谱图理论中几个相关问题的研究

The spectral graph theory is mainly concerned with the relation between the spectral and structural properties of graphs；it overlaps graph theory and combinatorial matrix theory. The spectral graph theory has close relationship with extremal graph theory and Turán theory. One uses the eigenvalues of graphs as an important tool to study the structural properties of graphs. The problem of characterizing graphs with least eigenvalue -2 was　one of the original problems of spectral graph theory. The techniques　used in the investigation of this problem have continued to be useful　in other contexts including forbidden subgraph techniques as well as geometric methods involving root systems. On the other hand, the relationship between the eigenvalues and some parameters of graphs has wide applications in theoretical physics, quantum chemistry and theoretical computer science and so on. The study of various combinatorial objects including distance regular and distance transitive graphs, association schemes, and block designs have made use of eigenvalue techniques, usually as a method to show the nonexistence of objects with certain parameters. The basic method is to construct a graph which contains the structure of the combinatorial object and then to use the properties of the eigenvalues of this graph. The content of this proposal contains the following：Firstly, based on the signless Laplacian spectra, together with the spectra graph theory with the graphic structure including forbidden subgraph, we study several kinds of Brualdi-Solheid-Turán type problems in extremal graph theory. Secondly, we study the relation between the largest (resp. smallest) eigenvalue of the signless Laplacian matrix with structural properties (resp. parameters) of (weighted) graphs. Thirdly, we study the spectral moment of the signless Laplacian matrix of graphs.Our purpose is to order graphs according to their sequences of signless Laplacian spectral moments of a given class of graphs. Finally, we study the relationship between the Laplacian coefficients of graphs and the total number of subtrees of trees. The research of this proposal will extend the connotation of the study on spectral graph theory, which will improve the research level of algebraic graph theory and combinatorial matrix theory in China.

谱图理论主要通过图矩阵来研究图的结构特征与代数性质，是图论与组合矩阵论的交叉领域。谱图理论与图论中的极图理论和Turán理论都有着紧密的联系。图的特征值一方面是研究图的结构特征的重要工具；另一方面它与一些图参数之间的内在联系在理论物理、量子化学、理论计算机科学等领域有着广泛的应用背景。本项目研究内容涉及到：将谱图研究与不含某些禁用子图的图结构研究有机结合起来，利用研究无符号Laplace谱来研究各类Brualdi-Solheid-Turán 型问题的极图理论；研究（赋权）图的无符号Laplace最大特征值、最小特征值与图的结构以及图参数之间的关系；研究图的无符号Laplace矩阵的谱矩并根据谱矩序列对图进行排序；研究图的Laplace特征多项式系数与树的子树的计数二者之间的内在联系。本项目的研究将拓展谱图理论研究的内涵，进一步推动我国代数图论与组合矩阵论的研究水平．

图中结构与图的参数之间的关系是图论研究的一个热点，对其进行研究不但有重大的理论意义，而且在理论物理、量子化学、理论计算机科学等领域有着广泛的应用背景。谱图理论主要通过图矩阵来研究图的结构特征与代数性质，是图论与组合矩阵论的交叉领域。谱图理论与图论中的极图理论和Turán 理论都有着紧密的联系。图的特征值一方面是研究图的结构特征的重要工具；另一方面它与一些图参数之间也有着一些内在的联系。本项目研究内容涉及到：将谱图研究与不含某些禁用子图的图结构研究有机结合起来，利用研究无符号Laplace 谱来研究各类Brualdi-Solheid-Turán 型问题的极图理论；研究（赋权）图的无符号Laplace 最大特征值、最小特征值与图的结构以及图参数之间的关系；研究图的无符号Laplace 矩阵的谱矩并根据谱矩序列对图进行排序；研究图的Laplace 特征多项式系数与图的支撑树的计数二者之间的内在联系。本项目的研究将拓展谱图理论研究的内涵，进一步推动我国代数图论与组合矩阵论的研究水平．