Internal vertex

Proof, (i) Take an arbitrary (r, )-polycycle P. We tile each r-gon into 4-gons by connecting its center with the midpoints of the sides. Then the number of 4-gons in each r-gon is equal to r and the number of 4-gons, incident to any internal vertex, is equal to q. Then the number of 4-gons, incident only to internal vertices of the polycycle P, is equal to vintq, while the total number of 4-gons is equal to rpr. Hence, vin,q < rpr. 

Since the angle of a regular r-gon is equal to r- n and the number of regular r-gons that meet at an internal vertex of the polycycle P is equal to q, the curvature of any internal vertex of the polycycle P is equal to ... 

In other words, a cycle in a catacondensed benzenoid is of the size 6 or 10 or 14 or 18. .. This is because catacondensed systems (by definition [3]) possess no internal vertices. On the other hand, according to Theorem 1 the existence of a cycle whose size is divisible by four implies the existence of at least one internal vertex. 

Since a catacondensed system possesses no internal vertex its perimeter embraces all the vertices. Consequently, the perimeter of a catacondensed system is a Hamiltonian cycle. In other words, all catacondensed benzenoid systems are Hamiltonian. 

The distinction between catacondensed and pericondensed systems is applicable to all the classes of polyhexes treated above (Fig. 2). A catacondensed polyhex is defined by the absence of internal vertices. An internal vertex is a vertex shared by three hexagons. A pericondensed polyhex possesses at least one internal vertex. In terms of dualists a pericondensed polyhex reveals itself by the presence of at least one three-membered cycle (triangle). A dualist of a circulene has a cycle larger than a triangle. 

The pericondensed extreme-left benzenoids constitute a subclass of the strictly pericondensed benzenoids in the sense of Dias [12, 15, 19, 21, 55-57] they are defined by having all their internal vertices connected and no catacondensed appendages. Phenalene, C13H9, which has only one internal vertex, is reckoned among the strictly pericondensed benzenoids. An equivalent definition in a most succint form reads ... 

Since the paths and P2 contain as internal vertex the stable synthon S13b4), the stable synthon S3(T) is not an element of the stable neighborhood of the synthon S12(A). The stable neighborhood of S12(/l) is S(S12(/1)) = S9(/l), S13(v4). In other words, a stable neighbourhood of a synthon S(T) is composed only of those synthons S (A), for which there exists no shortest path between S( 4) and S ( 4) in graph %D(A) containing as an intermediate some stable synthon. 

Figure 7 The structure of [Fe aM(0) i o(OH) o(02CPh)2o] showing the four M4O2 butterflies with full lines, and the internal vertex-sharing bitetrahedron of metals as dashed lines.

Integrate over all internal vertex coordinates (xiTi), and sum over all internal spin coordinates. 

Perifusenes with One Internal Vertex - A Complete Mathematical Solution. 

Systems with One Interned Vertex. The formulas for the single coronoids with = 1 are found in the second formula column of Tabid 5.3. Here C35H17 has a unique position. Its isomers can only be generated by the attachments of C H according to C32H16 C35H17. The... 

Here A° = 6, nfi = 0. The pertinent corona hole benzenoid has one fjord, but when a hexagon is immersed a cove is created. As a result, one obtains A = A - 1, = 1 (the internal vertex is... 

Cyvin SJ, Zhang FJ, Brunvoll J (1992) Enumeration of Perifiisenes with One Internal Vertex — A Complete Mathematical Solution. J Math Chem 11 283... 

The same argument may be repeated. In fourth order there is a second internal vertex, which can lead to any one of three types of factor in the numerator, together with a further D factor in the denominator. On summing in a similar way, we find that 4 is obtained from 3 simply by inserting another factor It is then clearly possible to sum the... 

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