The Research On Spectral Determination Of Some Graphs

The theory of graph spectra is originated in chemical field in the fifties of the last century. Its research areas mainly concerned with the adjacency spectrum, the Laplacian spectrum and the signless Laplacian spectrum. About the spectral determination theory of graph, in early time, it is believed that every graph is determined by its spectrum. As cospectral graphs are been found, the question "which graphs are determined by their spectra?" by attention of more and more scholars.In the first chapter of this dissertation, we introduce the applications and research status of the theory of graph spectra. In chapter2, we introduce some basic knowledge about graph theory. In chapter3, we investigate some properties of some graphs and the spectra of these graphs. In chapter4, we investigate spectral characterizations of some graphs. The main results are as follows.(1) The properties of the signless Laplacian matrix of PS graph are obtained, so the structure of signless Laplacian spectrum of a PS graph is given.(2) We introduce the corona of two graphs and give the relationship between the spectrum of the corona of two graphs and the spectra of two graphs.(3) The Laplacian spectrum of the sun graph (Cn。2K1)is obtained, and the sun graph is proved to be determined by its Laplacian spetrum.(4) The signless Laplacian spetrum of the double starlike tree is given and the double starlike tree is proved to be determined by its Laplacian spetrum.(5) The properties of the spetrum of Tn2graph are obtained and it is proved that Tn3is determined by its signless Laplacian spectrum.