Signless Laplacian eigenvalue problems of Nordhaus-Gaddum type

Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$denotethesignlessLaplacianeigenvaluesof$G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) #P3.6] showed that $q_1(G)+q_1(\bar{G})\le 3n-4$, with equality holding if and only if $G$ or $\bar{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the following problem: for $n\ge 6$, does $q_2(G)+q_2(\bar{G})\le 2n-5$ 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 $q_2(G)+q_2(\bar{G})\ge n-2$, and the extremal graphs are also characterized.