A cospectral family of graphs for the normalized Laplacian found by toggling

We give a construction of a family of (weighted) graphs that are pairwise cospectral with respect to the normalized Laplacian matrix, or equivalently, probability transition matrix. This construction can be used to form pairs of cospectral graphs with different number of edges, including situations where one graph is a subgraph of the other. The method used to demonstrate cospectrality is by showing the characteristic polynomials are equal.