Characterization of graphs with some normalized Laplacian eigenvalue of multiplicity n-3

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 ${\rho }_{n-1}(G)\mathrm{\ne }-1$ and $\nu (G)\mathrm{\ne }2$. Moreover, it is proved that these graphs are determined by their spectrum.