On the embedding of graphs into graphs with few eigenvalues

A graph is called of type k if it is connected, regular, and has k distinct eigenvalues. For example graphs of type 2 are the complete graphs, while those of type 3 are the strongly regular graphs. We prove that for any positive integer n, every graph can be embedded in n cospectral, non-isomorphic graphs of type k for every k ≥ 3. Furthermore, in the case k ≥ 5 such a family of extensions can be found at every sufficiently large order. Some bounds for the extension will also be given. © 1996 John Wiley & Sons, Inc.