Isoscattering strings of concatenating graphs and networks.

We identify and investigate isoscattering strings of concatenating quantum graphs possessing n units and 2n infinite external leads. We give an insight into the principles of designing large graphs and networks for which the isoscattering properties are preserved for [Formula: see text]. The theoretical predictions are confirmed experimentally using [Formula: see text] units, four-leads microwave networks. In an experimental and mathematical approach our work goes beyond prior results by demonstrating that using a trace function one can address the unsettled until now problem of whether scattering properties of open complex graphs and networks with many external leads are uniquely connected to their shapes. The application of the trace function reduces the number of required entries to the [Formula: see text] scattering matrices [Formula: see text] of the systems to 2n diagonal elements, while the old measures of isoscattering require all [Formula: see text] entries. The studied problem generalizes a famous question of Mark Kac "Can one hear the shape of a drum?", originally posed in the case of isospectral dissipationless systems, to the case of infinite strings of open graphs and networks.