  We consider the spin-$J$ XXZ-Hamiltonian on general graphs $\mathcal{G}$ and
show its equivalence to a direct sum of discrete many-particle Schr\"odinger
type operators on what we call "$N$-particle graphs with maximal local
occupation number $M$", where the kinetic term is described by a weighted
Laplacian. Generalizing previous results for the spin-$1/2$ case, we give
sufficient conditions for the existence of spectral gaps above the low-lying
droplet band when the underlying graph $\mathcal{G}$ is (i) the chain and (ii)
a strip of width $L$.
