On Vietoris--Rips complexes (with scale 3) of hypercube graphs

For a metric space $(X,d)$ and a scale parameter $r\ge 0$, the Vietoris-Rips complex $\mathcal{V}\mathcal{R}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $\sigma \subseteq X$ is a simplex if and only if diameter of $\sigma$ is at most $r$. For $n\ge 1$, let ${\mathbb{I}}_n$ denotes the $n$-dimensional hypercube graph. In this paper, we show that $\mathcal{V}\mathcal{R}({\mathbb{I}}_n;r)$ has non trivial reduced homology only in dimensions $4$ and $7$. Therefore, we answer a question posed by Adamaszek and Adams recently. A (finite) simplicial complex $\mathrm{\Delta }$ is $d$-collapsible if it can be reduced to the void complex by repeatedly removing a face of size at most $d$ that is contained in a unique maximal face of $\mathrm{\Delta }$. The collapsibility number of $\mathrm{\Delta }$ is the minimum integer $d$ such that $\mathrm{\Delta }$ is $d$-collapsible. We show that the collapsibility number of $\mathcal{V}\mathcal{R}({\mathbb{I}}_n;r)$ is $2^r$ for $r\in \{2,3\}$.