树的最大特征值的上界的一个注记

设T是一个树,V是T的顶点集．记dv是υ∈V的度,△是T的最大顶点度．设υ∈V且dw=1．记k=ew+1,这里ew是w的excentricity．设δj′= max{dυ:dist(υ,w)=j},j=1,2,…,k-2,我们证明和这里μ1(T)和λ1(T)分别是T的Laplacian矩阵和邻接矩阵的最大特征值．特别地,记δo′=2．

A Note on the Upper Bound of the Largest Eigenvalue of Trees

Let T be a tree with vertex set V. Let dυdenote the degree ofυ∈V and let A denote the largest vertex degree of T. Let w∈V such that dw = 1. Let k = ew + 1 where ew is the eccentricity of w. For j = 1,2,..., k - 2, letδ′j =max{dυ: dist (υ,w) = j}. We prove thatμ1(T) < max1 andλ1 (T) < max1, whereμ1(T) andλ1(T) are the largest eigenvalue of the Laplacian matrix and the adjacency matrix of T. Specially, we denote thatδ′o = 2.