On the homology of independence complexes

TL;DR

This paper develops a spectral sequence approach to analyze the homology of independence complexes of graphs, revealing connections between maximal independent sets and homology vanishing, applicable to paths and cycles.

Contribution

It introduces a deformation-based double complex and spectral sequence framework to study independence complex homology, linking combinatorial properties to topological invariants.

Findings

Spectral sequence converges to zero, revealing homology structure.

Relation between maximal independent sets and homology vanishing.

Applicable to paths and cyclic graphs.

Abstract

The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of $G$ and various subgraphs of $G$, obtained by removing independent sets and their neighborhoods from $G$. It is shown that this spectral sequence may be used to study the homology of $\mathrm{Ind}(G)$. Furthermore, a careful investigation of the sequence's first page exhibits a relation between the cardinality of maximal independent sets in $G$ and the vanishing of certain homology groups of the independence complexes of some subgraphs of $G$. This relation is shown to hold for all paths and cyclic graphs.