图的谱能量及其相关问题的研究

发布时间：2018-06-18 02:23

本文选题：图谱理论 + 谱能量　；参考：《兰州理工大学》2017年硕士论文

【摘要】：图谱理论是图论研究的重要分支,其中对图能量的研究是近年的热点。图能量是用图谱表示图中的量,其在计算机科学、物理、化学、生命科学、控制工程等领域均有重要应用。图的谱与图结构密切相关。设G = (V(G),E(G))是一个简单无向图,其中V(G)和E(G)分别对应顶点集和边集。图G对应的邻接矩阵记为A(G)、拉普拉斯矩阵记为L(G)、无符号拉普拉斯矩阵记为Q(G),它们的特征根构成图G的邻接谱、拉普拉斯谱、无符号拉普拉斯谱。图能量的研究对象是图的各类矩阵及其谱。本文研究的几类图:由图G1和图G2得到的剖分冠点图G1◇G2、剖分冠边图G1%紾2,由图G和图H1,H2,…,Hn得到的广义R-冠点图R(G)(?)∧inHi,阿基米德晶格图等。通过图的广义矩阵得到图的广义特征多项式,进而求得其A-谱、L-谱、Q-谱,解决了这几类复杂图难于求谱的问题。作为应用,计算了图的生成树数目及Kirchhoff指数,求得了几类能量。提出了一种计算机求解能量的方法,得到了阿基米德晶格图在有限点内的能量指标。本文的主要成果如下:(1)计算并证明了广义R-冠点图R(G)(?)∧inHi的广义特征多项式及一些特定情况下的生成树数目和Kirchhoff指数,同时构造了一类广义同谱图;(2)用计算机编程的方法给出了阿基米德晶格图的三类谱能量的具体能量值;(3)给出了剖分冠点图G1◇G2、剖分冠边图G1%紾2的广义特征多项式并扩大了选图范围;(4)用计算机统计Estrada指数的数值范围,找到Estrada几乎同能图,同时统计了HOMO-LUMO距离和指数。
[Abstract]:Map theory is an important branch of graph theory, among which the study of graph energy is a hot topic in recent years. The energy of graph is represented by graph, which has important applications in computer science, physics, chemistry, life science, control engineering and so on. The spectrum of a graph is closely related to its structure. Let G = V ~ (+) G ~ (+) be a simple undirected graph, where V _ (G) and E _ (G) correspond to vertex set and edge set, respectively. The adjacent matrix corresponding to graph G is denoted as Agna Gi, Laplace matrix is denoted as Ln Gn, and unsigned Laplace matrix is denoted as QG. Their characteristic roots form the adjacent spectrum, Laplace spectrum and unsigned Laplace spectrum of graph G. The research object of graph energy is the matrix of graph and its spectrum. In this paper, we study several kinds of graphs: G _ 1 G _ 2, G _ 1% G _ 2, G _ 1 and H _ 1H _ 2 obtained from G _ 1 and G _ 2. The generalized R- crown-point graph R _ G _ (G ~ (+) A _ (H _ n) in Hi, Archimedean lattice diagram and so on. The generalized characteristic polynomial of a graph is obtained by the generalized matrix of a graph, and its A-spectrum L- spectrum Q- spectrum is obtained, which solves the problem that it is difficult to obtain the spectrum of these complex graphs. As an application, the number of spanning trees and Kirchhoff exponents of graphs are calculated, and several kinds of energy are obtained. In this paper, a computer method for solving energy is proposed, and the energy index of Archimedes lattice graph at finite point is obtained. The main results of this paper are as follows: 1) in this paper, we calculate and prove the generalized characteristic polynomials of the generalized R- crown graph R ~ (1) and the number of spanning trees and the Kirchhoff exponent in some special cases. At the same time, a class of generalized isospectral graphs is constructed. The specific energy values of three kinds of spectral energy of Archimedean lattice graphs are given by computer programming.) the generalized characteristics of G _ 1 ~ G _ 2 and G _ 1% ~ (2) C _ 2 are given. The range of Estrada exponent is calculated by computer, and the numerical range of Estrada exponent is calculated by computer. The Estrada almost isomorphic graph is found and the HOMO-LUMO distance and exponent are also calculated.
【学位授予单位】：兰州理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TP391.41;O157.5

【参考文献】