Generalized single coronoid

Definition 2.S A generalized single coronoid is either a single coronoid or a special degenerate single coronoid. 

If one wants to avoid the term "special degenerate single coronoid", then one has the synonymous designation "degenerate generalized single coronoid" at hand. 

In this chapter some graph—theoretical proofs are conducted for generalized single coronoids. Some of these proofs are closely connected with the discussion of Sect. 2.2. In particular, the rheo conjecture is proved. 

Also the following lemma (like Lemma 8.1), here formulated for single coronoids G, applies to generalized single coronoids (Zhang and Zheng 1992). 

Assume now that G is a degenerate generalized single coronoid, and A is a set of nonadjacent vertices, all of them on the outer or on the inner perimeter of G, but not on the intersection of the two perimeters. 

Theorem 8.S An essentially disconnected single coronoid G has at least two effective units. One of them is a normal benzenoid and an other a normal generalized single coronoid (degenerate or non—degenerate). 

Theorem 8.S A generalized single coronoid G is normal if and only if there are two Kekule structures and M2 of G such that C and C of G are M - and M2-alternating cycles,... 

It is natural to consider polyhexes as a class since many general properties of these systems can be formulated and can, in the best cases, be expressed by general functions of g. Here we are especially interested in such general properties, while specific treatment of single coronoids is the main subject of the remaining chapters. The present chapter is only an introduction to the theory of polyhexes. 

Assume now that the single coronoid C can be circumscribed k times, and define as Anfold circumscribed C cf. eqn. (12). Then the generalization of eqn. (23) reads... 

Eqn. (42) is general for single coronoids. It is also valid when c(C) is not a coronoid. 

Let (TijjjP symboUze the formula for a Aj-fold circumscribed gP system. Then this formula (for k > 0) pertains to the perfect higher members of single coronoids. From (3) and (5.52) one obtains the general formulation ... 

The term polyhex has been used about benzenoids and coronoids together [18]. Benzenoids are polyhexes without holes, while a single or multiple coronoid is a polyhex with one or more holes, respectively. There is also a tendency to use the term polyhex in a more general sense [16]. Here we shall use it as a universal term for all hexagonal systems which are to be considered. 

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