树的最大与次小Laplacian特征值和的上界

随着计算机技术和网络技术的不断发展,图的谱被广泛应用于网络拓扑结构的特征分析,Laplacian矩阵的谱(特别是最大特征值和次小特征值)在网络结构中扮演重要角色.设G=(V,E)是一个具有n个顶点的简单图,A(G)为G的邻接矩阵,D(G)为G的度对角矩阵.定义G的Laplacian矩阵为L(G)=D(G)-A(G),设L(G)的特征值为μ1(G)≥μ2(G)≥…≥μn-1(G)≥μn(G)=0,最大特征值μ1(G)称为图G的Laplacian谱半径;次小特征值μn-1也称作图G的代数连通度.本文讨论了树的L(G)的最大与次小特征值和μ1(G)+μn-1(G)的上界,得到几个有意义的结论.

Upper Bounds for the Sum of Laplacian Eigenvalues of Trees

With the development of computer technology and network technology, the spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associating with the network. Let G be a simple graph with n vertices, The Laplacian matrix L（G） = D（G） -A（ G）, where A（G） is the adjacency matrix of G and D（G） is the diagonal matrix of the vertex degrees of G. The eigenvalues of L（G） will be denoted by/μI≥...≥μn-1≥μn = 0. The largest eigenvalue/μ1 （G） is called the Laplacian spectral radius of the graph G, the second smallest Laplacian eigenvalue μn-1 （G） is also called the algebraic connectivity of G. we discuss upper bounds of the sum of the largest and second smallest eigenvalues of trees, and obtain some significant results.