On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G = (V, E) be a simple graph. Denote by D (G) the diagonal matrix of its vertex degrees and by A (G) its adjacency matrix. Then the Laplacian matrix of G is L (G) = D (G) - A (G) and the signless Laplacian matrix of G is Q (G) = D (G) + A (G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetković et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.