The Study Of Spectral Properties Of Graph

The graph spectra theory is a new field in graph theory. It originated from the techniques, which was first used by theoretic chemists and physicists, of seeking approximate numerical solution for certain partial differential equations. The fundamental papers^[2](1957) of L.Collatz and U.Sinogowitz are usually considered as the starting point of the study of graph spectra theory. In the short past 50 years, it has been developed into a systematic, integrated theory. Many monographs on this field ([1,3-5,13]) have been published. Literatures on this field are increasing with the speed of thousands of papers per year.Two kinds of graph spectra — the spectra of adjacency matrices and the spectra of Laplacian matrices, has been widely studied in the past several decades. They have great significance not only in graph theory, but also in some fields of chemistry and physics (refer to Chapter 8 in [3]). As has been pointed out in [3], chemists and physicists know the structure of the graph and they are looking for the corresponding spectrum, whereas, in general, graph theorists and combinatorialists assume that the spectrum is known and they try to say something about the graph structure.In this paper, three kinds of spectra of graphs defined in §1.2 are involved.1. In Chapter 2, we consider the nullity of graphs. For unicyclic graphs and bicyclic graphs, we showed that the nullity set both are [0,n - 4](n ≥ 5 for unicyclic graphs and n ≥ 6 for bicyclic graphs). We determined the graphs with η(G) = n - 4,for unicyclic graphs and bicyclic graphs. We found some graphs with large nullity for general graphs. This is instructive and helpful to the study of nullity of general graphs. We also gave some conditions for a graph is singular or non-singular.2. In Chapter 3, some estimation problems are concerned. Let g_n,k(k < n) and g_n,k'(k < n) be the set of graphs of order n with k independent vertices and the set of graphs of order n with k independent edges, respectively.