The Castelnuovo-Mumford regularity of binomial edge ideals

TL;DR

This paper proves a conjectured upper bound on the Castelnuovo-Mumford regularity of binomial edge ideals for graphs, showing it is at most n-1 for non-path graphs, and explores how regularity behaves under graph join operations.

Contribution

It establishes a proven upper bound for the regularity of binomial edge ideals and analyzes its behavior under graph join operations, confirming a conjecture by Matsuda and Murai.

Findings

Regularity of J_G is at most n-1 for non-path graphs.

Confirmed the conjectured upper bound for the regularity.

Analyzed the effect of graph join on the regularity.

Abstract

We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that $\mathrm{reg}(J_G)\leq n-1$ for any graph $G$ with $n$ vertices, which is not a path. Moreover, we study the behavior of the regularity of binomial edge ideals under the join product of graphs.