On the second largest normalized Laplacian eigenvalue of graphs

Let G = (V, E) be a simple graph of order n with normalized Laplacian eigenvalues rho(1)>= rho(2) >= ... rho(n-1) >= rho(n) = 0. The normalized Laplacian spread of graph G, denoted by rho(1) - rho(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 rho(2) of graphs. Moreover, we give a lower bound on rho(2) 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 extrernal graphs. (C) 2018 Published by Elsevier Inc.