S5 graphs as model systems for icosahedral Jahn–Teller problems

The degeneracy of the eigenvalues of the adjacency matrix of graphs may be broken by non-uniform changes of the edge weights. This symmetry breaking is the graph-theoretical equivalent of the molecular Jahn–Teller effect (Ceulemans et al. in Proc Roy Soc 468:971–989, 2012). It is investigated for three representative graphs, which all have the symmetric group on 5 elements, S5, as automorphism group: the complete graph K5, with 5 nodes, the Petersen graph, with 10 nodes, and an extended K5 graph with 20 nodes. The spectra of these graphs contain fourfold, fivefold, and sixfold degenerate manifolds, respectively, and provide model systems for the study of the Jahn–Teller effect in icosahedral molecules. The S5 symmetries of the distortion modes of the quintuplet in the Petersen graph yield a resolution of the product multiplicity in the corresponding H⊗(g +2h) icosahedral Jahn–Teller problem. In the extended Petersen graph with 20 nodes, a selection rule prevents the Jahn–Teller splitting of the sextuplet into two conjugate icosahedral triplets.