On There To The Whole Spectrum And Gauss Spectra

A graph is called integral if and only if all the zeros of its characteristic polynomial are integral. The notion of integral graphs was first introduced by Harary and Schwenk in 1974. Since then, a lot of papers have been published by many scholars who engaged in the research on this field. But most of their results are limited on non-directed graphs and integral trees, and few works about integral directed graphs are obtained. In fact, it is helpful to the development of graph theory to discover special kinds of integral directed graphs.In this dissertation, some important theorems are generalized and new results are successfully derived by investigating the operations of non-directed graphs and directed graphs in detail. Most of all, a new method to construct integral non-directed and directed graphs is presented and we can construct infinite integral non-directed or directed graphs by this way. The main contents of the dissertation are as follows:1. In chapter one: current development of integral graphs and the results of this dissertation are briefly introduced.2. In chapter two, special operations of a special kind of graphs is studied. And characteristic polynomials of graphs is obtained by means of calculating some matrices and determinants, which is used to be the base of a farther study on the condition that graphs are integral.3. In chapter three, the spectra of the new graphs is calculated by using the spectra of the known graphs. For example, if the spectra of the known graphs is integral, then so is the spectra of the new graphs which are constructed by special methods.4. In chapter four, a new method to construct integral directed graphs is discovered through the research on the relationship between the zeros of the characteristic polynomial of the single level and multi-level circulant matrices and the parameters of the matrices. The method can be used to constructinfinite integral directed graphs.5. In chapter five, generalized line graph method, another method to construct integral graphs is introduced by using conclusion of other researchers. The method can also be used to construct infinite integral directed graphs. For example, if graph G is even-degree integral, and i is free parameter(i≥ 1), then the generalized line graph LBi(G) is also even-degree integral. Besides, the common characteristics of the spectra of directed graphs and the charac teristics of the graphs which has "Zeros" in the spectra are studied.6. In chapter six, the main results of this dissertation are listed in detail and the further objects for research and the problems which are not yet solved are introduced.