  The principles of density-functional theory are studied for finite lattice
systems represented by graphs. Surprisingly, the fundamental Hohenberg-Kohn
theorem is found void in general, while many insights into the topological
structure of the density-potential mapping can be won. We give precise
conditions for a ground state to be uniquely v-representable and are able to
prove that this property holds for almost all densities. A set of examples
illustrates the theory and demonstrates the non-convexity of the pure-state
constrained-search functional.
