Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages

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 co-planar the usual deenition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omni-capping) is developed that enables to extend the deenition of IQ and compute it for any 3D drawing. The IQ's 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.