Bounds Of The Signless Laplacian Eigenvalues Of Graphs

Graph theory is an important branch of mathematics, and has a broad background in practical application. In order to study a graph, we introduce some matrices, such as adjacency matrix, distance matrix, Laplacian matrix, signless Laplacian matrix, etc. Up to isomorphism, they are all similar respectively. So the properties of these matrices are closely related with the structure of a graph.This paper focuses on the eigenvalues of signless Laplacian matrix of a graph(Q-eigenvalues). It’s divided into 4 chapters.In chapter one, we give the definitions, symbols and signs of graphs, also summarize the history and current status of the signless Laplacian of graphs. At the end,some main research results are listed.In chapter two, some matrix knowledge and lemmas are presented.In chapter three, we study the signless Laplacian spectra(Q-spectra) of graphs, and show an interlacing relation between a graph and its subgraph which is obtained by vertices deleting. Moreover, bounds that() 1( 1, 2,,)i iq G ≥d-i +i = Ln are given,where()iq G denotes the i-th largest Q-eigenvalue of the graph G of order n, andid denotes the i-th largest degree. Also, the sufficient condition for() 1( 2,3,,)i iq G =d-i = Lk and the necessary conditions for the equalities are given.In chapter four, the least signless Laplacian eigenvalue(the least Q-eigenvalue) of a class of graph was studied. Its sharp upper bound of the least Q-eigenvalue is 1, and a class of graphs with least Q-eigenvalue of 1 is constructed. In addition, a method of using least Q-eigenvalue of1 H ∨Kto decide whether H has a perfect match is presented. At last, a class of graphs with any integer or δ-1 as the least Q-eigenvalue are constructed.