On The Q-eigenvalues And The Structures Of Graphs

Let A（G） and D（G） be respectively the adjacency matrix and the degree matrix of a graph G. Then the Q-matrix of G is defined to be Q（G）= D（G）+A（G）. The Q-eigenvalues of G are those of matrix Q（G）, and the Q-spectrum of G is a multiset consisting of the Q-eigenvalues together with their multiplicities.Cvetkovic, Rowlinson and Simic pointed that the Q-matrix is more superior han other matrices of a graph, so this thesis mainly discusses the Q-matrix, par-icularly investigates the relations between the Q-eigenvalues and the structures of graphs. The thesis is divided into the following four chapters:1. In Chapter 1, we introduce the development of spectral graph theory, the research background and summarize the main results of the thesis.2. In Chapter 2, we characterize the connected graphs with the second largest Q-eigenvalues no more than l, where l= 3.2470 is the largest root of the equation q³-5q²+6g-1= 0.3. In Chapter 3, we identify the connected graphs whose third largest Q-eigenvalues is no more than (3+(5^1/2))/2.4. In Chapter 4, we determine the connected graphs with the fourth largest Q-eigenvalue at most 1.