On the second smallest and the largest normalized Laplacian eigenvalues of a graph

Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$. Let ${\lambda }_k(G)$ be the $k$-th smallest normalized Laplacian eigenvalue of $G$. Denote $\rho (A)$ the spectral radius of the matrix $A$. In this paper, we study the behaviors of ${\lambda }_2(G)$ and $\rho (\mathcal{L}(G))$ when the graph is perturbed by three operations.