Topology and Stability of Trivalent Polyhedral Clusters

An algorithm to generate the molecular graphs of the trivalent clusters is presented. All the polyhedral trivalent cages with 4-16 vertices are enumerated, with particular attention being given to their symmetry, spectra and stability. The relative stability of isomers is not decided solely by their symmetry. A new kind of spectral phenomenon is reported. This is the occurrence of graphs having the same HOMO-LUMO gap, the same E and the same even-order moments. We refer to it as 'pseudo-isospectrality'.