【作者】 张静；

【导师】 郭继明；

【作者基本信息】 中国石油大学(华东) ， 数学， 2014， 硕士

【摘要】 图谱理论是图论中的一个很重要的分支领域,在物理,化学,计算机科学等领域都有重要的应用。图谱理论与图的结构之间有着自然的联系,能反映图的图论性质,因此图谱的研究正越来越引起人们的关注,是当前图论研究中的一个热点问题。研究图的拉普拉斯特征谱主要有三种方法:代数方法,几何方法和概率方法。本文主要运用了代数和几何方法对图的第二大拉普拉斯特征值进行了研究。主要内容如下:1.介绍了图理论及拉普拉斯特征值的背景及意义,给出了相关的符号概念和有关拉普拉斯特征值的国内外研究现状。2.总结了相关的基本理论和图的运算对拉普拉斯特征值的影响。3.分别研究了含圈图和树的第二大拉普拉斯特征值不超过2?2的图,最后给出了在n?7的情况下第二大拉普拉斯特征值不超过2?2的所有的连通图。更多还原

【Abstract】 The theory of the graph spectra is an important branch in graph theory, there are many important applications in the fields of physics, chemistry, computer science and other fields.There is a natural connection between the theory of the graph spectra and the structure of the graph. The theory of the graph spectra can reflect the nature of graph theory. Therefore graph spectra is paid more and more attention by people. It is a hot issue in current study of graph theory.The study on the Laplacian spectrum of graphs has three kinds of methods: algebraic method, geometric method and probability method. In this paper, we study the second largest Laplacian eigenvalue of the graph by using the method of algebra and geometry.The main contents are as follows:1. In this paper, it introduces the background and significance of Laplacian eigenvalue and the graph theory, and gives the concept of symbols and research status at home and abroad about Laplacian eigenvalue.2. In this paper, it summarizes the related basic theories and the influence diagram operations on the Laplacian eigenvalue of the graph.3. In this paper, it respectively studied the graph including circle and tree that their second largest Laplacian eigenvalue no more than 2 ?2. Finally, in the case of n ?7, it gave all the connected graph of the second largest Laplacian eigenvalue no more than 2 ?2.更多还原