Normalized Laplacian, cospectral graphs, scaling, twins.
The spectrum of the normalized Laplacian matrix cannot determine the number of edges in a graph, however finding constructions of cospectral graphs with differing number of edges has been elusive. In this paper we use basic properties of twins and scaling to show how to construct such graphs. We also give examples of families of graphs which are cospectral with a subgraph for the normalized Laplacian matrix.
"Using twins and scaling to construct cospectral graphs for the normalized Laplacian",
Electronic Journal of Linear Algebra,