On graphs with some normalized Laplacian eigenvalue of extremal multiplicity

Let $G$ be a connected simple graph on $n$ vertices. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$ and $\rho_{n-1}(G)$ be the second least eigenvalue of $\mathcal{L}(G)$. Denote by $\nu(G)$ the independence number of $G$. Recently, the paper [Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$, arXiv:1912.13227] discussed the graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$. However, there is one remaining case (graphs with $\rho_{n-1}(G)\neq 1$ and $\nu(G)= 2$) not considered. In this paper, we focus on cographs and graphs with diameter 3 to investigate the graphs with some normalized Laplacian eigenvalue of multiplicity $n-3$.