  Schultz \cite{S98} proved dispersive estimates for the wave equation on
lattice graphs $\mathbb{Z}^d$ for $d=2,3,$ which was extended to $d=4$ in
\cite{BCH23}. By Newton polyhedra and the algorithm introduced by Karpushkin
\cite{K83}, we further extend the result to $d=5:$ the sharp decay rate of the
fundamental solution of the wave equation on $\mathbb{Z}^5$ is
$|t|^{-\frac{11}{6}}.$ Moreover, we prove Strichartz estimates and give
applications to nonlinear equations.
