Regularity comparison of symbolic powers, integral closure of powers and powers of edge ideals

TL;DR

This paper investigates the regularity properties of symbolic powers, integral closures, and ordinary powers of edge ideals in graphs, revealing conditions under which these regularities coincide.

Contribution

It provides new results on the regularity of integral closures of powers of edge ideals, especially for graphs with at most two odd cycles.

Findings

Regularity of integral closures matches that of powers for graphs with up to two odd cycles.

Established relationships between symbolic powers, integral closures, and regularity.

Extended understanding of algebraic properties of edge ideals in graph theory.

Abstract

We study the regularity of small symbolic powers and integral closure of small powers of edge ideals. We also prove that the regularity of integral closure of powers of edge ideals of graphs with at most two odd cycles is the same as the regularity of their powers.