Research On Some Kinds Of Spectra Of Graphs

The spectral graph theory is an active and important research area in algebraic graph theory.The main work of the spectral theory is to study the structure properties of graphs by means of the matrix theory.In this thesis,we mainly focus on three important topics in the spectral graph theory,i.e.,the adjacency spectrum(A-specttum),Laplacian speetrum(L-spectrum)and signless Laplacian speetrum(Q-spectrum).We obtain the main results as follows:(1)We research the upper bounds for the adjacency spectra radius λ and signless Laplacian spectral radius y.Several new upper bounds for λ and γ are obtained in terms of maximum degree,minimum degree,the number of edge,order,degree and average 2-degree etc.Moreover,we present some examples to illustrate that our bounds improve some known results.(2)We define two new graph operations,namely the SSG-vertex join graph G1*G2 and RG-vertex join graph G1 ⊙ G2;and we present the A-spectrum,L-spectrum and Q-spectrum of them;as applications,we construct infinite pairs of A-cospectral graphs,L-cospectral graphs and Q-cospectral graphs;moreover,the number of spanning trees and the Kirchhoff index of G1*G2 and G,O G,are determined.(3)We investigate the ordering of the Laplacian and signless Laplacian spectral radius for maximal outer planner graphs and connected nonregular graphs,and we find it can be transferred to the ordering of their maximum degrees A.Besides,we give examples to illustrate our bounds for A are best possible in some cases.