  Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a
generalized Chern-Simons-Higgs equation $$ \Delta u=-\lambda e^{g(u)}\left(
e^{g(u)}-1\right)^2+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$ on $G$, where
$\lambda$ is a positive constant; $g(u)$ is the inverse function of
$u=f(\upsilon)=1+\upsilon-e^{\upsilon}$ on $(-\infty, 0]$; $N$ is a positive
integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and
$\delta_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there is
critical value $\lambda_c$ such that the generalized Chern-Simons-Higgs
equation has a solution if and only if $\lambda\geq \lambda_c$ . We also prove
the existence of solutions to the Chern-Simons-Higgs equation
  $$ \Delta u=\lambda e^{u}(e^{u}-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$
on $G$ when $\lambda$ takes the critical value $\lambda_c$ and this completes
the results of An Huang, Yong Lin and Shing-Tung Yau (Commun. Math. Phys. 377,
613-621 (2020)).
