On The Signless Laplacian Spectrum Of Graphs

The spectra of graph theory is a very important research area in algebra graph theory, which study on eigenvalues and applications about associated matrices of graphs, such as the Adjacency matrix, the Incidence matrix, the Laplacian matrix, the signless laplacian matrix and so on. In this paper graphs which are discussed are simple connected. We use A(G), L(G), Q(G) to denote the adjacency matrix, the laplacian matrix and the signless laplacian matrix of a graph G, respectively, where L(G)=D(G)-A(G), Q(G)=D(G)+A(G) and D(G) is the diagonal matrix of vertex degrees of a graph G.A large number of lovers of graph theory are interested in the integral graphs since the quest for integral graphs was initiated by F.Harary and A.J.Schwenk in1973[19], and the study of A-integral graphs is extended to the study of L-integral graphs and other associated matrices of integral graphs. But the graphs with integral Q-spectrum are hardly studied so far. Therefore, we will investigate the study of Q-integral graphs.This paper is divided into three parts and the main structure of the article is as follows.In chapter l:we introduce the fundamental concepts and properties of graphs, also the domestic and foreign researching situations.In chapter2:we have characterised the connected Q-integral graphs with qi<5and there are12graphs like this.In chapter3:we obtain only partial connected Q-integral graphs with q1=6. Firstly we have characterised13connected Q-integral graphs with q1=6and maximum degree3, then we have characterised only one connected Q-integral graph with q1=6and maximum degree5, finally, there are4connected Q-integral graphs with two main signless Laplacian eigenvalues and q1=6, maximum degree4which we have characterised.