$\mathbb{R}^d$中的{ \em忠实（单位）距离图}是其顶点集是$d$维欧几里得空间的有限子集的图，其中两个顶点相邻当且仅当它们之间的欧氏距离正好是$1$。$\mathbb{R}^d$中的{ \em（单位）距离图}是这样的图的任何子图。在本文的第一部分，我们专注于这两类图之间的差异。特别地，我们证明了对于任何固定的$d$，$\mathbb{R}^d$中$n$标号顶点上的忠实距离图的个数是$2^{(1+o(1)) d n \log_2 n}$，并给出了已知事实$\mathbb{R}^d$中$n$标号顶点上的距离图的个数是$2^{(1-1/\lfloor d/2 \rfloor +o(1))n^2/2}$的一个简短证明.我们还研究了涉及这些图的几个Ramsey型量的行为。%和高围长图。本文的第二部分讨论了在$\mathbb R^d$中确定与忠实距离图不同构的图的最小可能边数的问题。
英文摘要：
A {\em faithful (unit) distance graph} in $\mathbb{R}^d$ is a graph whose set of vertices is a finite subset of the $d$-dimensional Euclidean space, where two vertices are adjacent if and only if the Euclidean distance between them is exactly $1$. A {\em (unit) distance graph} in $\mathbb{R}^d$ is any subgraph of such a graph. In the first part of the paper we focus on the differences between these two classes of graphs. In particular, we show that for any fixed $d$ the number of faithful distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1+o(1)) d n \log_2 n}$, and give a short proof of the known fact that the number of distance graphs in $\mathbb{R}^d$ on $n$ labelled vertices is $2^{(1-1/\lfloor d/2 \rfloor +o(1))n^2/2}$. We also study the behavior of several Ramsey-type quantities involving these graphs. % and high-girth graphs from these classes. In the second part of the paper we discuss the problem of determining the minimum possible number of edges of a graph which is not isomorphic to a faithful distance graph in $\mathbb R^d$.
A recent generalization of the Erdős Unit Distance Problem, proposed by Palsson, Senger and Sheffer, asks for the maximum number of unit distance paths with a given number of vertices in the plane and in $3$-space. Studying a variant of this question, we prove sharp bounds on the number of unit distance paths and cycles on the sphere of radius $1/\sqrt{2}$. We also consider a similar problem about $3$-regular unit distance graphs in $\mathbb{R}^3$.
We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.
We introduce a new graph minimization method, in which it is required to preserve some graph property and there is an effective procedure for checking this property. We applied this method to minimize 5-chromatic unit-distance graphs and obtained a graph with 509 vertices and 2442 edges.
In this paper we compute bounds for signless Laplacian energy, distance signless Laplacian eigenvalues and signless Laplacian energy of unitary addition Cayley graph G_{n}. We also obtain distance Laplacian eigenvalues and distance Laplacian energy of G_{n}.
This paper is devoted to the development of algorithms for finding unit distance graphs with chromatic number greater than 4, embedded in a two-dimensional sphere or plane. Such graphs provide a lower bound for the Nelson-Hadwiger problem on the chromatic number of the plane and its generalizations to the case of the sphere. A series of 5-chromatic unit distance graphs on 64513 vertices embedded into the plane is constructed. Unlike previously known examples, these graphs do not contain the Moser spindle as a subgraph. The construction of 5-chromatic graphs embedded in a sphere at two values o…
The unit distance graph $G_{\mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $\mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{\mathbb{R}^2}^1$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G_{\mathbb{R}^d}^1$ to closed convex subsets $X$ of $\mathbb{R}^d$. We show that the graph $G_{\mathbb{R}^d}^1[X]$ is connected precisely when the radius o…
We prove that a graph on up to 9 vertices is a unit-distance graph if and only if it does not contain one of 74 so-called minimal forbidden graphs. This extends the work of Chilakamarri and Mahoney (1995), who provide a similar classification for unit-distance graphs on up to 7 vertices.
We investigate the box dimensions of compact sets in $\mathbb{R}^2$ that contain a unit distance in every direction (such sets may have zero Hausdorff dimension). Among other results, we show that the lower box dimension must be at least $\frac{4}{7}$ and can be as low as $\frac{2}{3}$. This quantifies in a certain sense how far the unit circle is from being a difference set.
