The Study Of Some Problems On The Laplacian Spectral Radius Of Trees

The theory of graph spectra has a wide range of applications in operations research,quantum chemistry, theoretical physics, computer science, communication networkand information science. The study on the problem of extremal graphs on theLaplacian spectrum of trees and the study on the problem of estimating the upperbounds on the Laplacian spectral radius of trees are important research contents in thetheory of graph spectra. So it has a good practical and theoretical significance tostudy some problems on the Laplacian spectral radius of trees.The main results of this paper are as follows:Firstly, we give the definition of the standard Laplacian spectral vector of graphsand obtain some properties and applications on the standard Laplacian spectral vectorof graphs.Secondly, we study the structure on the maximal Laplacian spectrum tree in thetree set with given the number of the vertices n and the number of the pendantvertices k, and obtain some upper bounds on the Laplacian spectral radius of treeswith given the number of the vertices n and the number of the pendant vertices k.Thirdly, we study the property and structure on the maximal Laplacian spectrumtree in the tree set with given the number of the vertices n and the maximal degreeΔ, obtain some necessary conditions on the maximal Laplacian spectrum tree in thetree set with given the number of the vertices n and the maximal degree Δ, and wedescribe the structure of the maximal Laplacian spectrum tree in this tree set.Finally, we study the problem of estimating the upper bounds on the Laplacianspectral radius of trees with given the number of the vertices n and the maximaldegree Δ, obtain a new upper bound on the Laplacian spectral radius of trees for3≤Δ≤n2. This upper bound is the best upper bound expressed by the number ofthe vertices n and the maximal degree Δ, at present. And we do furtherimprovement on the upper bounds of the Laplacian spectral radius of trees for Δ=3.