“different” quantum graph models. With this aim in mind, it would be natural to start by recalling briefly the standard quantum graphs, their description, properties, and numerous applications. In this volume, however, this would be clearly to bring owls to Athens[^2] and we refrain from doing that referring to the other articles in these proceedings, or to [@BCFK] as another rich bibliography source.

To motivate a need to look for alternative description of graph-like structures, let us observe that — despite its mathematical simplicity, beauty, and versatility — the standard quantum-graph model has also some drawbacks. In our opinion, the following two are the most important:

-   the presence of *ad hoc* parameters in the boundary conditions which describe how the wave functions are matched at the graphs vertices

-   the fact that particles are strictly confined to graph edges. While this is often a reasonable approximation when dealing, say, with electrons in semiconductor quantum wires, such a model neglects *quantum tunneling* which can play role once such wires are placed close to each other. One consequence is that in such a description, with the graph embedded in $\mathbb{R}^\nu$, spectral properties reflect the topology while the graph geometry enters only 