# Upper bounds for the regularity of symbolic powers of certain classes of edge ideals

## TL;DR

This paper establishes upper bounds for the regularity of symbolic powers of edge ideals in graphs and shows cases where symbolic and ordinary powers have equal regularity.

## Contribution

It provides new upper bounds for the regularity of symbolic powers and identifies classes of graphs where symbolic and ordinary powers share the same regularity.

## Key findings

- Upper bounds for regularity of symbolic powers derived
- Regularity of symbolic and ordinary powers coincide for certain graph classes
- Enhanced understanding of the algebraic properties of edge ideals

## Abstract

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal in a polynomial ring over a field $\mathbb{K}$. In this paper, we obtain upper bounds for the Castelnuovo-Mumford regularity of symbolic powers of certain classes of edge ideals. We also prove that for several classes of graphs, the regularity of symbolic powers of their edge ideals coincides with that of their ordinary powers.