Exploring Homological Properties of Independent Complexes of Kneser Graphs

Abstract

We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n,k))$, with $n\ge 3$ and $k\ge 1$. By identifying one kind of maximal simplices through projective planes, we obtain homology generators for the $6$-dimensional homology of the complex Ind(KG$(3,k))$. Using cross-polytopal generators, we provide lower bounds for the rank of $p$-dimensional homology of the complex Ind(KG$(n,k))$ where $p=1/2\cdot \left(\genfrac{}{}{0ex}{}{2n+k}{2n}\right)$. Denote ${\mathcal{F}}_n^{[m]}$ to be the collection of $n$-subsets of $[m]$ equipped with the symmetric difference metric. We prove that if $\ell$ is the minimal integer with the $q$th dimensional reduced homology ${\tilde{H}}_q(\mathcal{V}\mathcal{R}({\mathcal{F}}_n^{[\ell ]};2(n-1)))$ being non-trivial, then $$\text{rank}({\tilde{H}}_q(\mathcal{V}\mathcal{R}({\mathcal{F}}_n^{[m]};2(n-1)))\ge \sum _{i=\ell }^m\left(\genfrac{}{}{0ex}{}{i-2}{\ell -2}\right)\cdot \text{rank}({\tilde{H}}_q(\mathcal{V}\mathcal{R}({\mathcal{F}}_n^{[\ell ]};2(n-1))).$$ Since the independence complex Ind(KG$(n,k))$ and the Vietoris-Rips complex $\mathcal{V}\mathcal{R}({\mathcal{F}}_n^{[2n+k]};2(n-1))$ are the same, we obtain a homology propagation result in the setting of independence complexes of Kneser graphs. Connectivity of these complexes is also discussed in this paper.