I will review the recent achievements in simulation of quantum graphs by microwave networks. I will show that bidirectional microwave networks can simulate quantum graphs with time reversal symmetry whereas the directed microwave networks consisting of microwave circulators can be used to simulate quantum graphs for which the time reversal symmetry is broken.

I will discuss a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such as quantum graphs and microwave networks. It addresses an important mathematical problem whether scattering properties of such systems are uniquely connected to their shapes?

I will consider an important problem of incomplete spectra and discuss how to use short- and long-range correlation functions to cope with this situation.

Finally, I will demonstrate that there exist graphs which do not obey the Weyl’s law N(R) = LR/π + O(1), where O(1) is a function which for R going to infinity is bounded by a constant.

Acknowledgements: This work was supported in part by the National Science Centre, Poland, Grant No. 2016/23/B/ST2/03979 and the Operational Programme Innovative Economy Grant No. POIG.01.01.02-00-008/08

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