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

Fenglei Tian, Yiju Wang

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)…