Spectral analysis of graphs by cyclic automorphism subgroups

The theory of spectral decomposition modulo subgroups of the graph automorphism group is extended to cyclic configurations of arbitrary rotational order. By regarding graphs with cyclic automorphisms as composite polymers of relatively simple monomeric structural units, it is shown that the spectrum of eigenvalues of many prominent molecular and nonmolecular families devolves to consideration of a single monomer-derived reduction network. As the only parameter associated with this network is the set of simple circuit eigenvalues, a direct connection is forged between the spectrum of a circuit and the spectrum of any cyclic array of the same periodicity.In addition to simplifying determination of individual graph spectra, the role of the automorphism reduction network in organizing and uniting disparate aspects of spectral theory is stressed. Systems sharing a subspectrum of identical eigenvalues are readily recognized from the graphic nature of networks. As previously, symbolic and notational devices are devised for greatest economy in the spectral analysis.Part 4 of the series Unified Theory of Graph Spectral Reduction Networks