图谱理论是图论与代数的交叉研究领域，它是通过研究图的特征值与结构参数之间的关系来揭示图的性质。Laplacian矩阵是图的一种经典表示矩阵，Laplacian特征值与图结构联系密切，而且在复杂网络等领域有着重要的应用。本项目主要围绕图的Laplacian特征值和若干关键结构参数展开研究，目的是建立Laplacian特征值与图结构之间新的联系。本项目拟开展以下三个方面的研究：(1)构造图的前k大Laplacian特征值之和的更好上界并研究Brouwer猜想；(2)刻画三种圈指标条件下代数连通度的极大值和极图；(3)建立最小控制图的Laplacian谱刻画。本项目通过多个角度对图的Laplacian特征值与结构参数之间的关系进行探讨，争取在解决Brouwer猜想上取得新的进展，同时做出一系列创新性的研究成果，从而推动Laplacian谱理论的发展以及在其它领域的应用。

Spectral graph theory is a cross-disciplinary research area of graph theory and algebra, its core content is to reveal the properties of graphs by studying relations between the eigenvalue and structure of graphs. Laplacian matrix is a classical representation matrix of graphs. Laplacian eigenvalue is closely connected with the structure of graphs, and it plays important role in complex networks. In this project, the Laplacian eigenvalues and some crucial structural parameters will be considered, and the aim is to obtain some new relations between the Laplacian eigenvalue and structure of graphs. The following three research contents will be investigated: (1) Constructing the upper bound of the sum of the first k largest Laplacian eigenvalues, and studying Brouwer conjecture. (2) Determining the maximum algebraic connectivity and extremal graph under three cycle indexes. (3) Establishing the Laplacian spectral characterization of the domination minimum graphs. The relations of Laplacian eigenvalues and structural parameters are discussed from several angles. The project will make progress of Brouwer conjecture, obtain innovative results, and promote the development and the application of Laplacian spectra.