  Suppose that $G=(V, E)$ be a locally finite and connected graph with
symmetric weight and uniformly positive measure, where $V$ denotes the vertex
set and $E$ denotes the edge set. We are concered with the following problem $$
\begin{cases}-\Delta u+h u=f(x, u), & \text { in } \Omega, \\ u=0, & \text { on
} \partial \Omega,\end{cases} $$ on the graph, where $h: \Omega \rightarrow
\mathbb{R}$, $f: \Omega \times \mathbb{R} \rightarrow \mathbb{R}$ and $u:
\Omega \rightarrow \mathbb{R}$. When $ f $ and $ h $ satisfies certain
assumption conditions, we can ascertain the existence of one or two nontrivial
solutions on the graph.
