The Spectrum Of Corona Graphs And Property R(SR) Of Nonsingular Graphs

The theory of graphs spectra is an important research project, it is a crossing area between graph theory and algebra. This field mostly research the spectral character of graphs, the graphic character of graph spectrum and the corresponding questions. In literature [2] and [3], there has been some results about the spectrum of corona graphs and property R(SR) of nonsingular graphs. For the connected graph G1 and a regular graph G2, the adjacency spec-trum of G1(?)G2 has been determined. In the case of G2 being any graph, the L-spectrum of G1(?)G2 has been completely characterized, but it is quite difficult to determine the adjacency spectrum and Q-spectrum of G1(?)G2. This paper is focus on the adjacency spectrum and Q-spectrum of corona graphs G1(?)Km1,m2, we also further research property R(SR/NR) of nonsingular graphs. The main results are as follows:In the first chapter, introduction, we review the research history and present situation. Then we give the basic definitions, symbols and notations about graph spectrum. We also give the corresponding concepts about corona graphs and property R(SR) of nonsingular graphs. We know that some corona graphs have property R(SR), but the graphs which have property R(SR) are not always corona graphs.The second chapter consists of two sections. In the first section, we completely charac-terize the adjacency spectrum of the corona graph G1(?)Km1,m2 by the method of classified discussion. In the second section, we characterize the Q-spectrum of G1(?)Km1,m2 by the similar method. Where G1 is any graph, Km1,m2 is a complete bipartite graph. We also give examples and tables to explain our results.The third chapter consists of two sections. In the first section, we list some special graphs which have property R(SR/NR) according to the known research results. In the second section, we characterize general graphs which have property R(NR) according to the corresponding results of sign-inverse graph and balanced sign-graph, and research the conditions which these graphs satisfy.