1. Clar numbers of leapfrog fullerenesJack E. Graver, Elizabeth J. Hartung, 2025 Abstract: A fullerene is a 3-regular plane graph with only hexagonal and pentagonal faces. The Fries number of a fullerene G, F(G), is the maximum number of benzene rings over all possible Kekulé structures for G. The Clar number of G, C(G), is the maximum number of independent benzene rings possible over all possible Kekulé structures for G. In this paper, we show that for leapfrog fullerenes, any set of faces attaining the Clar number is a subset of faces attaining the Fries number. This property is false for fullerenes in general (as shown in paper from E. J. Hartung in 2014). We then show that if L(G) is the leapfrog of a fullerene G, then the Clar number of L(G) is equal to the vertex independence number of G.
Keywords: chemical graph theory, fullerenes, leapfrog fullerenes, Clar number, Fries number, Kekulé structure, perfect matching
Published in RUP: 27.07.2026; Views: 176; Downloads: 3
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2. The Clar-Fries mysteryJoshua Fenton, Jack Edward Graver, Elizabeth J. Hartung, 2026, original scientific article Abstract: A fullerene is a 3-regular plane graph whose faces are hexagons and pentagons. The Fries number of a fullerene is the largest number of benzene rings over all possible Kekulé structures while the Clar number of a fullerene is the largest number of independent benzene rings over all possible Kekulé structures. One question was whether it is always the case that a largest set of independent benzene rings, giving the Clar number, must be a subset of some largest set of benzene rings giving the Fries number. This question is still open for benzenoids, but was answered negatively for fullerenes, with the first counterexample given in paper from E. J. Hartung in 2014. In 2016 in paper from J. E. Graver and E. J. Hartung, the authors constructed a family of fullerenes with the property that the set of benzene rings giving the Clar number was actually disjoint from the set of benzene rings giving the Fries number. Fowler and Myrvold then developed a program for computing the Clar number directly and discovered a significant number of fullerenes in which the Clar sets were not a subset of any Fries set and most of these were not of the type constructed in paper from J. E. Graver and E. J. Hartung in 2016. Exactly why this occurs is somewhat of a mystery. In her Ph.D. thesis, Hartung developed the concept of Clar chains to describe the Kekulé structure giving the Clar sets; in his Ph.D. thesis, Fenton developed the concept of a Fries mesh to describe the Kekulé structure giving the Fries sets. Comparing these two constructions enables us to shed some light on this mystery.
Keywords: fullerene, Clar number, Fries number
Published in RUP: 23.03.2026; Views: 403; Downloads: 32
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3. Clar and Fries structures for fullerenesPatrick W. Fowler, Wendy Myrvold, Rebecca L. Vandenberg, Elizabeth J. Hartung, Jack E. Graver, 2026, original scientific article Abstract: Fries and Clar numbers are qualitative indicators of stability in conjugated π systems. For a given Kekulé structure, call any hexagon that contains three double bonds benzenoid. The Fries number is the maximum number of benzenoid hexagons, whereas the Clar number is the maximum number of independent benzenoid hexagons, in each case taken over all Kekulé structures. A Kekulé structure that realises the Fries (Clar) number is a Fries (Clar) structure. For benzenoids, it is not known whether every Fries structure is also a Clar structure. For fullerenes C_n, it is known that some Clar structures in large examples correspond to no Fries structure. We show that Fries structures that are not Clar occur early: examples where some Fries structure is not Clar start at C_34, and examples where no Fries structure is Clar start at C_48. Hence, it is unsafe to use fullerene Fries structures as routes to Clar number. However, Fries structures often describe the neutral fullerene better than a Clar structure, e.g. in rationalising bond lengths in the experimental isomer of C_60. Conversely, an extension of Clar sextet theory suggests the notion of anionic Clar number for fullerene anions, where both pentagons and hexagons may support sextets.
Keywords: chemical graph theory, fullerenes, benzenoids, Clar, Fries, Kekule, perfect matching
Published in RUP: 22.12.2025; Views: 696; Downloads: 3
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4. Tight upper bounds for the p-anionic Clar number of fullerenesAaron Slobodin, Wendy Myrvold, Gary MacGillivray, Patrick W. Fowler, 2026, original scientific article Abstract: A fullerene is an all-carbon molecule with a polyhedral structure where each atom is bonded to three other atoms and each face is either a pentagon or a hexagon. Fullerenes correspond to 3-regular planar graphs whose faces have sizes 5 or 6. The p-anionic Clar number C_(p)(G) of a fullerene G is equal to p + h, where h is maximized over all choices of p + h independent faces (exactly p pentagons and h hexagons) the deletion of whose vertices leave a graph with 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 with charge −p, compete with the hexagons to host ‘Clar sextets’ of six electrons, and pentagons will preferentially acquire the p 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 cutsets 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.
Keywords: chemical graph theory, anionic Clar number, fullerenes
Published in RUP: 21.12.2025; Views: 837; Downloads: 2
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