In this talk, I will show a method to get time decay rates for the fundamental solutions of dispersive equations on the discrete graph Z^d. As in the setting of Euclidean space, it is necessary to deal with the asymptotic behaviour of oscillatory integral with parameters. Newton polyhedra and an algorithm dated back to Karpushkin will be introduced as two important parts in the proof. In a more general sense, our method will focus on obtaining uniform decay estimate for highly degenerate oscillatory integrals. This is a joint work with Cheng Bi and Prof. Bobo Hua.