  A lattice version of the widely used Functional Renormalization Group (FRG)
for the Legendre effective action is solved - in principle exactly - in terms
of graph rules for the linked cluster expansion. Conversely, the FRG induces
nonlinear flow equations governing suitable resummations of the graph
expansion. The (finite) radius of convergence determining criticality can then
be efficiently computed as the unstable manifold of a Gaussian or non-Gaussian
fixed point of the FRG flow. The correspondence is tested on the critical line
of the L\"{u}scher-Weisz solution of the $\phi^4_4$ theory and its $\phi_3^4$
counterpart.
