Quantum graphs attracted a lot of interest recently. There are several reasons for that. On one hand these models are useful as descriptions of various structures prepared from semiconductor wires, carbon nanotubes, and other substances. On the other hand they provide a tool to study properties of quantum dynamics in situations when the system has a nontrivial geometrical or topological structure. Quantum graph models contain typically free parameters related to coupling of the wave functions at the graph vertices, and to get full grasp of the theory one has to understand their physical meaning. A natural approach to this question is to investigate fat graphs, that is, systems of thin tubes built over the skeleton of a given graph, and to analyze its limit as the tube thickness tends to zero.