Graph rules for the linked cluster expansion of the Legendre effective action

Graph rules for the linked cluster expansion of the Legendre effective action $\mathrm{\Gamma }[\varphi ]$ are derived and proven in $D\ge 2$ Euclidean dimensions. A key aspect is the weight assigned to articulation vertices which is itself shown to be computable from labeled tree graphs. The hopping interaction is allowed to be long ranged and scale dependent, thereby producing an in principle exact solution of $\mathrm{\Gamma }[\varphi ]$'s functional renormalization group equation.