Research On The Signless Laplacian Spectra Of Some Graphs

The theory of graph spectra is a popular research field, it has important applications in physics, chemistry and computer science. In [D. Cvetkovic, P Rowlinson, S K Simic. Signless Laplacians of finite graphs. Linear Algebra Appl.2007.423:155-171P], D. Cvetkovic et al. discussed the property of the signless Laplacian spectra of graphs. After that, the signless Laplacian spectra attracts more and more attention.Let A(G) be the adjacency matrix of a graph G, D(G) be the diagonal matrix of vertex degrees of G. The matrix Q(G)=D(G)+A(G) is called the signless Laplacian matrix of G. The spectrum of Q(G) is called the signless Laplacian spectrum of G. The maximum eigenvalue of Q(G) is called the signless Laplacian spectral radius or the Q-index of G.In the first chapter of this dissertation, we introduce the background and applications of the theory of graph spectra. In chapter 2, we introduce some basic knowledge about graph theory and linear algebra. In chapter 3, we investigate the limit points of the Q-indices of some graphs. In chapter 4, we investigate spectral characterizations of some graphs, the main results is as follows.(1) The limit point of the Q-indices of T-shape trees is obtained, so the bounds of the Q-index of a T-shape tree is given.(2) The limit point of the Q-indices of lollipop graphs is obtained, so the bounds of the Q-index of a lollipop graph is given.(3) The limit point of the Q-indices of H(x,y,z) graphs is obtained, so the bounds of the Q-index of a H(x,y,z) graph is given.(4) For any graph G, G attachs two paths and G attachs a cycle have the same limit point of the Q-index.(5) The union of stars are proved to be determined by its signless Laplacian spetrum.(6) Some caterpillars are proved to be determined by their signless Laplacian spetra.