图的Laplace特征值与Laplace能量的研究

图的Laplace特征值的研究是图谱理论的重要研究课题之一。Brouwer猜想图的前k大Laplace特征值之和不超过该图的边数与组合数C(k+1,2)之和。图的Laplace能量与图的前k大Laplace特征值之和以及揭示图的Laplace特征值分布的一个代数不变量b密切相关。寻找给定图类中具有最大Laplace能量的图是当前研究图的Laplace能量的一个中心问题。本项目主要研究图的前k大Laplace特征值之和的上界以及代数不变量b，并利用所得结果进一步推进Brouwer猜想和图的Laplace能量的最大值问题的研究进展。

The study of the Laplacian eigenvalues of graphs is one of the important research topics in the theory of graph spectra. Brouwer conjectured that the sum of the k largest Laplacian eigenvalues of a graph does not exceed its edge number plus the combination number C(k+1,2). The Laplacian energy of a graph is closely related to the sum of the k largest Laplacian eigenvalues of the graph and an algebraic invariant b which sheds light on the distribution of the Laplacian eigenvalues of the graph. For a given class of graphs, finding the graphs with the maximum Laplacian energy is one of the central problems in the recent study of the Laplacian energy of graphs. In this project, we mainly study the upper bounds on the sum of the k largest Laplacian eigenvalues of a graph as well as the algebraic invariant b, by which we further push forward the progress of the research on Brouwer's conjecture and the problem of the maximum Laplacian energy of graphs.