Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues

Let $G$ be a simple connected graph and $\mu_1(G) \geq \mu_2(G) \geq \cdots \geq \mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $\overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $\mu_{n-1}(G)+\mu_{n-1}(\overline{G})\geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $\mu_k(G)+\mu_k(\overline{G})\geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $\overline{G}$ is isomorphic to $K_{n-k}\vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $k\leq \frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $n\geq 2c+8$.