On the second largest normalized Laplacian eigenvalue of graphs

摘要

Let G=(V,E) be a simple graph of order n with normalized Laplacian eigenvalues ρ1≥ρ2≥⋯≥ρn−1≥ρn=0. The normalized Laplacian spread of graph G, denoted by ρ1−ρn−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.