A graph theoretic approach to atomic displacements in fullerenes

We consider a classical mechanics approach to atomic displacements in fullerene molecules. The problem is reduced to the study of the graph Laplacian spectra by deriving an analytical expression for the atomic displacement due to thermal vibrations/oscillations. We then use the concepts of graph isoperimetric constant and graph expansion to prove that "among all graphs on n nodes, those with good expansion properties display the smallest topological displacements of their nodes." Consequently, fullerenes with the property of being Ramanujan graphs, i.e., Ramafullerenes, are those that exhibit the smallest atomic displacements due to thermal movement. We show that fullerenes with the smallest atomic perturbations due to thermal effects are the most stable ones. Then, relationships between atomic displacements, spectral gap, and energy are presented for different families of fullerenes.