The monomial ideal of independent sets associated to a graph

TL;DR

This paper introduces a monomial ideal linked to a graph's independent sets, analyzing its algebraic properties and relating them to the graph's combinatorial features, including Betti numbers and Cohen-Macaulay conditions.

Contribution

It defines a new monomial ideal associated with independent sets and provides formulas and characterizations connecting algebraic invariants to graph combinatorics.

Findings

Computed the minimal primary decomposition of the ideal

Characterized when the ideal is Cohen-Macaulay

Derived a formula for Betti numbers based on the independence polynomial

Abstract

Independent sets play a key role into the study of graphs and important problems arising in graph theory reduce to them. We define the monomial ideal of independent sets associated to a finite simple graph and describe its homological and algebraic invariants in terms of the combinatorics of the graph. We compute the minimal primary decomposition and characterize the Cohen--Macaulay ideals. Moreover, we provide a formula for computing the Betti numbers, which depends only on the coefficients of the independence polynomial of the graph.