Eigenvalue spectra of leapfrog polyhedra

Huckel magic-number rules for fullerenes, open nanotubes and polyhex tori, though superficially different, are shown to have a common explanation in eigenvalue theorems for the spectra of the trivalent polyhedra produced by leapfrogging (omnicapping then dualising) trivalent polyhedral parents. A leapfrog polyhedron L, considered as a neutral all-carbon pi system, will have a properly closed, meta-closed or open shell, accordingly as the trivalent polyhedral parent P has at least one face of a size not divisible by three, or has at least one face of a size not divisible by six, or is an alternant with all face sizes divisible by six. Non-alternant L with all face sizes divisible by six are of genus at least two and have meta-closed shells. Thus, leapfrog fullerenes have closed shells, leapfrog polyhex tori have open shells and leapfrog infinite nanotubes are metallic in the Huckel model.