On the second minimum algebraic connectivity of the graphs whose complements are trees

For a graph Γ the algebraic connectivity denoted by a(Γ), is the second smallest eigenvalue of the Laplacian matrix of Γ. In Jiang et al. (2015), proved a unique graph with first minimum algebraic connectivity among the graphs which belong to a class of graphs whose complements are trees. In this paper, we characterize the unique graph with second minimum algebraic connectivity in the same aforesaid class of graphs.