On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Let G be a simple connected graph with order n . Let ℒ(G) and 𝒬(G) be the normalized Laplacian and normalized signless Laplacian matrices of G , respectively. Let λ k ( G ) be the k -th smallest normalized Laplacian eigenvalue of G . Denote by ρ ( A ) the spectral radius of the matrix A . In this paper, we study the behaviors of λ 2 ( G ) and ρ(ℒ(G)) when the graph is perturbed by three operations. We also study the properties of ρ(ℒ(G)) and X for the connected bipartite graphs, where X is a unit eigenvector of ℒ(G) corresponding to ρ(ℒ(G)) . Meanwhile we characterize all the simple connected graphs with ρ(ℒ(G))=ρ(𝒬(G)) .