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{\bf Steve Butler and Jason Grout }
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{\bf A Construction of Cospectral Graphs for the Normalized Laplacian}
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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.

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