Quantum graphs have become popular systems in many areas of Physics and Mathematics, e.g., in quantum chaos and spectral geometry. We first discuss many-particle quantum systems on graphs and introduce two types of singular two-particle interactions. The first type is associated with the vertices, and the second type is a realisation of delta-type contact interactions on the edges. In the second part of the talk we discuss under what circumstances Bose-Einstein condensation (BEC) occurs for non-interacting Bose gases on graphs. We also address the problem of BEC for interacting systems and prove the absence of BEC for repulsive hardcore interactions
