  We study the low-energy properties of Heisenberg antiferromagnetic spin-1/2
and spin-1 systems on various clusters made up of triangles. Some of the
clusters have a geometry similar to representative pieces of the Kagome
lattice, while others have the geometry of a sawtooth chain. While the ground
state always has the lowest possible spin (0 or 1/2), the nature of the
low-energy excitations depends on the geometry and the site spin. For the
Kagome clusters with spin-1 sites, the lowest excitations are gapped, with
singlet and triplet excitations having similar gaps. This is in contrast to
Kagome clusters with spin-1/2 sites where there are several low-energy singlet
excitations (possibly gapless in the thermodynamic limit), while triplet
excitations have a gap. For the sawtooth chain with spin-1 sites, the lowest
excitations are triplets with a gap; the gap to singlet excitations is about
twice the triplet gap.
