Dendrimer eigen-characteristics

Eigen-Persistence in Graphs

“…The much studied [40][41][42][43][44][45] eigen-solution to the "Bethe tree" manifests high degeneracies. The more general related class of dendrimers also has been much studied, both experimentally [46][47][48][49][50][51] & theoretically [52][53][54][55][56][57] -with the theory revealing high degeneracies for the associated adjacency matrices, though often the associated high degeneracies are not noted. Another special topic concerns eigen-features of a subgraph "persisting" up to the parent graph (containing the subgraph), such work appearing (seemingly first) with George Hall [58][59][60] (and others [61][62][63]), framing his ideas in terms of "local" reflection symmetries.…”

“…Indeed such connections were early on recognized in "free radicals", first identified by Gomberg over a century ago, and reviewed in [74], though many more similarly connected radicaloid, or poly-radicaloid, species [49][50][51][75][76][77][78]. Their potential for high degeneracies has more recently been emphasized [57] (in the context of "local symmetries"). In fact, the simplest so connected case has been much studied in the case of the so-called Bethe tree [43][44][45]79], (often dealing with the Ising model or lattice gas model, rather than the Hückel model) and also a more general class of dendrimers [57].…”

“…Their potential for high degeneracies has more recently been emphasized [57] (in the context of "local symmetries"). In fact, the simplest so connected case has been much studied in the case of the so-called Bethe tree [43][44][45]79], (often dealing with the Ising model or lattice gas model, rather than the Hückel model) and also a more general class of dendrimers [57]. In our next theorem concerning such connectors, it is useful for a given set of numerical parameters {x 1 , x 2 , .…”