  We identify a fundamental challenge for non-perturbative linked cluster
expansions (NLCEs) resulting from the reduced symmetry on graphs, most
importantly the breaking of translational symmetry, when targeting the
properties of excited states. A generalized notion of cluster additivity is
introduced, which is used to formulate an optimized scheme of graph-based
continuous unitary transformations (gCUTs) allowing to solve and to physically
understand this fundamental challenge. Most importantly, it demands to go
beyond the paradigm of using the exact eigenvectors on graphs.
