  Let $G=(V,E)$ be a connected finite graph. We study the existence of
solutions for the following generalized Chern-Simons equation on $G$
\begin{equation*}
  \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi
\sum_{s=1}^{N} \delta_{p_{s}} \quad , \end{equation*} where $\lambda>0$,
$\delta_{p_{s}}$ is the Dirac mass at the vetex $p_s$, and $p_1, p_2,\dots,
p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there
exists a critial value $\hat{\lambda}$ such that when $\lambda >
\hat{\lambda}$, the generalized Chern-Simons equation has at least two
solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons
equation has a solution, and when $\lambda < \hat\lambda$, the generalized
Chern-Simons equation has no solution.
