  Using the Floquet-Bloch transform, we show that $\mathbb{Z}^d$-periodic
graphs have finitely many finite support eigenfunctions up to translations and
linear combinations and show that this can be used to calculate the density of
eigenvalues. We study the Kagome lattice to illustrate these techniques and
generalize the claims to amenable quasi-homogeneous graphs whose acting group
has Noetherian group algebra (this includes all virtually polycyclic groups).
Finally, we provide a formula for the von Neumann dimension of eigenvalues of
$\mathbb{Z}^d$-periodic graphs using syzygy modules.
