Eigenvalues And Structural Parameters Of Bicyclic Graphs

The theory of graph spectra is a very important area in graph theory. There are wide-ranging applications in the fields of quantum chemistry, computer science, communication networks and so on. In the theory of graph spectra, the adjacency matrix and the Laplacian matrix are employed frequently to learning the structures of the graph. One of the main problem of graph spectra theory is to determine precisely properties of graphs how are reflected in the algebraic properties. espe-cially properties about eigenvalues, such as maximal eigenvalue i.e. spectral radius, spectral spread, energy and so forth.This thesis mainly investigates the spectral radius and the spreads for adjacency matrices of graphs and tries to build some connections between them and some structure variables of corresponding graphs by algebraic method. The main content of this thesis are as follows.(i) In Chapter 1, we first introduce the evolvement of graph theory, and look back the backgrounds and research progresses of some questions on graph spectra theory that we study. Besides, some definitions and notations for the corresponding questions are given.(ii) In Chapter 2, we study the extremal graph with maximal adjacency spec-tral radius among all bicyclic graphs with given grith and the number of pendent vertexes. Here, the extremal graph with maximal adjacency spectral radius is deter-mined among all oo bicyclic graphs of above characters, and some properties of the extremalθbicyclic graph are given.(iii) In Chapter 3, we investigate the unique graph with maximal spectral spread is determined among allθbicyclic graphs with given girth. Moreover, we obtain the graphs with maximal spectral spread among all bicyclic graphs with given girth.