Dispersion on certain Cartesian products of graphs

In this short note we prove a sharp dispersive estimate $\mathrm{\parallel }{\mathrm{e}}^{\mathrm{i}tH}f{\mathrm{\parallel }}_{\mathrm{\infty }}<t^{-d\mathrm{/}3}\mathrm{\parallel }f{\mathrm{\parallel }}_1$ for any Cartesian product ${\mathbb{Z}}^d{G}_F$ of the integer lattice and a finite graph. This includes the infinite ladder, $k$-strips and infinite cylinders, which can be endowed with certain potentials.