单圈图的无符号拉普拉斯最小特征值

发布时间：2018-03-10 09:33

本文选题：无符号拉普拉斯矩阵　切入点：特征方程　出处：《华东理工大学》2017年硕士论文　论文类型：学位论文

【摘要】：图的谱理论作为图论和组合矩阵理论的一个组成部分,已经得到了越来越多研究者的关注,并且在量子化学、统计力学、计算机科学、通信网络以及信息科学中均有着广泛的应用.近年来,对于图的特征值的研究也是十分活跃的课题.大量文章集中于对图的无符号拉普拉斯矩阵、拉普拉斯矩阵以及邻接矩阵相关性质的的研究,比如对于无符号拉普拉斯矩阵的特征值的研究,最大特征值以及最小特征值的界的研究以及扰动下的无符号拉普拉斯矩阵最小特征值的一些性质。在由E.R.van Dam,W.H Haemers 2003年写的《Which graphs are determined by their spectrum》一书中指出在区分非同构图中无符号拉普拉斯矩阵的谱比拉普拉斯矩阵的谱或者邻接矩阵的谱能更好的反应图的性质,同时图的无符号拉普拉斯矩阵的最小特征值是图的二部性的一个重要判断指标。文献[9]给出了无符号拉普拉斯矩阵的最小特征值达到最小时的极图,即三角形在其某一端点处悬挂一条路.本篇论文主要对无符号拉普拉斯矩阵的最小特征值进行了研究.给出了在所有的非二部单圈图中,无符号拉普拉斯矩阵的最小特征值达到最大的极图。主要是利用移接变形,二次型,特征方程等方法对无符号拉普拉斯谱的一些结论更深层次的探讨。
[Abstract]:As a component of graph theory and combinatorial matrix theory, the spectral theory of graphs has been paid more and more attention by researchers, and has been studied in quantum chemistry, statistical mechanics, computer science. In recent years, the research on the eigenvalues of graphs is also a very active subject. A large number of papers focus on the unsigned Laplace matrix of graphs. Studies on the related properties of Laplace matrices and adjacent matrices, such as the eigenvalues of unsigned Laplace matrices, The study of the bounds of maximum eigenvalue and minimum eigenvalue and some properties of minimum eigenvalue of unsigned Laplacian matrix under perturbation. In the book "Which graphs are determined by their spectrum", written by E.R. van DamW.H Haemers in 2003, it is pointed out in distinguishing nonisomorphism. The spectrum of the unsigned Laplacian matrix is better than the spectrum of the Laplace matrix or the spectrum of the adjacent matrix. At the same time, the minimum eigenvalue of the unsigned Laplacian matrix of a graph is an important criterion for the biparity of the graph. In reference [9], the pole graph of the unsigned Laplacian matrix when the minimum eigenvalue of the unsigned Laplacian matrix reaches the minimum is given. In this paper, the minimum eigenvalues of the unsigned Laplace matrix are studied. In all non-bipartite unicyclic graphs, the minimum eigenvalues of the unsigned Laplace matrix are studied. The minimum eigenvalue of unsigned Laplacian matrix reaches the maximum pole graph. Some conclusions of unsigned Laplacian spectrum are discussed deeply by means of shift deformation, quadratic form, characteristic equation and so on.
【学位授予单位】：华东理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.5

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