Abstract
AbstractWe observe that fullerene graphs are one-skeletons of polyhedra, which can be realized with all dihedral angles equal to π /2 in a hyperbolic 3-dimensional space. One of the most important invariants of such a polyhedron is its volume. We are referring this volume as a hyperbolic volume of a fullerene. It is known that some topological indices of graphs of chemical compounds serve as strong descriptors and correlate with chemical properties. We demonstrate that hyperbolic volume of fullerenes correlates with few important topological indices and so, hyperbolic volume can serve as a chemical descriptor too. The correlation between hyperbolic volume of fullerene and its Wiener index suggested few conjectures on volumes of hyperbolic polyhedra. These conjectures are confirmed for the initial list of fullerenes.
Highlights
Mathematical methods became a very strong research tool in biology and chemistry in the last decades
We refer to [1] for a modern survey on geometric, topological, and graph theoretical methods for the modeling and analysis of biomolecular data and to [2] for the classification of common chemical descriptors for QSAR/QSPR analysis used for machine learning in chemical informatics and drug discovery
We will show that any fullerene can be realized in a hyperbolic 3-dimensional space as a polyhedron with all dihedral angles equals π/2, see [5] for more information about hyperbolic geometry
Summary
Mathematical methods became a very strong research tool in biology and chemistry in the last decades. Some topological indices can be used, say the Wiener index, the pentagon signature, etc Another approach, which we present in this paper, is based on non-Euclidean geometry invariants. It demonstrates that hyperbolic volume can serve as a fullerene stability descriptor. These conjectures are based on the correlation between hyperbolic volumes and the Wiener index and are confirmed for the initial list of fullerenes
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