# Binomial edge ideals of regularity $3$

## TL;DR

This paper characterizes graphs with binomial edge ideals of regularity 3, provides an explicit formula for the regularity of join products, and discusses implications for Gorenstein properties and open problems.

## Contribution

It offers a complete characterization of graphs with binomial edge ideals of regularity 3 and derives a new formula for regularity of join products of graphs.

## Key findings

- Characterization of graphs with regularity 3 for binomial edge ideals
- Explicit formula for regularity of join product of two graphs
- Disproof of a conjecture on regularity of weakly closed graphs

## Abstract

Let $J_G$ be the binomial edge ideal of a graph $G$. We characterize all graphs whose binomial edge ideals, as well as their initial ideals, have regularity $3$. Consequently we characterize all graphs $G$ such that $J_G$ is extremal Gorenstein. Indeed, these characterizations are consequences of an explicit formula we obtain for the regularity of the binomial edge ideal of the join product of two graphs. Finally, by using our regularity formula, we discuss some open problems in the literature. In particular we disprove a conjecture in \cite{CDI} on the regularity of weakly closed graphs.