We 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.

No abstract

Получены новые верхние оценки объемов прямоугольных многогранников в пространстве Лобачевского $\mathbb{H}^3$ в следующих трех случаях: для идеальных многогранников, все вершины которых лежат на абсолюте, для компактных многогранников, все вершины которых конечны, и для многогранников конечного объема с вершинами обоих типов. Библиография: 23 названия.

By Andreev theorem acute-angled polyhedra of finite volume in a hyperbolic space H 3 are uniquely determined by combinatorics of their 1-skeletons and dihedral angles. For a class of compact rightangled polyhedra and a class of ideal right-angled polyhedra estimates of volumes in terms of the number of vertices were obtained by Atkinson in 2009. In the present paper upper estimates for both classes are improved.2010 Mathematics Subject Classification. 52B10. Key words and phrases. right-angled polyhedron, ideal polyhedron, hyperbolic volume.