由广义Bethe树和完全图所构造的图的谱(英文)

The Spectrum of a Graph Obtained from Copies of a Generalized Bethe Tree and a Complete Graph

广义Bethe树是指每层上的点都具有相同度的带根树.设Bk为具有k层的广义Bethe树,Kr为r阶完全图.把r个Bk的根点分别粘在Kr的r个点上,可获得一个图BkoKr.借助于k阶对称三对角矩阵,本文给出邻接矩阵A(BkoKr)和Laplace矩阵L(BkoKr)的特征值的简洁刻画.此外,本文还给出了A(BkoKr)的Perron向量的结构性质.

A generalized Bethe tree is a rooted tree for which the vertices in each level having equal degree. Let Вk be a generalized Bethe tree of k level, and let К^τ be a complete graph on r vertices. Then a graph В_κοК^τ is obtained from r copies ofВk and К^τ by appending r roots to the vertices of К^τ, respectively. In this paper, a simple way is introduced to characterize the eigenvalues of the adjacency matrix A（В_κοК^τ）and the Laplaeian matrix L（В_κοК^τ）of В_κοК^τ means of symmetric tridiagonal matrices of order k. In addition, the structure property of the Perron vectors of A（В_κοК^τ）is given.