On spectral constructions for Salem graphs

In recent years, people have begun studying Salem numbers by looking at the spectrum of the adjacency matrix of a graph. In this thesis we classify infinitely many new infinite families of Salem graphs using results about graph spectra. Our first method is to define a notion of how close a Salem graph is to being cyclotomic, the m-Salem graphs, and classify the whole family of 1-Salem graphs. The second method uses the Courant-Weyl inequalities in a novel way, partitioning the edges of a graph into two sets and considering the graphs they form. We exhaustively work through all possibilities to find even more families of Salem graphs. We also study when some of these graphs produce trivial Salem numbers, using a new extension of Hoffman and Smith's subdivision theorem.