On the spectral characterization of graphs with respect to the normalized Laplacian

A graph G is said to be determined by its normalized Laplacian spectrum if there is no other non-isomorphic graph with the same normalized Laplacian spectrum. To date, only a limited number of graphs have been proven to possess this property. In this paper, we obtain a new class of graphs that are determined by their normalized Laplacian spectra. Our approach is based on a characterization of all connected n -vertex graphs whose third smallest normalized Laplacian eigenvalue is at least n n − 1 .