The resonance graphs of nanotubes and toroidal polyhexes
Abstract
Coronoid systems, nanotubes and toroidal polyhexes (or fullerenes) can all be regarded as carbon networks composed of carbon atoms linked in hexagonal shapes. The resonance graphs of coronoid systems and nanotubes are not necessarily connected. For coronoid systems and elementary nanotubes, by using flow across cuts the present authors gave criteria for two perfect matchings lying in the same connected component of the resonance graph (Discrete Appl. Math. 395 (2026) 443-455). However, the sufficiency of such criterion does not hold for general nanotubes and toroidal polyhexes. In this paper we strengthen this requirement to obtain valid criteria for two perfect matchings of a nanotube (resp. toroidal polyhex) to lie in the same connected component of its resonance graph: they have the same flows across cuts along the x-axis (resp. longitude and latitude) and the same ladders. For toroidal polyhexes, our method uses homotopic classes of simple loops on the torus, and the above criterion can be simplified by using only simple flows, for the case in which two perfect matchings have alternating hexagons.
Related
- A Proof of the Tratnik-Ye Medianity Conjecture of Resonance Graphs on Surfaces
- Seifert type magnetohydrostatic equilibria in rotationally symmetric toroidal domains
- On orbits of the automorphism group of toroidal spherical varieties
- Entropy measures of degree-based topological indices for hyper carbon nanotubes
- Analytic toroidal 3D MHD equilibria and steady Euler flows with invariant surfaces
BibTeX
@misc{liang2026resonance,
title = {{The resonance graphs of nanotubes and toroidal polyhexes}},
author = {Lingmei Liang and Heping Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19766},
eprint = {2609.19766},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19766},
}
RIS
TY - UNPB
TI - The resonance graphs of nanotubes and toroidal polyhexes
AU - Lingmei Liang
AU - Heping Zhang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19766
UR - https://arxiv.org/abs/2609.19766
AB - Coronoid systems, nanotubes and toroidal polyhexes (or fullerenes) can all be regarded as carbon networks composed of carbon atoms linked in hexagonal shapes. The resonance graphs of coronoid systems and nanotubes are not necessarily connected. For coronoid systems and elementary nanotubes, by using flow across cuts the present authors gave criteria for two perfect matchings lying in the same connected component of the resonance graph (Discrete Appl. Math. 395 (2026) 443-455). However, the sufficiency of such criterion does not hold for general nanotubes and toroidal polyhexes. In this paper we strengthen this requirement to obtain valid criteria for two perfect matchings of a nanotube (resp. toroidal polyhex) to lie in the same connected component of its resonance graph: they have the same flows across cuts along the x-axis (resp. longitude and latitude) and the same ladders. For toroidal polyhexes, our method uses homotopic classes of simple loops on the torus, and the above criterion can be simplified by using only simple flows, for the case in which two perfect matchings have alternating hexagons.
ER -