Toroidal fullerenes with the Cayley graph structures

Abstract

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 2π √ 3n + O(n −2 ). Finally, we discuss how to realize those families in three dimensional space.

The topology of fullerenes

“…126 Kang analyzed the band gap in such toroids by graph theoretical means. 323 Deza et al pointed out that leapfrog toroids and Klein-bottles have equal numbers of positive and negative eigenvalues, but with 4 and 2 eigenvalues being zero respectively. 242 Interestingly, the leapfrog transformation performed on toroids (such as the one shown in Figure 24 ) results in an open shell toroid.…”

Eigenvalues of Cayley graphs

“…A fullerene is called toroidal if it lies on the torus. In [167], Kang classified all fullerenes which are Cayley graphs and determined their eigenvalues.…”

Untitled

“…Ye et al [10] have studied a k-resonance of toroidal polyhexes. Classifications of all possible structures of fullerene Cayley graphs is given in [11] by Kang. The atom-bond connectivity index (ABC) and geometric-arithmetic index (GA) of the toroidal polyhex are computed in [12] by Baca et al In [13][14][15][16][17][18][19][20][21][22][23], authors computed distance-based topological indices of eight infinite sequences of 3-generalized fullerenes.…”