  Assume $\alpha\geq p>1$. Consider the following $p$-th Yamabe equation on a
connected finite graph $G$: $$\Delta_p\varphi+h\varphi^{p-1}=\lambda
f\varphi^{\alpha-1},$$ where $\Delta_p$ is the discrete $p$-Laplacian, $h$ and
$f>0$ are fixed real functions defined on all vertices. We show that the above
equation always has a positive solution $\varphi$ for some constant
$\lambda\in\mathds{R}$.
