图的单形与谱

图的单形与谱（续）

Simplices and spectra of graphs, continued

本文证明了n维单形的余维2面的n-2维体积是其边长体积的代数无关量。证明涉及到Kneser图的特征值的计算。我们还构建家庭的非同胚单形不确定其余维2面积。

A zonotope and a product of two simplices with disconnected flip graphs

We give an example of a three-dimensional zonotope whose set of tight zonotopal tilings is not connected by flips. Using this, we show that the set of triangulations of $Δ^4 \times Δ^n$ is not connected by flips for large $n$. Our proof makes use of a non-explicit probabilistic construction.

Effective resistance is more than distance: Laplacians, Simplices and the Schur complement

This article discusses a geometric perspective on the well-known fact in graph theory that the effective resistance is a metric on the nodes of a graph. The classical proofs of this fact make use of ideas from electrical circuits or random walks; here we describe an alternative approach which combines geometric (using simplices) and algebraic (using the Schur complement) ideas. These perspectives are unified in a matrix identity of Miroslav Fiedler, which beautifully summarizes a number of related ideas at the intersection of graphs, Laplacian matrices and simplices, with the metric property o…

关于平面Rips复形及其对应单位圆盘图的分类

On the Classification of Planar-Rips complexes and their corresponding unit disk graphs

本文对可实现为平面点集Vietoris-Rips复形的伪流形与弱伪流形进行单纯同构分类，并同伦分类二维纯闭平面Rips复形，进而刻画一类所有极大团基数相同的单位圆盘图。