Bi-partite mode entanglement of bosonic condensates on tunneling graph

We study a set of $L$ spatial bosonic modes localized on a graph $\Gamma.$ The particles are allowed to tunnel from vertex to vertex by hopping along the edges of $\Gamma.$ We analyze how, in the exact many-body eigenstates of the system i.e., Bose-Einstein condensates over single-particle eigenfunctions, the bi-partite quantum entanglement of a lattice vertex with respect to the rest of the graph depends on the topology of $\Gamma.$