The Research Of Several Kinds Of Spectral Of Graphs

Graph theory is a branch of mathematical research,which takes graph as its object of study.In recent years,with the development of modern science and technology,graph theory has become an important basic theory of computer science and technology,modern information technology,physics,chemistry and many other disciplines.The most commonly used method to study graph theory is the combination of algebra and graph.For this reason,various matrices are introduced to reflect the properties of graphs by studying the properties of the eigenvalues of these matrices.The matrix eigenvalues and their multiplicities of graphs are called the spectra of corresponding matrices of graphs.The spectrum mainly include the adjacent spectra,Laplacian spectra,distance spectra,distance(signless)Laplacian spectra and normalized Laplacian spectra etc.In this paper,we study the matrix eigenvalues and eigenvectors of graphs to establish the relationship between the structure of graphs and the eigenvalues and eigenvectors of graphs.We use algebraic methods,combinatorial methods and analytical methods to study the topological properties of graphs.It mainly studies the following three aspects:(1)Using the principle of block-matrix and the method of solving eigenvalue of matrix in algebra,the distance spectrum,distance Laplacian spectrum and distance signless Laplacian spectrum of windmill graph are calculated;(2)According to the characteristics of graph construction and the principle of the block matrix,the distance spectra,distance Laplacian spectra and distance signless Laplacian spectra of double(neighbourhood)coronas are solved by mathematical induction based on the uniqueness of the eigenvalues DX=?X;(3)By the uniqueness of eigenvalue,the normalized Laplacian spectra of pentagonal iteration graphs are solved algebraically.At the same time,the normalized Laplacian spectra of pentagonal iteration graphs is applied to solve three important parameters:multiplicative degree-Kirchhoff index,Kemeny's constant and the number of spanning trees.