Possible energy spectra of systems constructed from finite bethe lattices

Abstract

The exact polynomial equations related to the calculation for the spectra of networks with a local Cayley-tree-like structure are presented. We firstly obtain the adjacency matrix and the characteristic polynomial as well as the spectra for a k-regular graph or (finite) Bethe lattice with k=3 (the largest eigenvalue being around 5). After verification of k=3 results with those obtained previously we then predict the adjacency matrix and the characteristic polynomial as well as the spectra for a k-regular graph or (finite) Bethe lattice with k=9 (the largest eigenvalue being around 17) which are still challenging now and might be relevant to the Platonic tessellations of genus g≡4. Our approach could be applied to inclusion compounds formed between guest polymers and host systems.