We prove that every set of $n$ points in $\mathbb{R}^3$ spans $O(n^{295/197+ε})$ unit distances. This is an improvement over the previous bound of $O(n^{3/2})$. A key ingredient in the proof is a new result for cutting circles in $\mathbb{R}^3$ into pseudo-segments.
A complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. %A complex unit gain graph($ \mathbb{T} $-gain graph) is a simple graph where each orientation of an edge is given a complex unit, and its inverse is assigned to the opposite orientation of the edge. In this article, we propose gain distance matrices for $ \mathbb{T} $-gain graphs. These notions generalize the corresponding known concepts of distance matrices and sig…
We consider the problem of estimating the distance between two bodies of volume $\varepsilon$ located inside a $n$-dimensional ball $U$ of unit volume for $n\to\infty$. Let $A$ be a closed set with a smooth boundary of the volume $\varepsilon$ ($0 \leq \varepsilon \leq 1/2$) inside a $n$-dimensional ball $U$ of unit volume that implements among all the sets of volume $\varepsilon$ is a set with the smallest possible free surface area, lying in one half-space with respect to a certain hyperplane that passes through the center of the ball. Then $A$ has the same free surface area as the set repre…
KU Leuven（荷语鲁汶大学）；Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences（奥地利科学院约翰·拉德纳计算与应用数学研究所）；Faculty of Information Technology, Czech Technical University in Prague（布拉格捷克理工大学信息技术学院）；School of Mathematical Sciences, Lancaster University, UK（兰开斯特大学数学科学学院）
我们证明了R^3中n个点所确定的单位距离的个数是O（n^{3/2}），稍微改进了Clarkson等人在1990年建立的界。新的证明使用了Guth和Katz最近引入的多项式分割技术[arXiv：1011.4105]。当这篇论文还处于草稿阶段时，Joshua Zahl [arXiv：1104.4987]在arXiv上发表了我们主要结果的类似证明。
We present a new method for reducing the size of graphs with a given property. Our method, which is based on clausal proof minimization, allowed us to compute several 553-vertex unit-distance graphs with chromatic number 5, while the smallest published unit-distance graph with chromatic number 5 has 1581 vertices. The latter graph was constructed by Aubrey de Grey to show that the chromatic number of the plane is at least 5. The lack of a 4-coloring of our graphs is due to a clear pattern enforced on some vertices. Also, our graphs can be mechanically validated in a second, which suggests that…
The Erd\H os unit distance conjecture in the plane says that the number of pairs of points from a point set of size $n$ separated by a fixed (Euclidean) distance is $\leq C_ε n^{1+ε}$ for any $ε>0$. The best known bound is $Cn^{\frac{4}{3}}$. We show that if the set under consideration is well-distributed and the fixed distance is much smaller than the diameter of the set, then the exponent $\frac{4}{3}$ is significantly improved. Corresponding results are also established in higher dimensions. The results are obtained by solving the corresponding continuous problem and using a continuous-to-d…
Let $G=(V,E)$ be a finite, combinatorial graph. We define a notion of curvature on the vertices $V$ via the inverse of the resistance distance matrix. We prove that this notion of curvature has a number of desirable properties. Graphs with curvature bounded from below by $K>0$ have diameter bounded from above. The Laplacian $L=D-A$ satisfies a Lichnerowicz estimate, there is a spectral gap $λ_2 \geq 2K$. We obtain matching two-sided bounds on the maximal commute time between any two vertices in terms of $|E| \cdot |V|^{-1} \cdot K^{-1}$. Moreover, we derive quantitative rates for the mixing ti…
A finite metric space is called here distance degree regular if its distance degree sequence is the same for every vertex. A notion of designs in such spaces is introduced that generalizes that of designs in $Q$-polynomial distance-regular graphs. An approximation of their cumulative distribution function, based on the notion of Christoffel function in approximation theory is given. As an application we derive limit laws on the weight distributions of binary orthogonal arrays of strength going to infinity. An analogous result for combinatorial designs of strength going to infinity is given.
