Graph-theoretic and chemical properties of anionic fullerenes
Date
2025
Authors
Slobodin, Aaron
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Abstract
A fullerene is an all-carbon molecule with a polyhedral structure where each atom is bonded to three others and each face is either a pentagon or a hexagon. Fullerenes correspond to cubic planar graphs whose faces have sizes 5 or 6. The p-anionic Clar number of a fullerene G is equal to p + h, where h is maximized over all choices of p + h independent faces (p pentagons and h hexagons) the deletion of whose vertices results in a graph that admits a perfect matching. This definition is motivated by the chemical observation that pentagonal rings can accommodate an extra electron, so that the pentagons of a fullerene p-anion compete with the hexagons to host ‘Clar sextets’ of six electrons, and pentagons preferentially acquire the excess electrons of the anion.
Tight upper bounds are established for the p-anionic Clar number of fullerenes for p > 0. The upper bounds are derived via graph theoretic arguments and new results on minimal cyclic-k-edge cuts in IPR fullerenes (fullerenes that have all pentagons pairwise disjoint). These bounds are shown to be tight by infinite families of fullerenes that achieve them.
A fullerene G is said to be k-anionic-resonant if the deletion of the vertices of any k independent pentagons in G results in a graph that admits a perfect matching. We prove necessary conditions for a fullerene to be 2-anionic-resonant and provide structural properties of fullerenes (should they exist) for which our necessary conditions are not sufficient.
Chemical aspects of the anionic Clar model and its utility are also explored in this work. These include the central question of the general comparison between predictions of the anionic Clar model and qualitative molecular-orbital theory for relative stability of charged fullerenes. H\"{u}ckel accounts of stability, including the chemical concepts of total pi energy, resonance energy, HOMO-LUMO gap and Coulson bond order, and the unified perspective offered by CSI (the recently defined charge stabilization index) are used in this analysis.
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Keywords
Fullerene, Anionic-Clar, Clar, Chemical graph theory, Clar number, Anionic-Clar number