缩影为正多面体一维骨架的图式流形同胚分类下界

Lower Bounds of Homeomorphic Classifications of Graphlike Manifolds Epitomized by the One-Dimensional Skeleton of Regular Polyhedrons

将一个简单图中所有顶点替换为有向圆周(称为结点),将所有边替换为圆柱面(称为管子),即得到一个缩影为这个简单图的图式流形。一个图式流形可对应一个伴随矩阵,若管子两端的圆周是同向的则为正管,矩阵对应元素为1;若是反向则为负管,对应元素为−1,若两结点间无管相连,则对应元素为0。利用伴随矩阵的特征多项式,可计算图式流形同胚类的下界。本文对于缩影为正多面体一维骨架的图式流形,分别计算出各情形的伴随矩阵特征多项式,并对其进行对比、去重,得到其同胚类下界。其中,结合正十二面体与正二十面体情形的伴随矩阵的稀疏性,利用Householder变换,先对伴随矩阵进行预处理,将其化为对称三对角矩阵,再用一种简便的递推算法编程计算其特征多项式,得到求同胚类下界的第二种方法,并计算出五种正多面体对应情形的同胚类下界分别为3个、6个、14个、46个和4184个。

By replacing all vertices in a simple graph with directed circles (called nodes) and all edges with cylinders (called tubes), you get a graphlike manifold epitomized by this simple graph. If the circles at both ends of the tube are in the same direction, it is called apositive tube;otherwise it is called a negative tube, corresponding to the matrix elementsbeing 1 or −1 respectively. If two nodes are not connected, the corresponding matrixelement is 0. Therefore, a graphlike manifold corresponds to a matrix, which is called the associated matrix of the graphlike manifold. The lower bound of the homeomorphic class of graphlike manifolds can be calculated by the characteristic polynomial of the associated matrix. In this paper, the homeomorphic classifications of graphlike manifolds epitomized by the one-dimensional skeleton of regular polyhedrons are presented by using the properties of the associated matrices of graphlike manifolds. For the cases of regular dodecahedron and regular icosahedron, based on the sparseness of the associated matrices, two algorithms for calculating the lower bounds of homeomorphic classifications are given by using the orthogonal transformation of the associated matrices. The results are that the lower bounds of the homeomorphic classes for the cases of five regular polyhedrons are 3, 6, 14, 46 and 4184, re-spectively.