Spectral gaps versus embedded eigenvalues from local graph structures

Determine whether attaching graph decorations or introducing internal vertex structures necessarily opens spectral gaps, thereby producing isolated eigenvalues rather than embedded eigenvalues, in the associated quantum-graph Laplacians.

Background

The spectrum of the tiling-graph Laplacians may contain exceptional energies in the discrete resonance set. Such energies can correspond either to isolated eigenvalues or to eigenvalues embedded in the rest of the spectrum.

The paper observes that local structures such as graph decorations and internal vertices can produce spectral gaps in some settings, but it does not establish whether these structures necessarily produce gaps in the general class under consideration. The unresolved issue is whether the resulting exceptional eigenvalues must be isolated or may remain embedded.