Let e a graph with vertices. We denote the signless Laplacian eigenvalues and Laplacian eigenvalues of , arranged decreasingly, by and , respectively. It is a conjecture on Laplacian spread of graphs that or equivalently , where denotes the complement graph of . We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph , . Aouchiche and Hansen conjectured that and . We prove the former and disprove the latter by constructing a family of graphs where is about of order .
