On the second largest normalized Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph of order n with normalized Laplacian eigenvalues ρ₁≥ρ₂≥⋯≥ρ_n−1≥ρ_n=0. The normalized Laplacian spread of graph G, denoted by ρ₁−ρ_n−1, is the difference between the largest and the second smallest normalized Laplacian eigenvalues of graph G. In this paper, we obtain the first four smallest values on ρ₂ of graphs. Moreover, we give a lower bound on ρ₂ of connected bipartite graph G except the complete bipartite graph and characterize graphs for which the bound is attained. Finally, we present some bounds on the normalized Laplacian spread of graphs and characterize the extremal graphs.