Eigen-Persistence in Graphs

Abstract

The idea of eigen-solution "persistence" from a suitable subgraph into a parent (molecular) graph G are formalized, in different ways for different cases. Most of it is based on the identification of suitably separated subgraphs sharing common eigenvalues, such that the subgraphs are all isomorphic. We recall Hall’s embedding method to identify adjacency-matrix eigen-solutions of a graph G as persistent from suitable disjoint subgraphs. General rigorous results are obtained for special embeddings, including cases where the subgraphs need not be isomorphic, but rather only share common eigenvalues. The question of accidental degeneracies is addressed, as well as the role of some sort of "local symmetries". Especially the mode of interconnection amongst a suitable family of subgraphs is addressed.