How To Enumerate the Connectional Isomers of a Toroidal Polyhex Fullerene

A simple algorithm is described for enumerating all isomers of a toroidal polyhex fullerene (a boundless network of three-valent carbon atoms embedded in the surface of an imaginary torus where all the rings or faces are hexagons). An implementation for a PC is convenient for structures of at least 10 000 atoms in size. An extension of this program allows fast and automatic compilation of an adjacency matrix and derivation of a closed trigonometric expression yielding eigenvalues for all structures with fewer than 7200 atoms and, apparently, most others, although a few evaluations beyond this size are more complicated. The simplicity of a strictly polyhex toroidal structure means that, in marked contrast to many chemical enumerations, the total number of isomers increases at only a modest rate and (within the range surveyed) does not exceed approximately 30% of the number of atoms.