Ground-state entanglement in interacting bosonic graphs

We consider a collection of bosonic modes corresponding to the vertices of a graph Gamma. Quantum tunneling can occur only along the edges of Gamma and a local self-interaction term is present. Quantum entanglement of one vertex with respect to the rest of the graph ( mode entanglement) is analyzed in the ground state of the system as a function of the tunneling amplitude tau. The topology of Gamma plays a major role in determining the tunneling amplitude tau(max) that leads to the maximum value of the mode entanglement. Whereas in most of the cases one finds the intuitively expected result tau(max)=infinity, we show that there exists a family of graphs for which the optimal value of tau is pushed down to a finite value. We also show that, for complete graphs, our bi-partite entanglement provides useful insights in the analysis of the crossover between insulating and super fluid ground states.