Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Vietoris--Rips complexes (with scale 3) of hypercube graphs   For a metric space $(X, d)$ and a scale parameter $r \geq 0$, the
Vitoris-Rips complex $\mathcal{VR}(X;r)$ 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 $n \geq 1$, let $\mathbb{I}_n$ denotes
the $n$-dimensional hypercube graph. In this paper, we show that
$\mathcal{VR}(\mathbb{I}_n;r)$ 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 $\Delta$ 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 $\Delta$. The collapsibility
number of $\Delta$ is the minimum integer $d$ such that $\Delta$ is
$d$-collapsible. We show that the collapsibility number of
$\mathcal{VR}(\mathbb{I}_n;r)$ is $2^r$ for $r \in \{2, 3\}$.

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The categories of this paper are: dg-ga
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The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
30
The categories of this paper are: q-alg
32
The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: stat
0
The categories of this paper are: hep-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: alg-geom