  We give a method to construct cospectral graphs for the normalized Laplacian
by a local modification in some graphs with special structure. Namely, under
some simple assumptions, we can replace a small bipartite graph with a
cospectral mate without changing the spectrum of the entire graph. We also
consider a related result for swapping out biregular bipartite graphs for the
matrix $A+tD$.
  We produce (exponentially) large families of non-bipartite, non-regular
graphs which are mutually cospectral, and also give an example of a graph which
is cospectral with its complement but is not self-complementary.
