Nordhaus-Gaddum type inequalities for the kth Laplacian

Let G be a simple connected graph and mu(1)(G)>=mu(2)(G)>=& ctdot;>=mu(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 mu(n-1)(G)+mu(n-1)(G)>= 1. Grij & ograve; et al. [Discrete Appl. Math., 267(2019), 176-183] conjectured that mu(n-2)(G)+mu(n-2)(G)>= 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)(G)>= 2 is true for some new graphs. Furthermore, we propose a more general conjecture that mu(k)(G)+mu(()(k)G)>= n-k holds for any graph G, with equality if and only if G or G is isomorphic to Kn-k boolean OR 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+1/2, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when n >= 2c+8.