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

Abstract

For a graph G, let μ2(G) denote its second smallest Laplacian eigenvalue. It was conjectured that μ2(G)+μ2(G‾)≥1, where G‾ 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⁡{μ2(G),μ2(G‾)}≥1−O(n−13), where n is the number of vertices of G.