Let G be a connected simple graph of order n. Let ${\rho }_1(G)\ge {\rho }_2(G)\ge \cdots \ge {\rho }_{n-1}(G)>{\rho }_n(G)=0$ 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 Sun and Das (2021) [9], $i.e.$, there is no graph with $m({\rho }_1)=n-3$ ($n\ge 6$) and $\nu (G)=2$. Moreover, we confirm that all the graphs with $m({\rho }_1)=n-3$ are determined by their normalized Laplacian spectra.

令G为n阶连通简单图。让${\rho }_1(G)\ge {\rho }_2(G)\ge \cdots \ge {\rho }_{n-1}(G)>{\rho }_n(G)=0$ 是归一化拉普拉斯矩阵的特征值 $\mathcal{升}(G)$的G。表示为$米({\rho }_{\mathrm{一世}})$ 归一化拉普拉斯特征值的重数 ${\rho }_{\mathrm{一世}}$. 让$\nu (G)$是G的独立数。在本文中，我们给出了具有重数的一些归一化拉普拉斯特征值的图的完整表征$n-3$，它回答了 Sun and Das (2021) [9] 中的一个遗留问题， $\mathrm{一世}.\mathrm{电子}.$，没有图 $米({\rho }_1)=n-3$ ($n\ge 6$） 和 $\nu (G)=2$. 此外，我们确认所有具有$米({\rho }_1)=n-3$ 由它们的归一化拉普拉斯光谱确定。