Characterization of Graphs with Some Normalized Laplacian Eigenvalue Having Multiplicity \(n{-}4\)

The spectrum of the normalized Laplacian matrix of a graph provides a lot of structural information of the graph, and it has applications in numerous areas and in different guises. In this paper, we completely characterize all connected graphs of order \(n\ge 25\) $n\ge 25$ with some normalized Laplacian eigenvalue \(\rho \in \big (0,\,\frac{n-1}{n-2}\big )\) $\rho \in (0,\frac{n-1}{n-2})$ having multiplicity \(n{-4}\) $n-4$ .