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 ${\mu }_1(G)\ge \cdots \ge {\mu }_{n-1}(G)\ge {\mu }_n(G)=0$. It is a conjecture on Laplacian spread of graphs that ${\mu }_1(G)-{\mu }_{n-1}(G)\le n-1$ or equivalently ${\mu }_1(G)+{\mu }_1(\text{Gb})\le 2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, ${\mu }_1(G){\mu }_1(\text{Gb})\le n(n-1)$. Aouchiche and Hansen [A survey of Nordhaus–Gaddum type relations, Discrete Appl. Math. 161 (2013), 466–546] conjectured that %for any graph $G$ with $n$ vertices, $q_1(G)+q_1(\text{Gb})\le 3n-4$ and $q_1(G)q_1(\text{Gb})\le 2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\text{ov}{H}_n)$ is about $2.15n^2+O(n)$.