Spectral Properties Of A Number Of Graphs

In this thesis, we will characterize the adjacency spectrum of the complete bipartite graphs Km,n. Besides, we will show that trees with fourth Laplacian eigenvalue less than two are determined by their Laplacian spectrum.There are four chapters in this thesis. In chapter 1, the background and development of the spectral graph theory are introduced; In chapter 2, some concepts and known results which will be used in the last two chapters of this thesis are given; In chapter 3, the adjacency spectrum of complete bipartite graphs Km,n are characterized; In chapter 4, we show that trees with fourth Laplacian eigenvalue less than two are determined by their Laplacian spectrum.