Strong-coupling expansion for the effective potential on a lattice

This is the first of three papers on the strong-coupling expansion of the renormalized effective potential in $g{\phi }^4$ quantum field theory in $d$-dimensional Euclidean space-time. In this paper we show how to express the effective potential in terms of one-particle-irreducible strong-coupling graphs. We describe two methods for obtaining analytic expressions for the vertices of these graphs. We also present an algebraic technique for evaluating the graphs on a Euclidean space-time lattice in arbitrary dimension $d$.