On correlation of hyperbolic volumes of fullerenes with their properties

Abstract

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.

Volume estimates for right-angled hyperbolic polyhedra

“…In Figure 2 dots presents volumes of compact right-angled hyperbolic polyhedra with at most 46 vertices, and lines present volume estimates from Theorem 2.3 (lower bound) and Theorem 2.4 (upper bound). Previously, volumes of the first 825 compact right-angled hyperbolic polyhedra were calculated in [6] and volumes of compact right-angled polyhedra with at most 64 vertices having only pentagonal and hexagonal faces were calculated in [4].…”

“…The upper estimate from Theorem 2.2 was generalized in [4] as in Proposition 4.1, Proposition 4.2 and Proposition 4.3 below. For the reader's convenience we present these statements with proofs.…”

On volumes of hyperbolic right-angled polyhedra

“…Recall the following volume bound from [10]. Lemma 4.1 (see [10], Corollary 3.2). Let P be a compact right-angled hyperbolic 3-polyhedron with V vertices.…”

On the Wiener (r,s)-complexity of fullerene graphs

“…Mathematical studies of fullerenes include applications of graph theory and topology methods, design of computational and combinatorial algorithms, information theory approaches, etc. (see selected publications [1,3,4,6,10,[12][13][14][15][22][23][24]).…”