Since there are many properties of graph which can be expressed in terms of both existence of graph homomorphisms and graph spectra I expect there are some papers exploring this connection between graph spectra and graph homomorphisms.
Any thoughts on this topic? Papers, books, people...?
There does not seem to be a large overlap; the basic problem is that homomorphisms generally destroy nearly all spectral information. There are important exceptions though. Thus in https://arxiv.org/abs/1601.00969, David Roberson resolves a conjecture of Cameron and Kazanidis by proving that if a strongly regular graph is not a core then its core is a complete graph; by my standards at least, his proof is a spectral proof.
