Let $G=(V,E)$ be a local finite connected weighted graph, $\mathrm{\Omega }$ be a finite subset of $V$ satisfying ${\mathrm{\Omega }}^{\circ }\ne \mathrm{\varnothing }$. In this paper, we study the nonexistence of the nonlinear wave equation

$${\mathrm{\partial }}_t^{2}u=\mathrm{\Delta }u+f(u)$$

on $G$. Under the appropriate conditions of initial values and nonlinear term, we prove that the solution for nonlinear wave equation blows up in a finite time. Furthermore, a numerical simulation is given to verify our results.