Given a graph G, the normalized Laplacian associated with the graph G, denoted ${\cal L}$(G), was introduced by F. R. K. Chung and has been intensively studied in the last 10 years. For a k-regular graph G, the normalized Laplacian ${\cal L}$ (G) and the standard Laplacian matrix L(G) satisfy L(G) = k {\cal L} (G), and hence they have the same eigenvectors and their eigenvalues are directly related. However, for an irregular graph G, ${\cal L} (G) and L(G) behave quite differently, and the normalized Laplacian seems to be more natural. In this paper, Cauchy interlacing-type properties of the normalized Laplacian are investigated, and the following result is established. Let G be a graph, and let H = G - e, where e is an edge of G. Let $\lambda_1\geq \lambda_2 \geq \cdots \geq \lambda_{n}=0$ be the eigenvalues of ${\cal L}(G)$, and let $\theta_1\geq \theta_2 \geq \cdots \geq \theta_{n}$ be the eigenvalues of ${\cal L}(H)$. Then, $\lambda_{k-1} \geq \theta_{k}\geq \lambda_{k+1}$ for each k = 1, 2, 3, \dots, n, where $\lambda_{0} =2$ and $\lambda_{n+1}=0$. Applications are given for eigenvalues of graphs obtained from special graphs by adding or deleting a few edges. A short proof is given of the result that G is a graph with each component a nontrivial bipartite graph if and only if $2-\lambda$ is an eigenvalue of ${\cal L}(G)$ for each eigenvalue $\lambda$ of ${\cal L }(G)$.