theorems & Definitions (53)

• Theorem 1.1: Theorem \ref{['bounds']}
• Theorem 1.2: Theorem \ref{['thm-main']}
• Definition 2.1
• Lemma 2.2: MR2724673
• Definition 2.3
• Remark 2.4
• Definition 2.5
• Theorem 2.6: Lang
• Remark 2.7
• Theorem 3.1