论文标题
超立方体图的Čech综合体
Čech complexes of hypercube graphs
论文作者
论文摘要
有限的简单图$ g $的čech综合体是图中的一块神经综合体，每个顶点都以一个球为中心。更准确地说，让čech复合物$ \ Mathcal {n}（g，r）$成为所有半径$ \ frac {r} {r} {2} $中心的封闭球的神经，以$ g $的顶点为中心，这些球以图形$ g $ g $ g $ g $ g $ g $ g $ g $ g $（配备了最简短的路径）。当$ r = 1 $时，简单的复杂$ \ MATHCAL {n}（g，r）$等于图$ g $，当$ r $小于$ g $的最短循环的一半时，均值等同于图$ g $。对于$ r $的较高值，$ \ MATHCAL {n}（g，r）$的拓扑并不理解。我们考虑$ n $ dimensional hypercube图$ \ mathbb {i} _n $，$ 2^n $ vertices。我们的主要结果如下。首先，当$ r = 2 $时，我们证明了čech复合物$ \ mathcal {n}（\ mathbb {i} _n，2）$是同质的，相当于所有$ n \ ge 1 $的2个楔形物，并且我们计算出在此楔形总和中出现的2个球的数量。其次，当$ r = 3 $时，我们表明$ \ nathcal {n}（\ mathbb {i} _n，3）$最多与尺寸的简单复合物相当于$ n \ ge 4 $，而对$ n \ ge 4 $降低了$ \ m narycal {在所有其他维度中为零。 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 HyperCube图的čech复合物。
A Čech 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 Čech 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 Čech 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 Čech complexes of hypercube graphs.
