  The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger model are
paradigmatic examples of non-Hermitian systems that host non-trivial boundary
phenomena. In this work, we use recently developed graph-theoretical tools to
design systems whose isospectral reduction -- akin to an effective Hamiltonian
-- has the form of either of these two models. In the reduced version, the
couplings and on-site potentials become energy-dependent. We show that this
leads to interesting phenomena such as an energy-dependent non-Hermitian skin
effect, where eigenstates can simultaneously localize on either ends of the
systems, with different localization lengths. Moreover, we predict the
existence of various topological edge states, pinned at non-zero energies, with
different exponential envelopes, depending on their energy. Overall, our work
sheds new light on the nature of topological phases and the non-Hermitian skin
effect in one-dimensional systems.
