谱图理论中几个相关问题的研究

项目名称： 谱图理论中几个相关问题的研究

项目编号： No.11271149

项目类型： 面上项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 李书超

作者单位： 华中师范大学

项目金额： 58万元

中文摘要： 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．

中文关键词： Laplace 谱；无符号Laplace谱；谱矩；特征多项式；图的支撑树

英文摘要： 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 thi

英文关键词： Laplacian spectrum；signless Laplacian spectrum；spectral moment；characteristic polynomial；spanning tree