Note on the role of symmetry in scattering from isospectral graphs and drums

We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in [1, 2]. It was shown in [3] that scattering matrices of such graphs have the same spectrum and polar structure, provided that infinite leads are attached in a way which preserves the symmetry of isospectral construction. In the current paper we compare this result with the conjecture put forward by Okada et al. [4] that the pole distribution of scattering matrices in the exterior of isospectral domains in R^2 are different.