Spectral Graph Theory

The spectral graph theory is one of the main branches in algebraic graph theory, it is mainly concerned with the adjacency spectrum and the Laplacian spectrum. The principal method for the research is by using the matrix representation of a graph, combined with matrix theory and the classical results in graph theory, to impulse the theoretical research of graph theory. The thesis is organized as follows.In Chapter 2, we mainly discuss the adjacency spectrum of graphs. Firstly, we present some sharp upper bounds on the adjacency spectral radius of graphs, and show that these bounds are somewhat better than the known ones. Secondly, we give a new upper bound on the kth largest eigenvalue of a graph. Finally, for the spectral radius of a graph with a cut vertex, we give an inequality concerning the spectral radius of the graph and its subgraphs.In Chapter 3, we consider a problem raised by Brualdi and Solheid [22]: For a given set of graphs (?), find an upper bound on the spectral radius of this set and characterize the graphs in which the maximal spectral radius is attained. There is a great of literature on this problem. In this chapter, we shall solve the problem for (?) to be the set of graphs with given chromatic number, matching number and diameter, respectively. We thus answer a problem raised by Hong [69] in the affirmative.