Tuples of homological invariants of edge ideals

TL;DR

This paper characterizes the combined homological invariants of edge ideals of graphs with a fixed number of vertices, providing a complete description of their dimension, depth, and regularity.

Contribution

It offers a comprehensive determination of the tuples of dimension, depth, and regularity for edge ideals of graphs with a fixed vertex count, filling a gap in understanding their homological properties.

Findings

Explicit formulas for invariants based on graph properties

Complete classification of invariant tuples for fixed vertex counts

Insights into the relationship between graph structure and algebraic invariants

Abstract

Let $G$ be a graph and $I(G)$ its edge ideal. In this paper, we completely determine the tuples $(\dim R/I(G), \depth (R/I(G)), \reg (R/I(G)))$ when the number of vertices is fixed for any graphs $G$.