关于度量空间距离中心对应的图
On the Graph Corresponding to Center of Distances of Metric Space
- Institute of Applied Mathematics and Mechanics of NASU（乌克兰国家科学院应用数学和力学研究所）
- University of Turku（图尔库大学）
- Donbas State Engineering Academy（顿巴斯国立工程学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究定义了度量空间的中心图，证明了有限度量空间及有限超度量空间的中心图边数下界，并刻画了达到边数下界的中心图的同构类。
AI中文摘要：
度量空间$(Y,\rho)$的距离中心$C(Y)$是满足对每个点$p\in Y$，方程$\rho(y,p)=t$都有解的所有$t\geq0$构成的集合。我们称简单图$CG_Y$是度量空间$(Y,\rho)$的中心图，当且仅当$Y$是$CG_Y$的顶点集，且$Y$中不同顶点$x,y$相邻当且仅当$\rho(x,y)\in C(Y)$。我们证明，对所有满足$|Y|\geq2$且$CG_Y$非空的有限度量空间$(Y,\rho)$，有$|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil$成立；同时证明，对每个有限超度量空间$(X,d)$，有$|E(CG_X)| \geq |X|-1$成立。我们还在图同构意义下，刻画了满足$|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil$的度量空间$(Y,\rho)$，以及满足$|E(CG_X)|=|X|-1$的超度量空间$(X,d)$的中心图。
英文摘要：
The center of distances $C(Y)$ of a metric space $(Y,ρ)$ is the set of all $t\geq 0$ for which the equation $ρ(y,p)=t $ has a solution for each point $p\in Y$. We say that a simple graph $CG_Y$ is the central graph of a metric space $(Y,ρ)$ if $Y$ is the vertex set of $CG_Y$ and distinct vertices $x,y\in Y$ are adjacent if and only if $ρ(x,y)\in C(Y)$. We prove the inequality $|E(CG_Y)| \geq \left\lceil \frac{1}{2}|Y| \right\rceil $ for all finite metric spaces $(Y,ρ)$ with $|Y|\geq 2$ and non-empty $CG_Y.$ It is also proved that the inequality $|E(CG_X)| \geq |X|-1 $ holds for each finite ultrametric space $(X,d)$. The central graphs of metric spaces $(Y,ρ)$ and ultrametric spaces $(X,d)$ satisfying $|E(CG_Y)|=\left\lceil \frac{1}{2}|Y| \right\rceil $ and, respectively, $|E(CG_X)|=|X|-1 $ are described up to graph isomorphism.
