INVARIANTS OF SINGLE CORONOIDS

In this chapter the findings for invariants of polyhexes (cf. Sect. 3.2) are summarized for the special case of single coronoids ( = 1) and extended in a more detailed treatment. It comes as a supplement to the treatment of invariants of single coronoids in Vol.I—3.2.2. In particular, the upper and lower bounds of several invariants are treated extensively. This leads naturally to a treatment of extremal single coronoids (cf. Par. 3.3.4), which invariably are perfect. 

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The invariants of single coronoids have integer values restricted to certain domains. The smallest (single) coronoid has h = 8, and also all values h > 8 are obviously possible. For n, zero and all positive integers are possible, and specifically n. = 0, 1, 2,. , for every h. Now,... 

Table 4.3. Inequalities for invariants of single coronoids in terms of h and...

We wish also to establish the connection between the formulas (n s) and (rip s ). The known relations between the different invariants of single coronoids (Table 4.1), combined with eqns. (19) and (21), give... 

The known relations between invariants of single coronoids (Table 4.1) yield... 

Table 4.2. Possible values of the invariants in single coronoids.

Table 3.1. Invariants of polyhexes benzenoids (p = 0), single coronoids g = 1) and multiple coronoids g > 1). ...

Assume that an invariant x for single coronoids is selected. Then it is of interest to determine the upper and lower bounds of x, if they exist, in terms of an other invariant, say y. Such an analysis may result in inequalities of the kind f y) < x < g(y), where f(y) = and y(y) = iinctions to be determined. 

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