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In [Steve Butler. A note about cospectral graphs for the adjacency andnormalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390,2010.], Butler constructed a family of bipartite graphs, which are cospectralfor both the adjacency and the normalized Laplacian matrices. In this article,we extend this construction for generating larger classes of bipartite graphs,which are cospectral for both the adjacency and the normalized Laplacianmatrices. Also, we provide a couple of constructions of non-bipartite graphs,which are cospectral for the adjacency matrices but not necessarily for thenormalized Laplacian matrices.

On the construction of cospectral graphs for the adjacency and normalized Laplacian matrices