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May 15, 2026

Exploring homological properties of independence complexes of Kneser graphs

Puntos clave

This study investigates the homological properties of the independence complexes associated with Kneser graphs.
Analyzed maximal simplices through projective planes.
Utilized cross-polytopal generators to establish lower bounds for homology ranks.
Investigated the relationship between independence complexes and Vietoris-Rips complexes.
Identified homology generators for 6-dimensional homology of $Ind(KG(3,k))$.
Established lower bounds for the rank of $p$-dimensional homology for $Ind(KG(n,k))$.
Showed that independence and Vietoris-Rips complexes are equivalent, enabling homology propagation results.

Resumen

We discuss the topological properties of the independence complex of Kneser graphs, Ind (KG (n, k) ), with n 3 and k 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 2n+k2n. Denote F₍^m to be the collection of n-subsets of m equipped with the symmetric difference metric. We prove that if l is the minimal integer with the qth dimensional reduced homology Hₐ (VR (F₍^l;2 (n-1) ) ) being non-trivial, then rank (Hₐ (VR (F₍^m;2 (n-1) ) ) ₈=₋^mi-2l-2 rank (Hₐ (VR (F₍^l;2 (n-1) ) ). Since we know the independence complex Ind (KG (n, k) ) and the Vietoris-Rips complex VR (F₍^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.

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