In this paper we propose quantum graphs as one-dimensional models with a
complex topology to study Bose-Einstein condensation and phase transitions in a
rigorous way. We fist investigate non-interacting many-particle systems on
quantum graphs and provide a complete classification of systems that exhibit
Bose-Einstein condensation. We then consider models of interacting particles
that can be regarded as a generalisation of the well-known Tonks-Girardeau gas.
Here our principal result is that no phase transitions occur in bosonic systems
with repulsive hardcore interactions, indicating an absence of Bose-Einstein
condensation
