Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages Tomaž Pisanski ,†Matjaž Kaufman ,*†Drago Bokal ,†Edward C. Kirby ,‡Ante Graovac§ Inštitut za matematiko, fiziko in mehaniko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia, Resource Use Institute, 14 Lower Oakfield, Pitlochry, Perthshire PH16 5DS, Scotland, UK
Abstract: The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same face may not be coplanar the usual definition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omni-capping) is developed that enables one to extend the definition of IQ and compute it for any 3D drawing. The IQs of fullerenes and other polyhedral cages are computed and compared for their NiceGraph and standard Laplacian 3D drawings. It is shown that the drawings with the maximal IQ values reproduce well the molecular mechanics geometries in the case of fullerenes and exact geometries for Platonic and Archimedean polyhedra.
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Dr. Claus Kiefer. Institut für Theoretische Physik Universität zu Köln … 0221 470-4300 (secretary). Fax: 0221 470-2189. E-mail: kiefer@thp.uni-koeln.de
(square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688
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