  We study systems of Kuramoto oscillators, driven by one pacemaker, on
$d$-dimensional regular topologies like linear chains, rings, hypercubic
lattices and Cayley-trees. For the special cases of next-neighbor and
infinite-range interactions, we derive the analytical expressions for the
common frequency in the case of phase-locked motion and for the critical
frequency of the pacemaker, placed at an arbitrary position on the lattice, so
that above the critical frequency no phase-locked motion is possible. These
expressions depend on the number of oscillators, the type of coupling, the
coupling strength, and the range of interactions. In particular we show that
the mere change in topology from an open chain with free boundary conditions to
a ring induces synchronization for a certain range of pacemaker frequencies and
couplings, keeping the other parameters fixed. We also study numerically the
phase evolution above the critical eigenfrequency of the pacemaker for
arbitrary interaction ranges and find some interesting remnants to phase-locked
motion below the critical frequency.
