Let G = (V,E) be a locally finite graph. Given any O ∈ V. Denote the function ρ(x) as follows: \(\rho(x)=\begin{cases}\text{dist}(x,O),\;x\neq{O}\\1,\;\;\;\;\;\;\;\;\;\;\;\;\;\;x = O,\end{cases}\) where dist(x,O) represents the distance between x and O. In this paper, we investigate a perturbed nonlinear biharmonic equation \(\Delta^{2}u-\text{div}(a(x)\nabla{u})+b(x)u=\frac{f(x,u)}{\rho(x)^\beta}+\epsilon{h}(x)\) on G = (V,E), where a(x) and b(x) are two functions with a positive lower bound defined on G, β ≥ 0, ε > 0. If h and f satisfy certain assumptions, we prove that there exists some positive constant ε1 > 0 such that for all ε ∈ (0, ε1), the above equation has two distinct nontrivial positive solutions.
