Extremal Betti numbers of edge ideals

TL;DR

This paper constructs specific graphs with prescribed regularity and a set number of extremal Betti numbers, advancing understanding of the algebraic properties of edge ideals.

Contribution

It provides a method to explicitly construct graphs with given regularity and extremal Betti number counts, filling a gap in the algebraic combinatorics of edge ideals.

Findings

Constructed graphs with specified regularity and extremal Betti numbers.

Established a link between graph structure and algebraic invariants.

Enhanced the classification of edge ideals based on Betti number properties.

Abstract

Given integers $r$ and $b$ with $1 \leq b \leq r$, a finite simple connected graph $G$ for which ${\rm reg}(S/I(G)) = r$ and the number of extremal Betti numbers of $S/I(G)$ is equal to $b$ will be constructed.