Algebraic properties of edge ideals via combinatorial topology

TL;DR

This paper uses combinatorial topology to analyze algebraic properties of edge ideals, providing new proofs, interpretations, and formulas for invariants like Betti numbers and projective dimension, applicable to various graph classes.

Contribution

It introduces a unified topological approach to study edge ideals, strengthening existing results and deriving new formulas for algebraic invariants.

Findings

Provided new proofs for Cohen-Macaulay properties of certain graph classes.

Derived recursive relations for Betti number generating functions.

Established formulas for the projective dimension of edge ideals.

Abstract

We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity, Betti numbers, and (sequentially) Cohen-Macaulay properties of edges ideals associated to chordal, complements of chordal, and Ferrers graphs, as well as trees and forests. Our approach unifies (and in many cases strengthens) these results and also provides combinatorial/enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph (adding whiskers and ears) as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs, (not necessarily pure) vertex decomposability, shellability, etc.