  Let $ G=(V,E) $ be a connected finite graph and $ \Delta $ the usual graph
Laplacian. In this paper, we consider a generalized self-dual Chern-Simons
equation on the graph $G$ \begin{eqnarray}\label{one1}
\Delta{u}=-\lambda{e^{F(u)}[e^{F(u)}-1]^2}+4\pi\sum_{i=1}^{M}{\delta_{p_{j}}},
\end{eqnarray} where \begin{equation} F(u)=\left\{\begin{array}{l}
  \widetilde{F}(u), \ \quad u\leq0,
  0, \quad \quad \quad u>0,
  \end{array} \right. \end{equation} $ \widetilde{F}(u) $ satisfies $
u=1+{\widetilde {F}(u)}-e^{\widetilde {F}(u)} $, $ \lambda>0 $, $M$ is any
fixed positive integer, $ \delta_{p_{j}} $ is the Dirac delta mass at the
vertex $p_j$, and $p_1$, $p_2$, $\cdots$, $p_M$ are arbitrarily chosen distinct
vertices on the graph. We first prove that there is a critical value
${\lambda}_c$ such that if $\lambda \geq{\lambda}_c$, then the generalized
self-dual Chern-Simons equation has a solution $u_{\lambda}$. Applying the
existence result, we develop a new method to construct a solution of the
equation which is monotonic with respect to $\lambda$ when $\lambda
\geq{\lambda}_c$. Then we establish that there exist at least two solutions of
the equation via the variational method for $\lambda>{\lambda}_c$. Furthermore,
we give a fine estimate of the monotone solution which can be applied to other
related problems.
