相关图的补图的谱半径.doc

中文摘要 在图论中, 人们为了从代数的角度来研究图的性质, 引进了各种矩阵与图建立联 系, 例如: 邻接矩阵, 距离矩阵, 拉普拉斯矩阵, 无符号拉普拉斯矩阵等等. 在上述的矩阵中, 人们最常研究的是邻接矩阵, 拉普拉斯矩阵和无符号拉普拉斯 矩阵. 相对于图的邻接矩阵, 拉普拉斯矩阵和无符号拉普拉斯矩阵包含了图的各点度 的信息, 更能反映图的某些性质. 本文对图的无符号拉普拉斯谱半径和谱半径进行了 研究, 主要内容分为四节: 第一节, 介绍了图的谱半径, 拉普拉斯谱半径以及无符号拉普拉斯谱半径的研究 背景及其发展现状. 第二节, 研究了图的补图的无符号拉普拉斯谱半径, 并找到了当无符号拉普拉斯 谱半径达到最大时的极图. 第三节, 研究了双圈图的补图的谱半径, 并找到了当谱半径达到最大时的极图. 第四节, 研究了带有k个悬挂点的双圈图的无符号拉普拉斯谱半径, 并找到了当无 符号拉普拉斯谱半径达到最大时的极图. 关键字: 邻接矩阵; 谱半径; 拉普拉斯矩阵; 拉普拉斯谱半径; 无符号拉普拉斯矩 阵; 无符号拉普拉斯谱半径 I Abstract In the graphs theory, people introduce various matrices, such as the adjacency matrix, the distance matrix, the Laplacian matrix, the signless Laplacian matrix and so on, to re- search the properties of graphs by studying the algebraic properties of matrices. The matrices that people most usually study are adjacency matrix, Laplacian matrix and signless Laplacian matrix. Comparing with adjacency matrix, Laplacian matrix and signless Laplacian matrix contain the information about the degree of all vertices. It can show some properties of graphs well. This thesis will research the problems of signless Laplacian matrix and adjacency matrix. The main content can be divided into four sections: In the ?rst section, we gives the research background and development of spectral ra- dius, Laplacian spectral radius and signless Laplacian spectral radius. In the second section, we study the signless Laplacian spectral radius of the complement of graphs, and also get the graph which attains the maximum spectral radius. In the third section, we study the spectral radius of the complement of bicyclic graph, and also get the graph which attains the maximum spectral radius. In the fourth section, we study the signless Laplacian spectral radius of bicyclic graphs with k pendant vertices, and also get the graph which attains the maximum spectral radius. Key Words: adjacency matrix; spectral radius; Laplacian matrix; Laplacian spectral radius; signless Laplacian matrix; signless Laplacian spectral radius II 目  录