Abstract: Algebraic graph theory methods play an important role in the network design. Spectrum of Laplacian matrix is associated with the synchronous ability of network. The algebraic connectivity is a depict important parameter of synchronous ability. In this paper, using a grafting method, it discusses the relationship between algebraic connectivity and diameter of a tree. For a special class of trees, the algebraic connectivity of the tree with a fixed number of vertices, is decreasing along with the increase of diameter. Moreover, using the Cauchy-Schwarz inequality as a guide, it also obtains bounds for the algebraic connectivity of a tree.

Key words:  tree, Laplace matrix, algebraic connectivity, diameter

摘要： 代数图谱理论方法在网络设计中发挥重要作用。网络拓扑图的Laplacian矩阵的谱与网络的同步能力有关，代数连通度就是一个刻画同步能力的重要参数。采用移接变形方法，讨论了树的代数连通度和直径之间的关系，获得了下面的结论：当树的顶点数固定时，树的代数连通度随着树的直径的增加而减少。进一步地，讨论了树的代数连通度的上界和下界。

关键词: 树, 拉普拉斯矩阵, 代数连通度, 直径