Trees which are cospectral with non-treesfor the normalized Laplacian

For the normalized Laplacian matrix it is possible for graphs with differing number of edges to have the same spectrum. This leads to the potential for there to be a tree and a non-tree which share the same spectrum and a well-known example of this are star graphs with other complete bipartite graphs. Previous to this work, this was the known infinite family with this property. We construct more families of graphs with this property