Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues 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 signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovic et al. (2007) have given conjectures on signless Laplacian eigenvalues of G (see also Aouchiche and Hansen (2010)[1], Oliveira et al. (2010) [14]). Here we prove two conjectures. (C) 2011 Elsevier B.V. All rights reserved.