  We concern in this paper the graph Kazdan-Warner equation \begin{equation*}
\Delta f=g-he^f \end{equation*} on an infinite graph, the prototype of which
comes from the smooth Kazdan-Warner equation on an open manifold. Different
from the variational methods often used in the finite graph case, we use a heat
flow method to study the graph Kazdan-Warner equation. We prove the existence
of a solution to the graph Kazdan-Warner equation under the assumption that
$h\leq0$ and some other integrability conditions or constrictions about the
underlying infinite graphs.
