On graphs whose signless Laplacian index does not exceed 4.5

Let A_G and D_G be respectively the adjacency matrix and the degree matrix of a graph G. The signless Laplacian matrix of G is defined as Q_G=D_G+A_G. The Q-spectrum of G is the set of the eigenvalues together with their multiplicities of QG. The Q-index of G is the maximum eigenvalue of Q_G. The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetković et al. [D. Cvetković, P. Rowlinson, S.K. Simić, Signless Laplacians of finite graphs, Linear Algebra Appl. 423 (2007) 155–171]. In the latter paper the authors determine the graphs whose Q-index is in the interval [0,4]. In this paper, we investigate some properties of Q-spectra of graphs, especially for the limit points of the Q-index. By using these results, we characterize respectively the structures of graphs whose the Q-index lies in the intervals (4,2+sqrt 5), (2+sqrt 5,ϵ+2) and (ϵ+2,4.5], where ϵ=1/3((54-6 sqrt 33)^1/3+(54+ 6 sqrt 33)^1/3)\approx 2.382975767. © 2009 Elsevier Inc. All rights reserved.