Some General Principles and Definitions

Common conventions of graph theory are followed here (see, e.g.. Refs. 8-10), but a rigid distinction is not always maintained between chemical and mathematical contexts. So the term graph is used interchangeably in both its strict sense as a mathematical object and as a shorthand term for its realization as an actual (usually carbon) molecule. Some equivalent pairs of terms are treated as being synonymous atom = vertex, bond = edge, valency = degree, and so on. 

For more specific cases where the surface network has a simple uniform pattern of rings, we specify the surface type and refer, for example, to a toroidal polyhex (also called a toroidal hexagonal system by John and Walther ) or a toroidal azulenoid, and so on. Others -prefer to speak more generally of torusenes or toroidal forms of carbon or of graphitic carbon or graphite. 

By a polyhex is meant a structure comprising an assembly of hexagons such that any two hexagons are either disjoint or have at least one common edge. (Note that this 

Some other terms are defined as they occur in the text. 

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