Matching numbers and the regularity of the Rees algebra of an edge ideal

TL;DR

This paper investigates the regularity of the Rees algebra of an edge ideal, establishing bounds based on graph invariants like matching numbers, especially under the condition of normality.

Contribution

It provides new bounds for the regularity of the Rees algebra of edge ideals using matching numbers, highlighting the role of normality and induced matchings.

Findings

Matching number is a lower bound for regularity when Rees algebra is normal.

Regularity is at most matching number plus one if the Rees algebra is normal.

Induced matching number serves as a lower bound for regularity in general.

Abstract

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.