【作者】 谭学忠；

【导师】 柳柏濂；

【作者基本信息】 华南师范大学 ， 基础数学， 2006， 博士

【摘要】 图谱理论是图论中的一个新兴领域。它起源于理论化学家和物理学家为寻求一类偏微分方程的近似解而建立起的一套离散的方法。1957年，L．COLLATZ和U．SINOGOWITZ的深刻论文[2]的发表被视为图谱理论诞生的标志。经过短短50年的发展，它已经形成了系统、完善的理论。许多专著（[1，3-5，13]）陆续问世。如今，图谱理论已经成为代数图论中的研究热点。 在近几十年中，图的邻接谱和Laplace谱得到了广泛的研究。这些成果不仅丰富了和发展了代数图论，而且在理论物理和理论化学中有着广泛的应用（参考Chapter 8 in[3]）。正如[3]中所指出，物理学家和化学家已知图结构去寻找它的谱，而图论学家由图的谱探索相应的图结构。 在本文中，我们涉及到§1．2中所定义的三种谱—邻接谱、Laplace谱和拟Laplace谱。 1．在第二章中，我们讨论了图的零维数（nullity）。对单圈图和双圈图，我们证明了零维数集（nullity set）均为[0，n-4]（单圈图n≥5，双圈图n≥6）。确定了η（G）=n-4的单圈图和双圈图。对于一般非空图，我们证明了零维数集为[0，n-2]，并证明nullity取得n-2的极图为完全二部图，以及取得n-3的图类为G₁（m，p，q）。 2．在第三章中，我们考虑了一些谱半径和特征值的估计问题。 ·设g_n，k（k＜n）和g′_n，k（k＜n）分别表示具有后阶点独立集和k阶边独立集的n介图。 （1） 设G∈g_n，k and m=n-k，那么 ρ（G）≤(m-1+（（m-1）²+4km）^1/2)/2，更多还原

【Abstract】 The graph spectra theory is a new field in graph theory. It originated from the techniques, which was first used by theoretic chemists and physicists, of seeking approximate numerical solution for certain partial differential equations. The fundamental papers^[2]（1957） of L.Collatz and U.Sinogowitz are usually considered as the starting point of the study of graph spectra theory. In the short past 50 years, it has been developed into a systematic, integrated theory. Many monographs on this field （[1,3-5,13]） have been published. Literatures on this field are increasing with the speed of thousands of papers per year.Two kinds of graph spectra — the spectra of adjacency matrices and the spectra of Laplacian matrices, has been widely studied in the past several decades. They have great significance not only in graph theory, but also in some fields of chemistry and physics （refer to Chapter 8 in [3]）. As has been pointed out in [3], chemists and physicists know the structure of the graph and they are looking for the corresponding spectrum, whereas, in general, graph theorists and combinatorialists assume that the spectrum is known and they try to say something about the graph structure.In this paper, three kinds of spectra of graphs defined in §1.2 are involved.1. In Chapter 2, we consider the nullity of graphs. For unicyclic graphs and bicyclic graphs, we showed that the nullity set both are [0,n - 4]（n ≥ 5 for unicyclic graphs and n ≥ 6 for bicyclic graphs）. We determined the graphs with η（G） = n - 4,for unicyclic graphs and bicyclic graphs. We found some graphs with large nullity for general graphs. This is instructive and helpful to the study of nullity of general graphs. We also gave some conditions for a graph is singular or non-singular.2. In Chapter 3, some estimation problems are concerned. Let g_n,k（k < n） and g_n,k’（k < n） be the set of graphs of order n with k independent vertices and the set of graphs of order n with k independent edges, respectively.更多还原