Extremal single coronoid

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. 

Maximum Number of Internal Vertices, and Extremal Single Coronoids... 

An extremal single coronoid (cf. Par. 3.3.4, especially Definition 3.4), A, is defined by having the maximum number of internal vertices (n ) for a given number of hexagons (h) =... 

In any extremal single coronoid, A, one has clearly h = min z however, this does not give the full information on the desired function, since A systems do not exist for every n - see... 

Fig. 4.4. The spiral walk for generating extremal single coronoids.

A single coronoid which has h = for a given s is clearly an extremal coronoid, A. However, the property h = cannot serve as a definition for extremal single coronoids. As... 

Parts of this chapter, like the preceding chapter, is a specialization of relations from Chapt. 3 for s 1, but also other aspects are treated. In particular, a condse treatise of the processes referred to as drcumscribing and excising is offered. These processes are espedally important when it comes to extremal polyhexes the extremal single coronoids is another main subject of this chapter. All the treatments in this chapter are (more or less) oriented towards expressions in terms of the coefficients of the formulas, viz. n and a, as is reflected in the title. 

Properties. Many properties of extremal benzenoids, A, can immediately be adapted to extremal single coronoids, A. Relevant properties of A are treated in several places (Cyvin SJ, Cyvin, BrunvoU, Gutman and John 1993 Cyvin SJ, Cyvin and BrunvoU 1993d 1993e) with more or less rigorous proofs therein. Here we shall not conduct special proofs for the adaptations of the different properties to the A coronoids. We are particularly interested in properties of A which pertain to formations (or absence of formations) on the perimeter of A. By the adaptation of these properties to A one correlates the outer perimeter of an A system with the (only) perimeter of an A. 

In other words, any extremal single coronoid (A) can be circumscribed without limitation. However, there also exist nonextremal single coronoids which are reproducible and therefore can be circumcscribed without limitation. Kekulene is an obvious example. 

Property 5.4. For any A, an extremal single coronoid, also c(A) is an extremal single coronoid. 

Definition 5.7 A single coronoid (P) is circumextremal if its associated benzenoid, P, is a drcumextremal benzenoid, and exds-P is the assodated benzenoid to a (smaller) extremal single coronoid. 

Some explanations should be attached to this somewhat complicated definition. It is dear that a drcumscribed extremal (single) coronoid is circumextremal write circum—A = P. Then it is also dear that excis-P = A. Assume now that the naphthalene hole of P is moved as near as possible to the outer perimeter, and call the new coronoid P. Then P and P are isomers, but P cannot be excised to give another coronoid excis—P becomes a degenerate system. Nevertheless, P falls under the definition of drcumextremal coronoids. An illustration is given below. 

In simple terms all the circumextremal single coronoids are obtained as circumscribed extremal single coronoids and their isomers. Once more, in other words a circumextremal single coronoid has the same formula as a drcumscribed extremal single coronoid. 

The class of drcumextremal single coronoids (P) is a subdass of the extremal single coronoids (A) cf. Property 5.4. 

The first (smallest) extreme single coronoids have the formulas C32H16, C35H17, C38H18, C37H17,... 

Notation and Preliminary Treatment. Let an extreme single coronoid which is not extremal be identified by X(n 5 ). These formulas occur for certain values of n- = n, but not... 

Fig. 6.6. One isomer each of the smallest nonextremal extreme single coronoids. The inscribed numerals indicate the number of last added hexagon during the modified spiral walk.

Fig. 5.7. Examples of nonextremal extreme single coronoids which are not naphthalenic.

Formula. In accord with the above description every nonextremal extreme single coronoid (X) corresponds to an extremal single coronoid, A( ), for which A(hr-l) -i A(h) is accompanied by An = 2. The formulas of these special A coronoids are obtained from eqn. (57), but starting the running index from h = 1 i.e. h = 1, 2, 3, 4,...These A coronoids are associated with the... 

In Table 3 "everything is perfect". The formulas at the extreme right of each row pertain to perfect extremal single coronoids, and therefore the right-hand boundary is a perfect staircase cf. Par. 3.7.3. 

The focussing on formulas for single coronoids is continued in this chapter. It starts with a treatment of sequences of associated formulas for extremal single coronoids. This study leads to another subdivision of the extremal coronoids ground forms and higher members. The pertinent sections can be considered as a preparation to the enumeration of coronoid isomers, which is treated in the subsequent chapter. Also the process of building-up (Sect. 6.5) is highly relevant to the enumeration of isomers. 

For extremal single coronoids we shall define ground forms and higher members in connection with a sequence of formulas (uq Sq), (n . , (nj. in analogy with the... 

DefimUian 6.1 A ground form (of single coronoids), G, is an extremal single coronoid which is not drcumextremal. 

Extreme Systems. The extreme single coronoids, E (Sect. 5.6), have formulas (n s ) situated at the tops of the columns of the table of formulas (cf. Fig. 5.8). Then it is obvious that... 

The extremal single coronoids, A (Par. 5.5.1), forming a subclass of the extreme systems,... 

Property 6.S An extremal single coronoid has never a cove and never a fjord in addition to the inner fjords of its naphthalene hole. 

Property 6,4- A nonextremal extreme single coronoid X has never a fjord except for the inner fjords of its naphthalene hole if the system is naphthalenic. 

All extremal single coronoids A are naphthalenic. The circular single coronoids 0 are also extremal they form a subclass of the A systems. Some expressions for circumscribing naphthalenic single coronoids, C, are given below. 

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