Applications Of Spectrum In Graph Energy And Graphs Ordering

The spectral graph theory mainly studies the relations between the spec-tral property and the structrue property of graphs, and uses the spectral prop-erty to characterize the structrue property of graphs. An important applicationof spectral graph theory in structural chemistry is as follows. By establishinggraph models of organic molecules and applying the eigenvalues of the graphs,We then have a quantitative analysis of energy levels and stabilities of theorganic molecules, which leads to the conception of graph energy.This thesis mainly studies (1) the variation of energy of a unicyclic graphby deleting an edge, and the integral formula of Laplacian energy of a graph,and (2)the order of unicyclic mixed graphs on the same number of verticesaccording to Laplacian spectral radius.Gutman and Hou et. al. have given clear characterizations of trees andunicyclic graphs with extreme energies. In this thesis, we discuss the variationof energy of a graph after a local change. For unicyclic graphs, we find that ingeneral energy decreases if deleting an edge. But it is not true to some specialunicyclic graphs.In 2005, Gutman and Zhou introduced the Laplace eigenvalues to thegraph energy, and gave the notion of Laplacian energy of a graph, which isconsistent with the energy of a graph for all regular graphs. In this thesis,we give some integral formulas of Laplacian energy based on Coulson integralformula.According to spectral radius of graphs (or other extreme eigenvalues),one can order some classes of graphs (for example, trees and unicyclic graphs),which will help us understand some extreme properties of graphs. Recently,Fan characterized the nonsingular unicyclic mixed graphs whose Laplacianspectral radius attains the maximum, and also characterized the bicyclic mixed graphs whose Laplacian spectral radius attains the maximum with Tam andZhou. In this thesis, we determine respectively the graphs whose Laplacianspectral radius are the largest, the second largest and the third largest amongall unicyclic mixed graphs containing the same number of vertices.The organization of this thesis is as follows. In Chapter 1, we introduce abackground of spectral graph theory and graph energy, some conceptions andnotations, and the research problems and results. In Chapter 2, we discussthe changes of energies of unicyclic graphs by deleting an edge, and give someintegral formulas of Laplacian energy of a graph. In Chapter 3, we orderunicyclic mixed graphs by their Laplacian spectral radius.