Toroidal polyhexes

Abstract

Some features of a boundless network of hexagons embedded on the surface of a torus are discussed. A systematic coding and classification scheme is suggested whereby any toroidal polyhex may be described by a unique string of three integers. This can be used to compile an adjacency matrix and to evaluate the eigen-spectrum by simple methods which are valid for all such structures with fewer than 7200 vertices (3600 hexagons). A general method, valid for all systems, is sketched out. Some eigenvalue regularities are pointed out, including cases of subspectrality. The enumeration of spanning trees and of Kekulé structures is discussed, and a Kekulé count published by the Galveston Group is confirmed.