  We compare a star and a ring network of interacting spins in terms of the
entanglement they can provide between the nearest and the next to nearest
neighbor spins in the ground state. We then investigate whether this
entanglement can be optimized by allowing the system to interact through a
weighted combination of the star and the ring geometries. We find that such a
weighted combination is indeed optimal in certain circumstances for providing
the highest entanglement between two chosen spins. The entanglement shows jumps
and counterintuitive behavior as the relative weighting of the star and the
ring interactions is varied. We give an exact mathematical explanation of the
behavior for a five qubit system (four spins in a ring and a central spin) and
an intuitive explanation for larger systems. For the case of four spins in a
ring plus a central spin, we demonstrate how a four qubit GHZ state can be
generated as a simple derivative of the ground state. Our calculations also
demonstrate that some of the multi-particle entangled states derivable from the
ground state of a star network are sufficiently robust to the presence of
nearest neighbor ring interactions.
