1. 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: 176; Downloads: 2
 Full text (843,13 KB) |
2. 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: 220; Downloads: 1
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3. Selected topics on Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2024, original scientific article Keywords: graph distance, Wiener index, average distance, topological index, molecular descriptor, chemical graph theory
Published in RUP: 26.05.2025; Views: 801; Downloads: 7
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4. Mathematical aspects of Wiener indexMartin Knor, Riste Škrekovski, Aleksandra Tepeh, 2016, original scientific article Abstract: The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
Keywords: Wiener index, total distance, topological index, molecular descriptor, chemical graph theory
Published in RUP: 03.01.2022; Views: 3944; Downloads: 49
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5. Edge-contributions of some topological indices and arboreality of molecular graphsTomaž Pisanski, Janez Žerovnik, 2009, original scientific article Abstract: Some graph invariants can be computed by summing certain values, called edge-contributions over all edges of graphs. In this note we use edge-contributions to study relationships among three graph invariants, also known as topological indices in mathematical chemistry: Wiener index, Szeged index and recently introduced revised Szeged index. We also use the quotient between the Wiener index and the revised Szeged index to study tree-likeness of graphs.
Keywords: mathematical chemistry, chemical graph theory, topological index, revised Szeged index
Published in RUP: 30.12.2021; Views: 3028; Downloads: 29
Full text (158,93 KB) |
