According to the bridged annulene model, macrocyclic aromaticity of a porphyrinoid species can be attributed to the annulene-like main macrocyclic conjugation pathway (MMCP). Macrocyclic aromaticity, however, is given theoretically as a sum of contributions from all macrocyclic circuits. We found that the aromaticity due to each macrocyclic circuit is determined formally but broadly by Hückel's [4n + 2] rule of aromaticity. Nitrogen atoms in the pyrrolic rings effectively suppress the variation in the number of π electrons staying along each macrocyclic circuit. As a result, all or most macrocyclic circuits in oligopyrrolic macrocycles are made aromatic (or antiaromaitc) in phase with the MMCP. Thus, the MMCP is not a determinant of macrocyclic aromaticity but can be regarded as a good indicator of this quantity. This is why the bridged annulene model appears to hold for many porphyrins.

The sequential line plot of topological resonance energy (TRE) against the number of π electrons (N(π)) for any polycyclic aromatic hydrocarbon (PAH) is very similar with the same number of extrema to that for benzene. Thus, global aromaticity of a PAH molecular ion strongly reflects that of a benzene molecular ion. Likewise, the N(π) dependence of TRE for any polycyclic π system formed by fusion of two or more rings of the same size reflects that for a monocyclic species of the same ring size. In general, TREs for such polycyclic π systems and their molecular ions can be interpreted consistently by reference to those for neutral and charged monocyclic species of the same ring size.

The global and macrocyclic aromaticity of porphyrinoids was characterized using our graph theory of aromaticity. The sequential line plots of topological resonance energy (TRE) against the number of π-electrons (N(π)) for different porphyrinoids are similar with four major extrema to those for five-membered heterocycles. This supports the view that five-membered rings are the main origin of global aromaticity in porphyrinoids. Macrocyclic circuits contribute significantly to macrocyclic π-circulation but modestly to global aromaticity. Macrocyclic aromaticity/antiaromaticity in oligopyrrolic macrocycles can be predicted by formally applying Hückel's [4n + 2] rule to an annulene-like main macrocyclic conjugation pathway (MMCP). This bridged annulene model can be justified by examining the contribution of individual macrocyclic circuits to macrocyclic aromaticity. A Hückel-like rule of macrocyclic aromaticity was found for porphyrinoid species.

Many fullerenes that violate the isolated pentagon rule (IPR) form stable metallofullerenes. In general, a fullerene cage is kinetically stabilized by acquiring a given number of electrons. Kinetic stability of negatively charged non-IPR fullerenes, including the recently isolated endohedral metallofullerene with a heptagonal face, was rationalized in terms of bond resonance energy (BRE). Interestingly, molecular anions of conventional fullerenes found in most isolated metallofullerenes are kinetically stable with large positive BREs for all CC bonds. As we pointed out in 1993, the IPR does not apply to charged fullerenes because π-bonds shared by two five-membered rings are aromatized to varying extents.

The energy of the lowest π molecular orbital in a planar boron cluster can be estimated from the connectivity of constituent boron atoms, or the mean valency of the boron atoms. Low-energy π molecular orbitals were then predicted to occur in all realistic planar polycyclic boron clusters. In fact, all such boron clusters studied have one or more π molecular orbitals with two or more π electrons. They are aromatic with positive topological resonance energies. π Conjugation and aromaticity must be totally or partially responsible for the planarity of low-energy boron clusters.

All isomers for sodium clusters up to Na9 were investigated by graph theory where a vertex and an edge correspond to atom and bond between adjacent atoms, respectively. All isomers, more than 10 million, for Na10 were listed. Structural features for sodium clusters up to Na9 were elucidated by using graph theory. Furthermore, geometrically forbidden graphs were excluded by using molecular mechanics.

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