Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q1(G) and Laplacian eigenvalues of G by µ1(G) ≥ · · · ≥ µn−1(G) ≥ µn(G) = 0. It is a conjecture on Laplacian spread of graphs that µ1(G) − µn−1(G) ≤ n − 1 or equivalently µ1(G) + µ1(G) ≤ 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, µ1(G)µ1(G) ≤ n(n − 1). Aouchiche and Hansen [A survey of Nordhaus-Gaddum type relations, Discrete Appl. Math. 161 (2013), 466-546] conjectured that q1(G) + q1(G) ≤ 3n − 4 and q1(G)q1(G) ≤ 2n(n − 2). We prove the former and disprove the latter by constructing a family of graphs Hn where q1(Hn)q1(Hn) is about 2.15n 2 + O(n).
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