A Characterization of Edge Ideals with $reg(R/I(G)) = 3$

TL;DR

This paper provides a complete characterization of graphs whose edge ideals have a regularity of exactly 3, advancing understanding of the algebraic properties linked to graph structures.

Contribution

It offers a novel, comprehensive classification of graphs with edge ideals of regularity 3, filling a gap in algebraic graph theory.

Findings

Identifies all graphs with $ eg(R/I(G))=3$

Establishes criteria linking graph structure to algebraic regularity

Completes the classification for this specific regularity value

Abstract

Let $G$ be a graph and $I(G)$ its edge ideal. In this paper, we give a complete characterization of the graphs $G$ for which $\reg(R/I(G)) = 3$.