A General Method for Extending Isoscattering Non-Isospectral Quantum Graphs

The present paper sets out a general method for extending the isoscattering, non-isospectral quantum graphs. The methodology is illustrated by considering two graphs, denoted by ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$, which have the same total metric length but different topologies. It is demonstrated that, despite the fact that their scattering matrices are identical, their eigenvalues, denoted by $k_n$, and Euler characteristics differ from one another. The next step in the process is to extend both graphs by affixing the same star graph to each of them. Utilising the Titchmarsh-Weyl M-function formalism, it is proven that the newly constructed, extended graphs are also isoscattering and non-isospectral.