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], Cvetković 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.