Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type

Let G be a graph of order n, and let q(1)(G) >= q(2)(G) >= center dot center dot center dot >= q(n)(G) denote the signless Laplacian eigenvalues of G. Ashraf and Tayfeh-Rezaie (2014) [3] showed that q(1)(G) + q(1)((G) over bar) <= 3n - 4, with equality holding if and only if G or (G) over bar is the star K-1,K- n-1. In this paper, we prove that q(2)(G) + q(2)((G) over bar) <= 2n - 4, where the equality holds if and only if G or (G) over bar is K-2, P-4 or C-4. Also, we discuss the following problem: for n >= 6, does q(2)(G) + q(2)((G) over bar) <= 2n - 5 always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining the bound. Moreover, we show that q(2)(G) + q(2)((G) over bar) >= n - 2, and the extremal graphs are also characterized. (C) 2019 Elsevier Inc. All rights reserved.