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 \(n \rightarrow \infty \). The theoretical predictions are confirmed experimentally using \(n=2\) 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 \(2n \times 2n \) scattering matrices \({\hat{S}}\) of the systems to 2n diagonal elements, while the old measures of isoscattering require all \((2n)^2\) 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.

我们确定并研究具有n个单元和2 n个无限外部引线的级联量子图的等散字符串。我们对设计大型图和网络的原理有深刻的了解，对于这些图和网络，\（n \ rightarrow \ infty \）保留了等散射特性。理论预测是使用\（n = 2 \）通过实验确定的单元，四引线微波网络。在实验和数学方法中，我们的工作超出了先前的结果，证明了使用跟踪函数可以解决直到现在为止尚未解决的问题：开放复杂图和具有许多外部引线的网络的散射特性是否唯一地与其形状相关。跟踪函数的应用将系统的\（2n \ times 2n \）散射矩阵\ {{\ hat {S}} \\}所需的条目数减少到2 n个对角元素，而等距的旧度量要求所有\（（（2n）^ 2 \）条目。研究的问题将马克·卡克（Mark Kac）的一个著名问题概括为“有人能听到鼓的形状吗？”，最初是在等谱无耗散系统的情况下提出的，是无限个开放图和网络字符串的情况。