Abstract
The adjacency spectrum of a trivalent parent cage determines completely the spectra of a range of formally related frameworks. Analytical generating relations are given for vertex truncation, vertex insertion and ten combined decorations, leading to the prediction of π stability for several series of molecular species. Expanded cages in which bonds are replaced by even-numbered carbon chains (e.g. fulleren-ynes), are predicted to have closed π shells for a closed-shell parent, e.g. a leapfrog or truncated fullerene. Spheriphanes and related polycoronands, in which parent vertices are replaced by benzenoid rings, are predicted to have closed π shells irrespective of the π configuration of the parent.
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