On the lower bound of the sum of the algebraic connectivity of a graph and its complement

For a graph G, let mu(2)(G) denote its second smallest Laplacian eigenvalue. It was conjectured that mu(2)(G) + mu(2)((G) over bar) >= 1, where (G) over bar is the complement of G. This conjecture has been proved for various families of graphs. Here, we prove this conjecture in the general case. Also, we will show that max{mu(2)(G), mu(2)((G) over bar)} >= 1 - O(n(-1/3)), where n is the number 1 of vertices of G. (C) 2021 Elsevier Inc. All rights reserved.