  In this paper, we study the gradient estimates for the positive solutions of
the weighted porous medium equation $$\Delta u^{m}=\delta(x)u_{t}+\psi u^{m}$$
on graphs for $m>1$, which is a nonlinear version of the heat equation.
Moreover, as applications, we derive a Harnack inequality and the estimates of
the porous medium kernel on graphs. The obtained results extend the results of
Y. Lin, S. Liu and Y. Yang for the heat equation [8, 9].
