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

TL;DR

This paper introduces a spectral graph theory-based method for classifying fullerene isomers using their facet graph structures, which correlates with stability and distinguishes isomers, demonstrated on C60 and C44.

Contribution

It presents a novel spectral clustering approach for fullerene isomers based on facet graphs, providing a formal stability criterion and applying it to various isomers.

Findings

Successfully classified all C60 isomers

Identified cospectral isomers of C44

Method correlates with quantum chemical stability data

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 $C_{40}, C_{60}$ 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. 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 MATLAB. 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.