  We establish that the spin-3/2 AKLT model on the honeycomb has a nonzero
spectral gap. We use the relation between the anticommutator of two projectors
and their sum, and apply it to related AKLT projectors that occupy plaquettes
or other extended regions. We analytically reduce the complexity in the
resulting eigenvalue problem and use a Lanczos numerical method to show that
the required inequality for the nonzero spectral gap holds. This approach is
also successfuly applied to several other spin-3/2 AKLT models on degree-3
semiregular tilings, such as the square octagon, star and cross lattices, where
the complexity is low enough that exact diagonalization can be used instead of
the Lanczos method. In addition, we also close the previously open cases in the
singly decorated honeycomb and square lattices.
