{
  "id": "2212.05871",
  "version": "v1",
  "published": "2022-12-12T13:17:56.000Z",
  "updated": "2022-12-12T13:17:56.000Z",
  "title": "Čech complexes of hypercube graphs",
  "authors": [
    "Henry Adams",
    "Samir Shukla",
    "Anurag Singh"
  ],
  "comment": "14 pages",
  "categories": [
    "math.CO",
    "math.AT"
  ],
  "abstract": "A \\v{C}ech complex of a finite simple graph $G$ is a nerve complex of balls in the graph, with one ball centered at each vertex. More precisely, let the \\v{C}ech complex $\\mathcal{N}(G,r)$ be the nerve of all closed balls of radius $\\frac{r}{2}$ centered at vertices of $G$, where these balls are drawn in the geometric realization of the graph $G$ (equipped with the shortest path metric). The simplicial complex $\\mathcal{N}(G,r)$ is equal to the graph $G$ when $r=1$, and homotopy equivalent to the graph $G$ when $r$ is smaller than half the length of the shortest loop in $G$. For higher values of $r$, the topology of $\\mathcal{N}(G,r)$ is not well-understood. We consider the $n$-dimensional hypercube graphs $\\mathbb{I}_n$ with $2^n$ vertices. Our main results are as follows. First, when $r=2$, we show that the \\v{C}ech complex $\\mathcal{N}(\\mathbb{I}_n,2)$ is homotopy equivalent to a wedge of 2-spheres for all $n\\ge 1$, and we count the number of 2-spheres appearing in this wedge sum. Second, when $r=3$, we show that $\\mathcal{N}(\\mathbb{I}_n,3)$ is homotopy equivalent to a simplicial complex of dimension at most 4, and that for $n\\ge 4$ the reduced homology of $\\mathcal{N}(\\mathbb{I}_n, 3)$ is nonzero in dimensions 3 and 4, and zero in all other dimensions. Finally, we show that for all $n\\ge 1$ and $r\\ge 0$, the inclusion $\\mathcal{N}(\\mathbb{I}_n, r)\\hookrightarrow \\mathcal{N}(\\mathbb{I}_n, r+2)$ is null-homotopic, providing a bound on the length of bars in the persistent homology of \\v{C}ech complexes of hypercube graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-12T13:17:56.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N31",
      "55U10",
      "05E45"
    ],
    "keywords": [
      "homotopy equivalent",
      "simplicial complex",
      "shortest path metric",
      "finite simple graph",
      "dimensional hypercube graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}