Let \(G=(V, E)\) be a locally finite graph. \(\Omega \) is a bounded domain of V with \(\Omega ^{\circ } \ne \emptyset \) . In this paper, we investigate the following nonlinear biharmonic equation: \(\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2} u-\textrm{div}(a(x) \nabla u )+(\lambda b(x)+1)u = c(x) |u|^{p-2}u\log {u^2}, \quad \text {in} \Omega ^{\circ },\\ u = 0, \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \text {on} \partial \Omega , \end{array} \right. \end{aligned}\) where \(\lambda >0\) and \(p>2\) are constants. Under some suitable assumptions, we establish the existence of a ground state solution to the equation by employing both the Brouwer degree theory and mountain-pass theorem. Additionally, we analyze the relationship between these two methods and explore the existence of an additional solution. Moreover, we investigate the convergence behavior of solutions as \(p \rightarrow 2\) and \(\lambda \rightarrow \infty \) . Finally, we conduct a simple numerical simulation to verify our theoretical results.
