We deal with the extinction of the solutions of the initial-boundary value problem of the discrete p-Laplacian equation with absorption 𝑢𝑡=Δ𝑝,𝜔𝑢−𝑢𝑞 with p > 1, q > 0, which is said to be the discrete p-Laplacian equation on weighted graphs. For 0 < q < 1, we show that the nontrivial solution becomes extinction in finite time while it remains strictly positive for 𝑝≥2, 𝑞≥1 and 𝑞≥𝑝−1. Finally, a numerical experiment on a simple graph with standard weight is given.