We consider
1/d expansions for classical spin systems based on the vertex renormalized linked cluster expansion (LCE). The free multiplicities of the LCE graphs on a hypercubic lattice in an arbitrary dimension d are calculated using generating functions. The technique is applied to the Ising model and to a two-component classical lattice gas corresponding to an extended Hubbard model at half filling in the zero-bandwidth limit. We use a resummation of the LCE to generate
1/d expansions for the equation of state and for the critical temperature. The method, which is rather general and applicable to a wide range of models, proves convenient for calculating asymptotic power series expansions in
1/d. The vertex renormalized expansion is shown to break down at low temperature in higher order approximations, barring attempts to construct simple approximations that are both self-consistent and exact to some finite order in
1/d.The equations of motion for the general many-time causal Green's functions for a fermion system are iterated and are shown not to lead to unlinked graphs, which is a general proof of the linked cluster theorem. An explicit expression is obtained for the perturbation expansion of an arbitrary Green's functions which is applied to the one- and two-particle Green's functions. By connecting lines systematically in a set of diagrams obtained from the equations of motion, the usual topologically different linked graphs and rules are generated.