Spectral clustering of combinatorial fullerene isomers based on their facet graph structure
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Spectral clustering of combinatorial fullerene isomers based on their facet graph structure Artur Bille1,2 · Victor Buchstaber2,3 · Evgeny Spodarev1 Received: 22 January 2020 / Accepted: 28 October 2020 © The Author(s) 2020
Abstract After Curl, Kroto and Smalley were awarded 1996 the Nobel Prize in chemistry, fullerenes have been subject of much research. One part of that research is the prediction of a fullerene’s stability using topological descriptors. It was mainly done by considering the distribution of the twelve pentagonal facets on its surface, calculations mostly were performed on all isomers of C40, C60 and C80. This paper suggests a novel method for the classification of combinatorial fullerene isomers using spectral graph theory. The classification presupposes an invariant scheme for the facets based on the Schlegel diagram. The main idea is to find clusters of isomers by analyzing their graph structure of hexagonal facets only. We also show that our classification scheme can serve as a formal stability criterion, which became evident from a comparison of our results with recent quantum chemical calculations (Sure et al. in Phys Chem Chem Phys 19:14296–14305, 2017). We apply our method to classify all isomers of C60 and give an example of two different cospectral isomers of C44. Calculations are done with our own Python scripts available at (Bille et al. in Fullerene database and classification software, https://www.uni-ulm.de/mawi/mawistochastik/forschung/fullerene-database/, 2020). The only input for our algorithm is the vector of positions of pentagons in the facet spiral. These vectors and Schlegel diagrams are generated with the software package Fullerene (Schwerdtfeger et al. in J Comput Chem 34:1508–1526, 2013). Keywords Convex polytope · Fullerene · Combinatorial isomer · Facet spectrum · C60 · Dual graph · Eigenvalue · Adjacency matrix Mathematics Subject Classification Primary: 52B12 · Secondary: 05C10 · 05C90 · 92E10
* Artur Bille artur.bille@uni‑ulm.de Extended author information available on the last page of the article
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Journal of Mathematical Chemistry … This spiritual experience, this discovery of what Nature has in store for us with carbon, is still ongoing. Richard E. Smalley Discovering the fullerenes, Nobel lecture, Dec. 7, 1996.
1 Introduction In this paper, we consider a fullerene Cn as a convex polytope modeling a closed three-dimensional carbon-cage with n atoms, cf. [1]. Each vertex is connected to exactly three other vertices, such that the facets are pentagons and hexagons only. Using the classical Euler relation and Eberhard’s theorem [19, Section 13.3] for simple three-dimensional convex polytopes one can conclude that the number of pentagonal facets is always equal to 12 and n is even. In this case, the number of hexagons is m6 = n2 − 10 , cf. [17]. Theoretically, Cn exists for n = 20 and all even n ≥ 24 , see [1]. In the sequel we call such a n feasible. However, up to now nly a few of them (some isomers of Cn with n = 60, 70, 76, 78
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