Unsigned And Regular Laplace Spectra Of Some Graphs

Graph theory is a branch of applied mathematics which develops very rapidly in recent years.As an important branch of algebraic graph theory,the spectral theory of graphs mainly studies the relationship between the combinatorial properties of graphs and the algebraic properties of matrices related to graphs（such as adjacency matrix,Laplacian matrix,signless Laplacian matrix,normalized Laplacian matrix and so on）.At present,the spectral theory of graphs has made great progress in its own theoretical research,but also become an important means to solve problems in other fields of mathematics,and its role is increasingly prominent.In addition,the spectral theory of graphs has been widely applied in many fields such as physics,chemistry,biology and information science.In this thesis,we describe the signless Laplacian spectra and normalized Laplacian spectra of arbitrary powers of lexicographic products of regular graphs,the weighted corona graphs and four types of double join operations of graphs,and obtain some new and meaningful results.The full text is divided into four chapters,the specific content is as follows:In Chapter 1,some basic concepts used in this paper are introduced firstly,then some research background and developments of the signless Laplacian spectra and nor-malized Laplacian spectra of graphs are introduced,and finally the main results of this paper are listed.In Chapter 2,we discuss the signless Laplacian spectra and normalized Laplacian spectra of arbitrary powers of lexicographic products of regular graphs.In Chapter 3,we mainly describe the signless Laplacian spectra and normalized Laplacian spectra of the weighted corona graphs.In Chapter 4,the signless Laplacian spectra and normalized Laplacian spectra of four types of double join operations of graphs G^S∨（C₁^·C₂^°）、G^Q∨（C₁^· C₂^°）、G^R∨（G₁^·,G₂^°）and G^T∨（G₁^·,G₂^°）are given.