Two classes of quantum spin systems that are gapped on any bounded-degree graph

We study translation-invariant quantum spin Hamiltonians on general graphs with non-commuting interactions either given by (i) a random rank-$1$ projection or (ii) Haar projectors. For (i), we prove that the Hamiltonian is gapped on any bounded-degree graph with high probability at large local dimension. For (ii), we obtain a gap for sufficiently large local dimension. Our results provide examples where the folklore belief that typical translation-invariant Hamiltonians are gapped can be proved, which extends a result by Bravyi and Gosset from 1D qubit chains with rank-$1$ interactions to general bounded-degree graphs. We derive the gaps by analytically verifying generalized Knabe-type finite-size criteria that apply to any bounded-degree graph.