Full characterization of graphs having certain normalized Laplacian eigenvalue of multiplicity $n-3$

Let $G$ be a connected simple graph of order $n$. Let $\rho_1(G)\geq \rho_2(G)\geq \cdots \geq \rho_{n-1}(G)> \rho_n(G)=0$ be the eigenvalues of the normalized Laplacian matrix $\mathcal{L}(G)$ of $G$. Denote by $m(\rho_i)$ the multiplicity of the normalized Laplacian eigenvalue $\rho_i$. Let $\nu(G)$ be the independence number of $G$. In this paper, we give a full characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, which answers a remaining problem in [S. Sun, K.C. Das, On the multiplicities of normalized Laplacian eigenvalues of graphs, Linear Algebra Appl. 609 (2021) 365-385], $i.e.,$ there is no graph with $m(\rho_1)=n-3$ ($n\geq 6$) and $\nu(G)=2$. Moreover, we confirm that all the graphs with $m(\rho_1)=n-3$ are determined by their normalized Laplacian spectra.