On The Spectral Radii And Energies Of Trees

Spectral graph theory is an important branch of graph theory,which mainlyutilizes tools and techniques of linear algebra and matrix theory,and isconcerned with the spectrum of a graph (i.e.,all eigenvalues of the adjacencymatrix of the graph),and the relation between spectrum and structure propertyor invariants of a graph.As the spectra of graphs have quite a few applicationsin quantum chemistry,electrical network theory and oscillation theory,particularly since the moment when the fact was discovered that the well-knownHückel molecular orbital theory (HMO)is fully equivalent to spectral graphtheory,it draws many attentions.Spectral technique has become an importanttool for the study of graph theory and combinatorics.The energy of a graph is an important parameter of the graph,which isdefined to be the sum of the absolute values of eigenvalues of the graph.Theconcept is from chemistry.For most hydrocarbons,the energies of theirmolecular graphs are linearly dependant with their totalπ-electronicenergies.Particularly one can get the resonance energies of acyclichydrocarbons by investigating the energies of trees.As the spectra and energies of graphs have remarkable applicationbackground in physics and chemistry,numerous chemists and mathematicians paytheir attention on them.It is hot to consider extreme graphs with respectto spectral radii or energies,to order graphs by their spectral radii orenergies,and to investigate the relations between the spectral radii orenergies and other parameters of graphs.The thesis mainly investigates the relation between the spectral radiiand maximum degrees of trees and trees with perfect matchings,and that betweenthe energies and maximum degrees of trees.It is proved that,for a tree or a tree with a perfect matching that possesses a larger maximum degree,thespectral radius is strictly increases with its maximum degree.With thisresult the 9th upto 13th trees in the order of trees by their largesteigenva]ues are given,as well astherange of the 8th upto 20th trees in theorder of trees with perfect matchings by their largest eigenvalues is given.The extreme trees with respect to energies are also determined in the thesis.