Graphs $4_n$ that are isometrically embeddable in hypercubes

On the Structure of Isometrically Embeddable Metric Spaces

Since its popularization in the 1970s the Fiedler vector of a graph has become a standard tool for clustering of the vertices of the graph. Recently, Mendel and Noar, Dumitriu and Radcliffe, and Radcliffe and Williamson have introduced geometric generalizations of the Fiedler vector. Motivated by questions stemming from their work we provide structural characterizations for when a finite metric space can be isometrically embedded in a Hilbert space.

关于可嵌入超立方体一层中的图及其极值数

On graphs embeddable in a layer of a hypercube and their extremal numbers

研究超立方体中可嵌入一层的图及其极值数，用边着色刻画分层图，证明多数非平凡细分有零图兰密度，存在周长为\(8\)的非分层立方体图有正图兰密度，还给出\(ex(Q_n, C_{10})\)的下界，揭示其与其他零图兰密度圈的不同。

Some properties of Bowlin and Brin's color graphs

Bowlin and Brin defined the class of color graphs, whose vertices are triangulated polygons compatible with a fixed four-coloring of the polygon vertices. In this article it is proven that each color graph has a vertex-induced embedding in a hypercube, and an upper bound is given for the hypercube dimension. The color graphs for $n$-gons up to $n=8$ are listed and some of their features are discussed. Finally it is shown that certain color graphs cannot be isometrically embedded in a hypercube of any dimension.

超立方体嵌入与判别排列交叠格中的中位结构

Hypercube Embeddings And Median Structure In The Intersection Lattice Of Discriminantal Arrangements B(n,k)

研究判别排列交叠格L(B(n,k))的度量结构，证明其覆盖图可嵌入超立方体，成为部分立方体和中位图，揭示超立方体几何。

所有凸环均具有相同长度的正则部分立方体的一个刻画

A characterization of regular partial cubes whose all convex cycles have the same lengths

本文研究了部分立方体的凸环性质，证明正则部分立方体为超立方体等图当且仅当其所有凸环具有特定相同长度，并推广了正则中位图的相关结论。

作为有限分配格的共振图的等距嵌入

Isometric embeddings of resonance graphs as finite distributive lattices

该研究针对平面二分图的共振图等距嵌入问题，证明了连通共振图的等距维数下界，给出等号成立的刻画，并设计算法生成对应二进制编码实现等距嵌入，解答了相关公开问题的基本情形。