Zoology of non-Hermitian spectra and their graph topology

We uncover the very rich graph topology of generic bounded non-Hermitian spectra, distinct from the topology of conventional band invariants and complex spectral winding. The graph configurations of complex spectra are characterized by the algebraic structures of their corresponding energy dispersions, drawing new intimate links between combinatorial graph theory, algebraic geometry, and non-Hermitian band topology. Spectral graphs conformally related belong to the same equivalence class, and are characterized by emergent symmetries not necessarily present in the physical Hamiltonian. The simplest class encompasses well-known examples such as the Hatano-Nelson and non-Hermitian SSH models, while more sophisticated classes represent novel multi-component models with interesting spectral graphs resembling stars, flowers, and insects. With recent rapid advancements in metamaterials, ultracold atomic lattices, and quantum circuits, it is now feasible to not only experimentally realize such esoteric spectra but also investigate the non-Hermitian flat bands and anomalous responses straddling transitions between different spectral graph topologies.?