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 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) 6 n − 1 or equivalently μ1(G) + μ1(G) 6 2n − 1. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph G, μ1(G)μ1(G) 6 n(n − 1). Aouchiche and Hansen [Discrete Appl. Math. 2013] conjectured that q1(G) + q1(G) 6 3n − 4 and q1(G)q1(G) 6 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 2.15n2 +O(n).