Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs

Let G=(V,E)G=(V,E) be a simple graph. Denote by D(G)D(G) the diagonal matrix of its vertex degrees and by A(G)A(G) its adjacency matrix. Then the signless Laplacian matrix of GG is Q(G)=D(G)+A(G)Q(G)=D(G)+A(G). In [5], Cvetković et al. (2007) have given conjectures on signless Laplacian eigenvalues of GG (see also Aouchiche and Hansen (2010) [1], Oliveira et al. (2010) [14]). Here we prove two conjectures.