In this dissertation we deal with some spectral problems for periodic differential operators originating from mathematical physics. We begin by using quantum graphs to model a particular graphyne and related nanotubes. The dispersion relations, and thus spectra, of periodic Schrodinger operators on these structures are analyzed. We find highly directional Dirac cones, which makes some types of graphynes fascinating. Then, we study a conjecture that has been widely assumed in solid state physics. Namely, the extrema of the dispersion relation of a generic periodic difference operator on a class of discrete graphs are proven to be non-degenerate. Here, by non-degeneracy we mean extrema having non-degenerate Hessian. Finally, we present a technique of creating and manipulating spectral gaps for a (regular) periodic quantum graph by inserting appropriate internal structures into its vertices.
