  Graphs with few distinct eigenvalues have been investigated extensively. In
this paper, we focus on another relevant topic: characterizing graphs with some
eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix
of a graph is considered here. Let $\rho_{n-1}(G)$ and $\nu(G)$ be the second
least normalized Laplacian eigenvalue and the independence number of a graph
$G$, respectively. As the main conclusions, two families of $n$-vertex
connected graphs with some normalized Laplacian eigenvalue of multiplicity
$n-3$ are determined: graphs with $\rho_{n-1}(G)=-1$ and graphs with
$\rho_{n-1}(G)\neq -1$ and $\nu(G)\neq 2$. Moreover, it is proved that these
graphs are determined by their spectrum.
