$ p $-Laplace elliptic inequalities on the graph

In this paper we study the following two $ p $-Laplace elliptic inequalities$ \triangle_pu(x)+u^{\tau}(x)\le 0 $and$ \triangle_pu(x)+(\Gamma_pu(x))^{\frac{\varrho}{p}}\le 0 $on a connected locally finite graph $ G = (V, E) $ for $ p>1 $. We derive Liouville type results for these inequalities when $ G $ satisfies some volume growth conditions for the case that $ \tau>p-1 $, or $ p-1<\varrho<p $ and $ \varrho\ge p $. We also give examples to show that the volume growth conditions are sharp.