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

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/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).

MINIMAL NONSIMPLE SETS OF VOXELS IN BINARY IMAGES ON A FACE-CENTERED CUBIC GRID

One can prove that a specified parallel thinning algorithm always preserves the topology of the input binary image by verifying that no iteration of that algorithm can ever delete a minimal non-simple ("MNS") set of 1's of an image. For binary images on a 3D face-centered cubic ("FCC") grid, we determine which sets of voxels can be MNS, and also determine which of those sets can be MNS without being a component of the 1's. These two problems are complicated by the fact that there are (at least) three reasonable ways of defining connectedness for sets of 1's and 0's in a binary image on an FCC grid, since one can: (a) use 18-connectedness for sets of 1's and 12-connectedness for sets of 0's; (b) use 12-connectedness both for sets of 1's and for sets of 0's; (c) use 12-connectedness for sets of 1's and 18-connectedness for sets of 0's. We solve the two problems in all three cases. The analogous problems for binary images on Cartesian grids were first solved by Ronse (in the 2D case) and Ma (in the 3D case). However, our treatment of simple 1's and MNS sets is rather different from theirs, in that it is based on the attachment sets of 1's in binary images. This concept was introduced in an earlier paper [T. Y. Kong, "On topology preservation in 2-D and 3-D thinning," Int. J. Pattern Recognition and Artificial Intelligence9 (1995) 813–844] and we use the same general approach to MNS sets as was used there. The voxels of an FCC grid are rhombic dodecahedra, which are rather more difficult to visualize and draw than the cubical voxels of a 3D Cartesian grid. An advantage of working with attachment sets is that such sets can be shown in a planar Schlegel diagram of a voxel, which is easy to draw.