Families of isospectral and isoscattering quantum graphs

A concept of germ graphs and the <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mi>M</a:mi></a:mrow></a:math>-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>M</b:mi></b:mrow></b:math>-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.