Spectral clustering of combinatorial fullerene isomers based on their facet graph structure

• Published in: Journal of mathematical chemistry Vol. 59; no. 1; pp. 264 - 288
• Main Authors: Bille, Artur, Buchstaber, Victor, Spodarev, Evgeny
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.01.2021; Springer Nature B.V
• Subjects: Algorithms; Buckminsterfullerene; Chemistry; Chemistry and Materials Science; Classification; Cluster analysis; Clustering; Combinatorial analysis; Fullerenes; Graph theory; Isomers; Math. Applications in Chemistry; Mathematical analysis; Original Paper; Physical Chemistry; Programming languages; Quantum chemistry; Software; Stability criteria; Theoretical and Computational Chemistry; Fullerene; 92E10; Combinatorial isomer; Convex polytope; Eigenvalue; 05C90; Adjacency matrix; Primary: 52B12; Secondary: 05C10; Facet spectrum; Dual graph
• ISSN: 0259-9791; 1572-8897
• DOI: 10.1007/s10910-020-01193-4

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 C 40 , C 60 and C 80 . 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 C 60 and give an example of two different cospectral isomers of C 44 . 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/mawi-stochastik/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).