# Invariants of the symbolic powers of edge ideals

## TL;DR

This paper studies the algebraic invariants of symbolic powers of edge ideals for specific graph classes, providing explicit descriptions, formulas for Waldschmidt constants, and regularity comparisons.

## Contribution

It offers explicit descriptions of symbolic powers for certain graphs and computes key invariants like Waldschmidt constants and resurgence, revealing regularity coincidences.

## Key findings

- Explicit description of symbolic powers for clique sums of odd cycles.
- Calculation of Waldschmidt constants for specific graph classes.
- Proof that Castelnuovo-Mumford regularity of symbolic and ordinary powers coincide for complete graphs.

## Abstract

Let $G$ be a graph and $I=I(G)$ be its edge ideal. When $G$ is the clique sum of two different length odd cycles joined at single vertex then we give an explicit description of the symbolic powers of $I$ and compute the Waldschmidt constant. When $G$ is complete graph then we describe the generators of the symbolic powers of $I$ and compute the Waldschmidt constant and the resurgence of $I$. Moreover for complete graph we prove that the Castelnuvo-Mumford regularity of the symbolic powers and ordinary powers of the edge ideal coincide.