Research On The Normalized Laplacian Spectra Of Some Graphs And Their Applications

Spectral theory is the intersection theory of graph theory and linear algebra.The research of spectral theory mainly combines graph theory and combinatorial mathematics theory,and uses algebraic methods and techniques to study the spectrum and its struc?tural properties of graphs.Computing the spectrum of graphs is just like determining the characteristic polynomials of graphs.It is a basic and meaningful work in spectral theory.The spectra and characteristic polynomials of graphs can help us to study some parametric properties,such as chromaticity,connectivity,matching number,etc.The eigenvalues of graph matrices can not only reflect the parametric properties of graphs,but also provide information related to graph energy.The normalized Laplacian eigenval-ues of graphs are one of them.In addition,comparing the coefficients and eigenvalues of two normalized Laplacian eigenvalues is a common and effective method to confirm the normalized Laplacian cospectral graphs.The normalized Laplacian spectra of the graphs studied in this paper mainly contain the following fields:bounds for the spectral radius of starlike tree and double-starlike tree,the normalized Laplacian spectrum of subdivision vertex-edge corona,the normalized Laplacian spectrum of subdivision vertex-edge join,the normalized Laplacian spectrum of two Hexagonal Systems.In this paper,the application of normalized Laplacian spectra is mainly focused on the degree-Kirchhoff index,the number of spanning trees and the construction of cospectral graphs.With the help of some graph operations(corona and join operations),we get some normalized Laplacian spectra of graphs,and give the number of spanning trees and the real value of the degree-Kirchhoff index related to graphs.Furthermore,searching for the normalized Laplacian cospectral invariants of graphs provides a powerful tool for judging the cospectral properties of two graphs.The division of concrete structure is mainly discussed in five chapters:In Chapter one,the research background of the normalized Laplacian spectrum and the basic concepts involved are given.In Chapter two,For starlike tree and double-starlike tree,by deleting cut points and cut edges,the upper bound of spectral radius is given from the relationship between the root of characteristic polynomial and the coefficient.Then,the bound of spectral radius of generalized starlike tree is derived from known conclusions.Finally,starting from changing maximum and second maximum which the inner path and the outer path of the generalized starlike tree is divided,the change of spectral radius is less than 1.In Chapter three,the normalized Laplacian spectrum of graph G1S o(G2V?G3E)is determined according to the characteristic roots of three connected regular graphs G1,G2 and G3.As applications,we construct some non-regular normalized Laplacian cospectral graphs.In addition,we also give the multiplicative degree-Kirchhoff index and the number of the spanning trees of G1S o(G2V?G3E).In Chapter four,we construct graph G1S(?)(G2V?G3E)by means of joint operation,and then the A-spectrum,the L-spectrum and the Q-spectrum of G1S(?)(G2V?G3E)are respectively determined in terms of the corresponding spectra for a regular graph G1 and two arbitrary graphs G2 and G3.Furthermore,when G1,G2 and G3 are regular graphs,the normalized Laplacian spectrum of graph G1S(?)(G2V?G3E)is also confirmed.As applications,we construct infinitely many pairs of A-cospectral graphs,L-cospectral graphs,Q-cospectral graphs and L-cospectral graphs.Finally,we give and compare two methods for calculating the number of spanning trees of G1S(?)(G2V?G3E).In Chapter five,we respectively obtain the normalized Laplacian polynomials of graphs Fn and Mn by the eigenvalues and the determinant of cyclic matrix,and also obtain the normalized Laplacian spectra of graphs Fn and Mn.As applications,we give the Randic energies and a sharp upper bound of RE(Fn)and RE(Mn),repectively.Finally,the number of spanning trees of the two graphs is determined.