Selected Topics Of Spectra Of Graphs

For a graph G, denote the adjacency matrix of G by A(G), the eigenvalues of A(G) are called the eigenvalues of G. The sequence of eigenvalues of G is called the spectra of graph G. The spectra of a graph is an important property of a graph. In the field of physics and chemistry, by the spectra of molecular graph, one can predict the property of this substance in physics and chemistry. And in the computer network, the spectra of the graph corresponding to the network will provide a useful algebra tool for studying this network. But for many graphs, we can not derive their spectra, thus, the estimation for the spectra of graph is an important problem in graph theory. In recent 30 years, there are many results about this problem. In this paper, we study the spectra of Cayley graphs, Bi-Cayley graphs and mixed Cayley graphs on Abelian group, give some algebraic properties of Bi-Circulant digraphs, and characterize the Gaussian integral circulant digraphs. In addition, we study the spectra of digraphs and the laplacian spread of quasi-tree graphs.In chapter 1, we introduce the background of our study, and give the def-initions of Cayley graphs, Bi-Cayley graphs, mixed Cayley graphs, spectra of graphs, laplacian spread, etc, and the main results in this paper. In chapter 2, we firstly study the spectra of folded hypercube and Bi-folded hypercube; Secondly, we study the adjacency matrix and spectra of the Cayley graph on Abelian group, and then give the spectra of Bi-Cayley graph and mixed Cayley graph on Abelian group and the definition of the Bi-Cayley digraph, furthermore, we derive the spectra of Bi-Cayley digraph and mixed Cayley digraph on Abelian group. Finally, we investigate the spectra of Bi-Circulant digraphs and some asymptotic enumeration theorem for the number of di-rected spanning trees in Bi-Circulant digraphs are presented. In chapter 3, we consider the relation of characteristic polynomials of regular bipartite digraph D and its bipartite complement D*. Further, we define the bipartite product of bipartite graphs and derive its spectra, and then generalize the definition and results to the bipartite digraphs. In chapter 4, we study the Laplacian spread of quasi-tree graph, characterize the unique quasi-tree graph with max-imum Laplacian spread among all quasi-tree graphs in the set Q(n, d) with 1≤d≤n-4/2. In chapter 5, we completely characterize the gaussian integral circulant digraphs with order k,2k,4k, and we find a class of gaussian integral circulant digraphs with order 2tk, where t>2 and k is odd.