Toroidal Fullerenes with the Cayley Graph Structures

A central issue in molecular orbital theory is to compute the HOMO-LUMO gap of a molecule, which measures the excitability of the molecule. Thus it would be of interest to learn how to construct a molecule with the prescribed HOMO-LUMO gap. In this paper, we classify all possible structures of fullerene Cayley graphs and compute their spectrum. For any natural number $n$ not divisible by three, we show there exists an infinite family of fullerene graphs with the same HOMO-LUMO gap of size $\frac{2\pi}{\sqrt{3}n}+O(n^{-2})$. Finally, we discuss how to realize those families in three dimensional space.