Construction of isoscattering non-isospectral quantum graphs using the concept of a germ graph

The recently developed concept of germ graphs (P. Kurasov et al., Phys. Rev. Research 7, L022071 (2025)) is employed to construct families of isocattering, non-isospectral graphs that exhibit dissipation. More specifically, a new pair of isoscattering non-isospectral graphs is constructed by connecting the existing isoscattering non-isospectral graphs, so called attachments, to the different contact vertices of the germ graph. The existing isoscattering non-isospectral graphs contained a two-loop graph labelled Γ A2 and a three-loop graph labelled Γ A3. These graphs have an equal total metric length, but different topologies. The resulting graphs Γ 3 and Γ 4 are verified as non-isospectral using their Euler characteristics, while their isoscattering properties are investigated by calculating the one-port scattering matrices numerically.