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

The recently developed concept of germ graphs P. Kurasov {it et al.}, Phys. Rev. Research {bf 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 Γ₃ and Γ₄ are verified as non-isospectral using their Euler characteristics, while their isoscattering properties are investigated by calculating the one-port scattering matrices numerically.