On Vietoris–Rips Complexes (with Scale 3) of Hypercube Graphs
Samir Shukla · SIAM Journal on Discrete Mathematics · 2023
Abstract. For a metric space [Formula: see text] and a scale parameter [Formula: see text], the Vietoris–Rips complex [Formula: see text] is a simplicial complex on vertex set [Formula: see text], where a finite set [Formula: see text] is a simplex if and only if the diameter of [Formula: see text] is at most [Formula: see text]. For [Formula: see text], let [Formula: see text] denote the [Formula: see text]-dimensional hypercube graph. In this paper, we show that [Formula: see text] has nontrivial reduced homology only in dimensions 4 and 7. Therefore, we answer a question posed by Adamaszek and Adams recently. A (finite) simplicial complex [Formula: see text] is [Formula: see text]-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most [Formula: see text] that is contained in a unique maximal face of [Formula: see text]. The collapsibility number of [Formula: see text] is the minimum integer [Formula: see text] such that [Formula: see text] is [Formula: see text]-collapsible. We show that the collapsibility number of [Formula: see text] is [Formula: see text] for [Formula: see text].