In this paper, we study the nonlinear Choquard equation $$\Delta^2u-\Delta u+(1+\lambda a(x))u=(R_{\alpha}*|u|^p)|u|^{p-2}u$$on a Cayley graph of a discrete group of polynomial growth with the homogeneous
dimension $N ≥ 1,$ where $α ∈ (0,N),$ $p>\frac{N+α}{N},$ $λ$ is a positive parameter and $R_α$ stands
for the Green’s function of the discrete fractional Laplacian, which has no singularity
at the origin but has same asymptotics as the Riesz potential at infinity. Under some
assumptions on $a(x),$ we establish the existence and asymptotic behavior of ground
state solutions for the nonlinear Choquard equation by the method of Nehari manifold.
