给定直径和悬挂点数的树的拉普拉斯系数

Laplacian coefficients of trees with given diameter and number of pendant vertices

令φ(T,λ)=∑nk=0(-1)kck(T)λn-k是一个n点树T的拉普拉斯矩阵的特征多项式。熟知,cn-2(T)和cn-3(T)分别等于T的维纳指标和修改超维纳指标。应用图的变换,确定给定直径和悬挂点数的树中所有拉普拉斯系数ck(T)最小的树。特别是确定了一些具有极端维纳指标、修改超维纳指标和Laplacian-like能量的树。

Let φ（T,λ）=∑k=0^n（-1）^kCk（T）λ^n-k be the characteristic polynomial of Laplacian matrix of a n-vertex tree T. It is k=0 well known that Cn-2（T） and Cn-3（T） are equal to the Wiener index and modified hyper-Wiener index of T, respectively. By applying some transformations of graphs, the trees with given diameter and number of pendant vertices were characterized which simultaneously minimize all Laplacian coefficients. In particular, some trees with extremal Wiener index, modified hyper-Wiener index and Laplacian-like energy were determined.