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 λ₂(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)).