  Let $G=(V,E)$ be a connected infinite and locally finite weighted graph,
$\Delta_p$ be the $p$-th discrete graph Laplacian. In this paper, we consider
the $p$-th Yamabe type equation $$-\Delta_pu+h|u|^{p-2}u=gu^{\alpha-1}$$ on
$G$, where $h$ and $g$ are known, $2<\alpha\leq p$. The prototype of this
equation comes from the smooth Yamabe equation on an open manifold. We prove
that the above equation has at least one positive solution on $G$.
