Entanglement of free fermions on Hamming graphs

Free fermions on Hamming graphs H(d,q) are considered and the entanglement entropy for two types of subsystems is computed. For subsets vertices that form subgraphs, an analytical expression obtained. corresponding to a neighborhood, i.e. set sites at fixed distance from reference vertex, decomposition in irreducible submodules Terwilliger algebra also yields closed formula entropy. Finally, made out multiple neighborhoods, it shown how construct block-tridiagonal operator which commutes with Hamiltonian. It identified as BC-Gaudin magnet Hamiltonian magnetic field diagonalized by modified algebraic Bethe ansatz.