Nordhaus-Gaddum type inequalities for Laplacian and signless Laplacian eigenvalues

Let G be a graph with n vertices. We denote the largest signless Laplacian eigenvalue of G by q(1)(G) and Laplacian eigenvalues of G by mu(1)(G) >= ... >= mu(n-1)(G) >= mu(n)(G) = 0. It is a conjecture on Laplacian spread of graphs that mu(1)(G) - mu(n-1)(G) <= n - 1 or equivalently mu(1)(G) + mu(1)((G) over bar) <= 2n - 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, mu(1)(G)mu(1)((G) over bar) <= n(n - 1). Aouchiche and Hansen [Discrete Appl. Math. 2013] conjectured that q(1)(G) + q(1)((G) over bar) <= 3n - 4 and q(1)(G)q(1)((G) over bar) <= 2n(n - 2). We prove the former and disprove the latter by constructing a family of graphs H-n where q(1)(H-n)q(1)((H-n) over bar) is about 2.15n(2) + O(n).