Comprehensive Study on Discrete Geometry

1.关于框架的刚性：我们在没有三角形的三维空间中提出了一种刚性单元-条形框架，在平面上提出了一种不能从边长和图结构数据构造的最小刚性框架。我们证明了如果平面上的完全二部框架K (m, n) （m >= 3, n >= 5）允许连续变形，则其中一个部分集位于直线L上，另一个部分集位于垂直于l2的直线上。关于结构的嵌入：证明了对于任意平面图G = (V，E)，存在一个嵌入f: V * R^2，使得x， y * V相邻当且仅当f (x)与f (y)之间的距离为整数。我们还证明了有理域上的每一个n维内积空间都可以等距嵌入到2n + 1维欧氏空间中。关于球体的排列：如果我们可以在桌子上放置实心球，那么图G可以用桌子上的球来表示，每个顶点对应一个球，因此只有当对应的顶点相邻时，两个球才相切。证明了可在桌子上用球表示的图族F与平面图族F不同，F不包含任何所谓的Petersen族的成员，并且证明了F中图的色数最大值为5或6.4。在随机图上，概率：我们确定了由圆上随机n个点生成的优势关系中最大常规比赛顺序的概率分布。我们将经典破产问题推广到三人博弈中，计算了固定赌徒a首先破产的概率，以及a是唯一幸存者的概率。

1.On the rigidity of frameworks : We presented a rigid unit-bar-framework in the 3-dimensional space that has no triangle, and a minimum rigid framework in the plane that cannot be constructed from the data of edge-lengths and graph structure. We proved that if a complete bipartite framework K (m, n) (m >= 3, n >= 5) in the plane admits a continuous deformation, then one of the partite-sets lies on a line L and the other partite-set lies on the line perpendicular to L.2.On embeddings of structures : We proved that for any planar graph G = (V,E), there is an emebedding f : V * R^2 such that x, y * V are adijacent if and only if the distance between f (x) and f (y) is an integer. We also proved that every n dimensional inner product space over the rational field can be isometrically embedded into 2n + 1 dimensional Euclidean space.3.On srrangements of spheres : A graph G is said to be representable by balls on the table, if we can place solid balls on a table, one ball for each vertex, so that two balls are tangent only when the corresponding vertices are adjacent. We proved that the family F of graphs representable by balls on a table is different from the family of planar graphs, and that F does not contain any member of the so-called Petersen family, and that the maximum value of the chromatic number of a graph in F is either 5 or 6.4.On random graphs, probability : We determined the probability distribution of the order of the maximum regular tournament in a dominance relation generated by a randam n points on a circle. We extended the classical ruin problem to 3 persons' game, and calculated the probability that a fixed gambler A ruined first, and the probability that A is the sole survivor.