Spectral Mathod In Graph And Hypergraph Theory

In this thesis we use the spectrum of matrix or tensor to characterize the struc-ture of graphs or hypergraphs. The thesis consists of three topics:the connectivity and extreme spectral property of graphs, the lower bounds of spectral radius of graphs subject to given clique number or domination number, spectral property of hypergraphs.The vertex (edge) connectivity is a key parameter to measure the graph con-nectivity. Denote by & (?)nk ((?)nk) the sets of graphs of order n with vertex (edge) connectivity k respectively.How to characterize the spectral radius of general graphs with fixed connectivity is one of our main work. We characterizes the maximum spectral radius of graphs among all graphs in (?)nk or (?)nk n, and also the graphs attaining the extreme bounds. As an application, we solve a conjecture posed by Aouchiche and Hansen on the relation between the adjacency spectral radius and vertex or edge connectivity. The upper bounds of spectral radius of graphs with fixed clique number or domination number have been given. In this theis we obtain the lower bounds of such graphs and also characterize the structures of the graphs.Relative to spectral radius the least eigenvalue of graphs was received less attention in past work. One main reason is that the method used in the discussion of spectral radius cannot be applied to least eigenvalue. Ifκ≤n/2 we completely characterize the lower bounds and the corresponding extreme graphs in the class (?)nk,(?)nk.Using the incidence matrix of hypergraphs, we introduce the adjacency ma-trix and signless Laplace of hypergraps. Analogous to the case of graphs, we get the structure of hypergraphs which have extrem spectral properties. In addition, we introduce the concept ofκ-vertex(edge) connectivity of hypergraphs, and char-acterize the maximum of spectral radius and the minimum of least eigenvalues of adjacency matrix and signless Laplace matrix of linear hypergraphs subject to fixed vertex (edge) connectivity. At the end of this thesis, we use supersymmetric tensor to representκ-uniform hypergraphs, and study the spectral properties of tensors, particularly the spectral radius.