论文标题
在越野式搭配上（带比例3）超立方体图
On Vietoris--Rips complexes (with scale 3) of hypercube graphs
论文作者
论文摘要
对于公制空间$（x，d）$和一个比例参数$ r \ geq 0 $，vietoris-rips complex $ \ nathcal {vr}（x; r）$是顶点$ x $的简单复杂，其中有限的$ nite $ nite $σ\ subseteq x $是unick x $，并且仅在$ nif diemeter of $ nif $ nif $ nif $ nif $ nist $ pess $。对于$ n \ geq 1 $，令$ \ mathbb {i} _n $表示$ n $ dimensional HyperCube图。在本文中，我们表明$ \ Mathcal {vr}（\ Mathbb {i} _n; r）$仅在尺寸$ 4 $和$ 7 $中具有非微不足道的同源性。因此，我们最近回答了Adamaszek和Adams提出的一个问题。 如果可以通过反复删除最多$ d $的面孔，则（有限的）简单复合物$δ$是$ d $ collapsible，如果可以将其简化为void Complex，而该面孔最多包含在$Δ$的独特最大面中。 $δ$的可折叠数是最小整数$ d $，因此$δ$是$ d $ collapsible。我们表明，$ \ mathcal {vr}的可折叠率数（\ mathbb {i} _n; r）$是$ 2^r $ for $ r \ in \ in \ {2，3 \} $。
For a metric space $(X, d)$ and a scale parameter $r \geq 0$, the Vietoris-Rips complex $\mathcal{VR}(X;r)$ is a simplicial complex on vertex set $X$, where a finite set $σ\subseteq X$ is a simplex if and only if diameter of $σ$ 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 $Δ$ 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 $Δ$. The collapsibility number of $Δ$ is the minimum integer $d$ such that $Δ$ is $d$-collapsible. We show that the collapsibility number of $\mathcal{VR}(\mathbb{I}_n;r)$ is $2^r$ for $r \in \{2, 3\}$.
