# Certain Algebraic Invariants of Edge Ideals of Join of Graphs

## TL;DR

This paper investigates algebraic invariants of edge ideals of graph joins, proving conjectures for specific graph classes, and exploring relationships between algebraic properties like regularity, depth, and multiplicity.

## Contribution

It introduces new results on the regularity of symbolic powers of edge ideals for join graphs and proves Minh's conjecture for several graph classes.

## Key findings

- Proves Minh's conjecture for wheel, complete multipartite, and certain co-chordal graphs.
- Identifies classes of graphs with edge ideals having regularity three.
- Shows multiplicity of edge ideals is independent of depth, dimension, regularity, and h-polynomial degree.

## Abstract

Let $G$ be a simple graph and $I(G)$ be its edge ideal. In this article, we study the Castelnuovo-Mumford regularity of symbolic powers of edge ideals of join of graphs. As a consequence, we prove Minh's conjecture for wheel graphs, complete multipartite graphs, and a subclass of co-chordal graphs. We obtain a class of graphs whose edge ideals have regularity three. By constructing graphs, we prove that the multiplicity of edge ideals of graphs is independent from the depth, dimension, regularity, and degree of $h$-polynomial. Also, we demonstrate that the depth of edge ideals of graphs is independent from the regularity and degree of $h$-polynomial by constructing graphs.