Excised internal structure

An excised internal structure or insular structure is a structure obtained by excising out the set of connected internal vertices usually associated with a strictly peri-condensed benzenoid hydrocarbon [3,13]. A strictly peri-condensed benzenoid PAH6 has all its internal third degree vertices mutually connected, has no cata-condensed... 

If a benzenoid excised internal structure is 1-factorable, 2-factorable, strictly peri-condensed, has one or more bay regions, and/or has one or more selective... 

Rule 1. Whenever the excised internal structure has an eigenvalue of zero, then the nonbisanthrene-like strictly peri-condensed benzenoid structure also has an eigenvalue of zero. 

Phenylenyl monoradical (C13H9) is a strictly peri-condensed PAH6 with the methyl radical as an excised internal structure and both have e = 0. The diradical C22H12 isomer, triangulene, is a strictly peri-condensed benzenoid hydrocarbon and has trimethylenemethane diradical as an excised internal structure both triangulene and trimethylenemethane diradicals have two eigenvalues of s = 0. 

The article reports investigations of the topological properties of benzenoid molecules which the author has performed in the last 20 years. Emphasis is given on recent developments and other scientists contributions to these researches. Topics covered in recent books and reviews are avoided. The article outlines spectral properties, some aspects of the study of Kekule and Clar structures, the Wiener index as well as a number of graphs derived from benzenoid systems (inner dual, excised internal structure, Clar graph, Gutman tree, coral and its dual). 

Inner Dual, Excised Internal Structure, Branching Graph... 

The subgraph spanned by the internal vertices of a benzenoid system was named by Dias [18-20] the excised internal structure (EIS). The subgraph spanned by the three-valent vertices of a benzenoid system was named by Kirby [12, 13] the branching graph (BG). Below are depicted the excised internal structure and the branching graph of the system 11 from Fig. 1 ... 

Kekulean/non-Kekulean nature of a benzenoid system is deduced from the properties of its excised internal structure. 

Theorem 13. The identity s(B, 1) = n(C(B), 1) holds if, and only if, the excised internal structure of the benzenoid system possesses a perfect matching. 

Strictly Pericondensed Benzenoid and Excised Internal Structure. . . 200... 

The excised internal structure [10,12,19,21,22,55] is defined in connection with strictly pericondensed benzenoids. It is the set of internal vertices and the edges... 

Fig. 2. A homolog series of hydrocarbons generated successively by two-contact additions (attachments of C3H). The excised internal structures are indicated by heavy lines (and the dot in Cl3H9)...

If A is a benzenoid which can be circumscribed, then it is the excised internal structure of circum-A. But an excised internal structure is not necessarily a benzenoid. In Fig. 2, for instance, one finds one vertex (corresponding to CH3) as the excised internal structure of phenalene (C13H9). Further, two connected vertices (corresponding to ethene, C2H4) is the excised internal structure of pyrene (C16H10). 

The circumscribing is obviously an important process in connection with excised internal structures. It is clear that a benzenoid with a cove or a fjord (or... 

In Fig. 6-13, a number of forms of different benzenoid isomers are displayed. The contours of excised internal structures for the strictly pericondensed systems are indicated in bold lines. The depictions of this type have proved to be very useful for the studies of benzenoid isomers in general, and especially for the extremal... 

The Kekulean systems belonging to constant-isomer benzenoid series are invariably normal (marked n in Fig. 11) and have A = 0. The non-Kekuleans are marked o with the A value indicated in a subscript (oA). In Fig. 11 also the excised internal structures [8] (see also below) are indicated by contours in bold. The same is found in Fig. 10 and subsequent figures. 

If C H5 has an excised internal structure [8, 32-34, 38], then its formula is C 2s+6Hs 6 in accord with Eq. (26). This is also the formula which appears in point (c ) of the special aufbau procedure (Sect. 11). In general an excised internal structure may be a non-benzenoid. Several examples of this phenomenon are found in Fig. 10. In connection with Eq. (27), however, when the excised internal structures of not too small ground forms (G) of constant-isomer benzenoid series are going to be considered, we shall only encounter benzenoids as such systems. Therefore, all we need to known about the definition of an excised internal structure here, is if G = circum-G0, then the benzenoid G° is the excised internal structure of G. Let again (n s) be the formula of G, and correspondingly (n° s°) of G°. Then... 

Table 8. Formulas for excised internal structures of ground forms of constant-isomer benzenoid series ...

Another interpretation of the location of the G° formula relates it to the circular benzenoid formula just above the one for j — 1,5. If O is the circular benzenoid with the formula given by (29), then its excised internal structure has the formula... 

It is immediately seen, on comparing with Eq. (40), that the formula (44) is obtained by subtracting CH from the one of G°. Hence the following alternative answer to our question. Subtract C3H from the formula of G to arrive at O find the formula for the excised internal structure of O, viz. O0 move one place up and one to the right in the periodic table to arrive at G°. 

The isomers with the formulas in Table 8 were enumerated by computer aid as far as possible. These systems are excised internal structures (G°) of ground forms for constant-isomer series of benzenoids when they are benzenoids themselves. They were classified according to A values and the symmetry groups to which they belong. The results are listed in Tables 9 and 10 for the even- and odd-carbon atom formulas, respectively. The documentation (given by footnotes) therein follow the previous review [8],... 

Dias JR (1990) A Formula Periodic Table for Benzenoid Hydrocarbons and the Aujbau and Excised Internal Structure Concepts in Benzenoid Enumerations. J Math Chem 4 17-30... 

Fig. 7.2. Forms of single coronoid isomers. Abbreviations 71 normal 0 non—Kekulean, with the A values indicated as subscript - 0. When the associated benzenoid is strictly peri condensed, the contour of its excised internal structure is given by heavy lines.

Gutman I, Dias JR (1990) The Excised Internal Structure of Hexagonal Systems [in] Contemporary Methods in Graph Theory (Bodendiek R, Edit). BI—Wissenschaftsverlag, Mannheim 249... 

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