Nordhaus-Gaddum Type Inequalities for the kth Largest Laplacian Eigenvalues

Let G be a simple connected graph and ₁ (G) ₂ (G) ₙ (G) be the Laplacian eigenvalues of G. Let G be the complement of G. Einollahzadeh et al. J. Combin. Theory Ser. B, 151 (2021), 235–249 proved that ₍-₁ (G) +₍-₁ (G) 1. Grijò et al. Discrete Appl. Math. , 267 (2019), 176–183 conjectured that ₍-₂ (G) +₍-₂ (G) 2 for any graph and proved it to be true for some graphs. In this paper, we prove ₍-₂ (G) +₍-₂ (G) 2 is true for some new graphs. Furthermore, we propose a more general conjecture that ₖ (G) +ₖ (G) n-k holds for any graph G, with equality if and only if G or G is isomorphic to K₍-₊ 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 n+12, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when n 2c+8.