给定阶、直径和悬挂点数的树的谱半径

On Spectral Radius of Trees Given Order,Diameter and Pendant Vertex Number

研究了在阶为n、直径为d且悬挂点数为s的所有树中,树具有最大的谱半径问题.令Pd+1是一个d+1阶的固定路,Tn,d,s表示通过在Pd+1的第r个顶点生成s-2条几乎等长的路得到的阶为n、直径为d且悬挂点数为s的树,其中r=r(d)是(d+1)/2的整数部分,则Tn,d,s具有最大谱半径.该结论推广了给定阶、直径或悬挂点数的树的谱半径的一些结果.借助该结论,也得到了树的谱半径与其独立数、覆盖数、边覆盖数和全独立数之间的关系.

Of all trees on given order n, diameter d and pendant vertices number s, which achieves the maximal spectral radius？ Let Tn,d,s be a tree with order n, diameter d and pendant vertex number s obtained by spanning s -2 paths of almost equal lengths at r th-vertex of fixed path Pd ＋ 1, where r = r （d） is the integer part on （d ＋ 1）/2. The maximal spectral radius is found to be obtained uniquely at Tn,d,s, which generalizes some results on spectralradius of trees given order, diameter or pendant vertex number. The relationship is also revealed between spectral radius of trees and independence number, covering number, edge covering number as well as total independence number.