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

Abstract: Let be a simple graph. Denote by the diagonal matrix of its vertex degrees and by its adjacency matrix. Then the signless Laplacian matrix of is . In , Cvetković et al. (2007) have given conjectures on signless Laplacian eigenvalues of (see also Aouchiche and Hansen (2010) , Oliveira et al. (2010) ). Here we prove two conjectures. [Copyright &y& Elsevier]