Some Graphic Characters Of The Laplacian Spectrum In Tricyclic Graphs

In a simple graph G = (V,E), Let A(G) the matrix be the adjacency matrix of G, eigen-values and spectrum of A(G) are called eigenvalues and spectrum of G. The correspondingrelation of the spectrum graphs and the structures of graphs is an active research direc-tion. Acorrdingly, people introduce the Laplacian spectrum of G, the Laplacian matrix of isL(G) = D(G) ? A(G), where D(G) is the diagonal matrix of vertex degrees. The Laplacianspectrum of G are called spectrum of L(G). There are abundance of related literature andresults, which re?ect plenty of structures of graphs. So people more and more pay attentionto Laplacian spectrum of graphs. This research not only intensifies the description of inher-ence relation of discrete structures, but also has actually far-reaching meaning in networkoptimizing and operational research, and so on.In the research of Laplace spectrum, the spectral radii is very important. We mainlyestimate its upper bound and determine structures of graphs when it is equal to its spectralradii. Thinking of this way, people try to determine structures of graphs when it is equal toits first some laplacian spectral radii. In this paper, we mainly obtain the first six largestLaplacian spectral radii among all the graphs in the class T (n)(n≥9) together with thecorresponding graphs based on the results of elder generation. The main results are asfollows:In the first chapter, introduction, we mainly look back Graph spectra theoretical researchhistory and present condition. We also list a few results that have been already had in theaspects of characterizing Laplacian maximal graphs.The second chapter consists of two sections. In the first section, we give the basicdefinitions, symbols and notations about graphs spectra. In the second section, we introducesome basic Lemmas to characterize Laplacian maximal graphs.In the third chapter, we obtain the first six largest Laplacian spectral radii among all thegraphs in the class T (n)(n≥9) together with the corresponding graphs.