Instability of Bose-Einstein condensation on quantum graphs under repulsive perturbations

In this Note we investigate Bose-Einstein condensation in interacting quantum many-particle systems on graphs. We extend previous results obtained for particles on an interval and show that even arbitrarily small repulsive two-particle interactions destroy a condensate in the non-interacting Bose gas. Our results also cover singular two-particle interactions, such as the well-known Lieb-Lininger model, in the thermodynamic limit.