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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 $μ_1(G)\ge\cdots\geμ_{n-1}(G)\geμ_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $μ_1(G)-μ_{n-1}(G)\le n-1$ or equivalently $μ_1(G)+μ_1(\Gb)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $μ_1(G)μ_1(\Gb)\le n(n-1)$. 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, $q_1(G)+q_1(\Gb)\le3n-4$ and $q_1(G)q_1(\Gb)\le2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\ov{H_n})$ is about $2.15n^2+O(n)$.