  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 the rest of the graph 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 $\tau^*$
which leads to the maximum ground-state entanglement. Whereas in most of the
cases one finds the intuitively expected result $\tau^*=\infty$ we show that it
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 cross-over between
insulating and superfluid ground states
