On The Signless Laplacian And Distance Spectrum Of Graph

The theory of graph spectra is an active and important area in graph theory. One of the main problems of algebraic graph theory is to determine precisely how, or whether, properties of graphs are reflected in the algebraic properties (especially eigenvalues) of the above matrices.This thesis mainly investigates the spectral radii for signless Laplacian matrices and distance matrices of graphs. The main conclusion of this paper is as follows.In Chapter2, we mainly investigate the problem about the signless Laplacian characteristic polynomial of graphs. We systematically investigate the signless Laplacian characteristic polynomials of graphs. It is make the signless Laplace matrix become block matrix cleverly, so we get some basic formulas about the computation of signless Laplacian characteristic polynomials.In Chapter3, we mainly investigate the problem about the distance spectra. We characterize trees with μn-1(T)∈[r,0](r≈-2.4295), unicyclic and bicyclic graphs with μn-1(G)∈[-2,0]. On the other hand, we discuss a closely-related graph parameter---the girth. We get the unicyclic graphs, which their spectral radius is the first four smallest, and some other unicyclic graphs, which their spectral radius is the first five largest. Finally, when the girth of the unicyclic graph is known, we also conjecture that the extremal graphs which attains the minimum and maximum distance spectral radius are S(n; k) and P(n; k), respectively.