【作者】 兰玲；

【导师】 周仲礼； 刘斌；

【作者基本信息】 成都理工大学 ， 数学， 2023， 硕士

【摘要】 图谱是图论的重要研究领域之一,其中蕴藏了大量关于图的结构信息.图谱主要是利用图的邻接矩阵、Laplacian矩阵等代数表示,采用组合矩阵论来探究图的拓扑性质和其确定性.图的特征多项式是研究图谱的基础,而利用图的广义特征多项式含参的性质可以将图的邻接、Laplacian等特征多项式整合在一起,进而减少计算量.本文主要研究了T-图、T-点冠图和修正T-点冠图、T-点邻接冠图和修正T-点邻接冠图这几类复合图,利用复合图的广义矩阵得到了这几类复合图的广义特征多项式,并求出了这几类复合图的(37)-谱、A-谱、L-谱和Q-谱,解决了这几类图的谱难以求出的问题.同时,计算了T-点冠图和修正T-点冠图的能量、生成树数目等指标;给出了T-点邻接冠图和修正T-点邻接冠图的一大类同谱图及其应用举例.本文的主要成果如下:(1)研究了T-图的广义特征多项式,并得到了T-图的A-谱、L-谱和Q-谱.在此基础上,利用这三类谱计算了T-图的能量、生成树数目和Kirchhoff指标.(2)研究了T-点冠图和修正T-点冠图的广义特征多项式,同时得到了这两类复合图的邻接、Laplacian、signless Laplacian和Normalized Laplacian等特征多项式及谱,并给出了T-点冠图和修正T-点冠图的能量、生成树数目、Kirchhoff指标和关联能量.(3)研究了T-点邻接冠图和修正T-点邻接冠图两类复合图的广义特征多项式,推广出了这两类复合图的邻接、Laplacian、signless Laplacian和Normalized Laplacian等特征多项式及谱,并利用其(37)-谱,构造了一些T-点邻接冠图和修正T-点邻接冠图的(37)-同谱图.更多还原

【Abstract】 Graph spectrum is an important area of research in graph theory,which contains a wealth of information about the structure of graphs.Graphs spectrum are mainly represented algebraically using adjacency matrices and Laplacian matrices of graphs,and combinatorial matrix theory is used to investigate the topological properties of graphs and their determinism.The characteristic polynomial of a graph is the basis for the study of graphs,and the generalized characteristic polynomial of a graph can be used to reduce the computational effort by combining the characteristic polynomials of a graph such as adjacency and Laplacian characteristic polynomial.In this paper,we study the generalized characteristic polynomials of composite graphs,including T-graph,T-vertex corona and modified T-vertex corona,T-vertex neighbourhood corona and modified T-vertex neighbourhood corona,using the generalized matrix of composite graphs.The(37)-spectrum,A-spectrum,L-spectrum and Q-spectrum of these graphs are solved,and the problem of difficulty in finding the spectra of these graphs is solved.The energy and the number of spanning trees of T-vertex corona and modified T-vertex corona are calculated,and a large class of cospectral graphs of T-vertex neighbourhood corona and modified T-vertex neighbourhood corona and simple application examples are obtained.The main results of this paper are as follows:(1)The generalized characteristic polynomial of the T-graph is studied,and A-spectrum,L-spectrum and Q-spectrum of the T-graph are obtained.On this basis,the energy,number of spanning trees and Kirchhoff metric of the T-graph are calculated using these three types of spectra.(2)The generalized characteristic polynomials of T-vertex corona and modified T-vertex corona are studied,and the adjoint,Laplacian,signless Laplacian and Normalized Laplacian characteristic polynomials and spectra of these two types of composite diagrams are obtained by using their covariance properties.The energy,number of spanning trees,Kirchhoff index and association energy of T-vertex corona and modified T-vertex corona are given by using their A-spectra,L-spectra and Q-spectra.(3)The generalized characteristic polynomials of T-vertex neighbourhood corona and modified T-vertex neighbourhood corona are obtained,and the characteristic polynomials and spectra of the adjoint,Laplacian,signless Laplacian and Normalized Laplacian of these two types of composite diagrams are generalized.Their spectra are used to construct some cospectral graphs of T-vertex neighbourhood corona and modified T-vertex neighbourhood corona.更多还原