In this paper, we first prove some propositions of Sobolev spaces defined on a locally finite graph G = (V, E), which are fundamental when dealing with equations on graphs under the variational framework. Then we consider a nonlinear biharmonic equationUnder some suitable assumptions, we prove that for any λ > 1 and p > 2, the equation admits a ground state solution u λ . Moreover, we prove that as λ → +∞, the solutions u λ converge to a solution of the equationwhere Ω = {x ∈ V : a(x) = 0} is the potential well and ∂Ω denotes the the boundary of Ω.

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