{
  "id": "2404.10566",
  "version": "v1",
  "published": "2024-04-16T13:40:22.000Z",
  "updated": "2024-04-16T13:40:22.000Z",
  "title": "Exploring Homological Properties of Independent Complexes of Kneser Graphs",
  "authors": [
    "Ziqin Feng",
    "Guanghui Wang"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We discuss the topological properties of the independence complex of Kneser graphs, Ind(KG$(n, k))$, with $n\\geq 3$ and $k\\geq 1$. 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 $p=1/2\\cdot {2n+k\\choose 2n}$. 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 $\\tilde{H}_q(\\mathcal{VR}(\\mathcal{F}^{[\\ell]}_n; 2(n-1)))$ being non-trivial, then $$\\text{rank} (\\tilde{H}_q(\\mathcal{VR}(\\mathcal{F}_n^{[m]}; 2(n-1)))\\geq \\sum_{i=\\ell}^m{i-2\\choose \\ell-2}\\cdot \\text{rank} (\\tilde{H}_q(\\mathcal{VR}(\\mathcal{F}_n^{[\\ell]}; 2(n-1))). $$ Since the independence complex Ind(KG$(n, k))$ and the Vietoris-Rips complex $\\mathcal{VR}(\\mathcal{F}^{[2n+k]}_n; 2(n-1))$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-16T13:40:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "kneser graphs",
      "exploring homological properties",
      "independent complexes",
      "dimensional homology",
      "homology propagation result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}