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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 $L$.

A Schrödinger operator approach to higher spin XXZ systems on general graphs