Trees with 4 or 5 distinct normalized Laplacian eigenvalues

We develop a tool for locating eigenvalues of the normalized Laplacian matrix of trees. This is obtained by extending an algorithm designed for the adjacency matrix, due to Jacobs and Trevisan (2011). As an application, we study the multiplicity of normalized Laplacian eigenvalues of small diameter trees. Our main result is the characterization of the trees that have 4 or 5 distinct normalized Laplacian eigenvalues. We also show that with a fixed diameter these trees are determined by their normalized Laplacian spectrum.