图的极小欧氏表示

非欧杆节点框架的无穷小刚度

Infinitesimal rigidity for non-Euclidean bar-joint frameworks

在非欧几里得空间（R^2，||.||_ q）是用（2，2）-紧图刻画的。具体地说，平面上一般放置的杆-关节框架（G，p）关于非欧l^q范数是极小无穷小刚性的当且仅当基础图G =（V，E）包含2条|V| -2边且每个子图H=（V（H），E（H））至多包含2条|V（H）|-2个边缘。

刚性图的欧氏嵌入的计数

Counting Euclidean embeddings of rigid graphs

一个图在$\mathbb{R}^d$中被称为（一般）刚性的，如果对于任意的充分一般的边长度的选择，它可以以有限数量的不同方式嵌入$\mathbb{R}^d$中，模刚性变换。在这里，我们处理的问题，确定的最大数量的平面欧氏嵌入的顶点数的函数。利用距离几何理论，我们得到多项式系统，完全捕捉给定图的结构。因此，计算给定刚性图的欧几里得嵌入的数目，减少到计算相应多项式系统的根的问题。

距离正则图在欧几里得空间中的最优嵌入

Optimal Embeddings of Distance Regular Graphs into Euclidean Spaces

本文给出了距离正则图在欧几里得空间中的最小失真嵌入的一个下界。我们使用下界来寻找汉明图、约翰逊图和所有强正则图的最小失真。我们的技术涉及半定规划和利用最优化问题的代数结构，使找到最小失真的下界的问题减少到一个关于正交多项式的分析问题。

随机几何图中图距离与欧氏距离的关系

On the relation between graph distance and Euclidean distance in random geometric graphs

给定随机几何图的任意两个顶点u，v，用d_E（u，v）表示它们的欧氏距离，用d_G（u，v）表示它们的图距离。用dE（u，v）求dG（u，v）的上界问题在文献中得到了广泛的关注。在本文中，我们改进了r=omega（sqrt（log n））的值的这些上界（即r高于连通性阈值）。我们的结果也改善了最有名的估计直径的随机几何图。我们还根据d_E（u，v）提供了d_G（u，v）的下界。

Embedding graphs in Euclidean space

The dimension of a graph $G$ is the smallest $d$ for which its vertices can be embedded in $d$-dimensional Euclidean space in the sense that the distances between endpoints of edges equal $1$ (but there may be other unit distances). Answering a question of Erdős and Simonovits [Ars Combin. 9 (1980) 229--246], we show that any graph with less than $\binom{d+2}{2}$ edges has dimension at most $d$. Improving their result, we prove that that the dimension of a graph with maximum degree $d$ is at most $d$. We show the following Ramsey result: if each edge of the complete graph on $2d$ vertices is c…

图上具有欧几里得边的最近邻马尔可夫点过程

Nearest-neighbour Markov point processes on graphs with Euclidean edges

本文提出了一种在具有欧几里得边的图上定义的最近邻马尔可夫点过程，并证明了其满足一致性条件。

无限图的刚性

The rigidity of infinite graphs

给出了R^d中可数无限单图关于l^p范数的欧几里德和非欧几里德配置的刚性理论，其中d>1且1<p< \infty .推广得到的Laman和Henneberg组合特征的一般无穷小刚性有限图在欧氏平面。此外，Tay的多图表征的刚性一般有限体杆框架在d维欧几里得空间推广到非欧lp范数和可数无限图。对所有维数和范数，证明了一般刚性可数单图是有限图的包含塔的直接极限，其中包含满足相对刚性性质.对于d>2，对于R^d中的一般位置是刚性的可数图可能会失去序列刚性的更强性质，而对于d=2，与序列刚性的等价性是从广义的拉曼特征得到的。应用程序给出的灵活性的非欧几里德凸多面体和无穷小和连续刚性紧凑的无限小面单纯多面体。

Embedding Euclidean Distance Graphs in $\mathbb{R}^n$ and $\mathbb{Q}^n$

For $S \subseteq \mathbb{R}$, positive integer $n$, and $d > 0$, let $G(S^n, d)$ be the graph whose vertex set is $S^n$ where any two vertices are adjacent if and only if they are Euclidean distance $d$ apart. The primary question we will consider in our work is as follows. Given $n$ and distance $d$ actually realized as a distance between points of the rational space $\mathbb{Q}^n$, does there exist a finite graph $G$ that appears as a subgraph of $G(\mathbb{Q}^n, d)$ but not as a subgraph of $G(\mathbb{R}^{n-1}, 1)$? We answer this question affirmatively for $n \leq 5$, and along the way, re…

Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can…