Benzenoid graphs

In Table 2 are given the sextet polynomials for the lower members of aromatic hydrocarbons together with their K(G) numbers. The sextet polynomials for larger benzenoid graphs are extensively tabulated and discussed [13, 14]. As already mentioned, for a thin benzenoid or coronoid system there is exactly a one-to-one relation between the Kekuie patterns and sextet patterns. In other words the following equality is obeyed,... 

In the theory known nowadays as the Clar theory of the aromatic sextet [12] a benzenoid system is represented by a Clar structure which is obtained by drawing circles in some of the hexagons of the corresponding benzenoid graph. These circles represent the aromatic sextets in the hydrocarbon. We consider here only Clar structures containing the maximum number of circles which are some times referred to as proper (or correct) Clar formulas. The rules for constructing such Clar structures are as follows [13]. 

It is worthy of observation that for both classes of benzenoid hydrocarbons represented in Figs. 2, 3 the corresponding trees (whose number of colorings = the number of Clar structures) coincide with the inner duals [ 19] of the benzenoid graphs. 

Since benzenoid graphs are bipartite, their characteristic polynomials can be written in the form... 

Special methods for the calculation of the characteristic polynomials of benzenoid graphs have recently been designed by Sachs and John [16, 17]. Their method is especially efficient in the case of catacondensed systems. 

From the Dewar — Longuet-Higgins formula, Eqs. (1) and (4), it is immediately seen that the above problem is equivalent to the question whether there exist zero eigenvalues in the spectrum of a benzenoid graph. Indeed, in computer-aided searches, constructions and classifications of benzenoid systems, the easiest and most efficient way to recognize non-Kekulean species is just to compute det A. At this point it should be mentioned that Hall [66] recently proposed a new easy method for rapid calculation of det A of a benzenoid system. 

It is really surprising that the Wiener indices have such number-theoretical properties. This kind of modular behavior was observed never before for any of the numerous topological indices studied in chemical graph theory. Results analogous to those given in Theorem 20 were later found for other classes of (non-benzenoid) graphs [130, 131], but their extension to pericondensed benzenoids was never accomplished. 

The definition of a benzenoid hydrocarbon/benzenoid system/benzenoid graph as well as a sufficient number of examples can be found in the preceding article [1] and elsewhere [12]. We shall not reintroduce the notation and terminology described in [1], except that for the readers convenience we list the most frequently employed symbols. Let BH be a benzenoid hydrocarbon and let B stand for the corresponding benzenoid system/benzenoid graph. Then ... 

Let us consider more closely the set of objects of Fig 7 associated with the same adjacency matrix A We will refer to these as the set T, A, B, P standing, respectively, for a caterpillar tree, a Clar graph (As L(T)) a benzenoid graph and a king polyomino The grai invariants that we will consider in each case are as follows ... 

Dobrynin, A.A. (1998a). Formula for Calculating the Wiener Index of Catacondensed Benzenoid Graphs. J.Chem.lnf.Comput.ScL, 38, 811-814. 

Dobrynin, A.A. (1998b) New congruence relations for the Wiener index of cata-condensed benzenoid graphs. J. Chem. Inf. Comput. Sci., 38, 405—409. 

Dobrynin, A.A. (1999b) Explicit relation between the Wiener index and the Schultz index of catacondensed benzenoid graphs. Croat. Chem. Aaa, 72, 869-874. 

Dobrynin, A.A. (2003) On the Wiener index decomposition for catacondensed benzenoid graphs. Indian J. Chem., 42, 1270-1271. 

Fajtlowicz, S., John, P.E. and Sachs, H. (2005) On maximum matchings and eigenvalues of benzenoid graphs. Croat. Chcm. Acta, 78,195-201. 

On Hosoya polynomials of benzenoid graphs. MATCH Commun. Math. Comput. Chem., 43, 49-66. 

Gutman, 1. and Markovic, S. (1993) Benzenoid graphs with equal maximum eigenvalues. /. Math. Chem., 13, 213-215. 

Vukicevic, D. and Trinajstic, N. (2004) Wiener indices of benzenoid graphs. Bull. Chem. Tech. Mac, 23, 113-129. 

Starting from the top left comer of the benzenoid graph, assign +1 or -1 to terminal rings according to the orientation of their aromatic sextets (Fig. 3). 

Euleric Corollary. For trivalent qua-benzenoid graphs cellularly embedded in a closed surface S, all = 0 for n 6 except for... 

Notably benzenoid graphs arise if and only if %(S) = 0, i.e., if 5 is a torus (or a suitable nonorientable surface discussed briefly in Section 9.4.4). The qua-benzenoids on spherically homeomorphic surfaces are usually i tmedfullerenes, for which many special results have been developed and will be discussed more explicitly before long. 

For the toroidal case all of the benzenoid graphs (such as are possible via the... 

The finding that for every benzenoid graph G another graph C G) can be constructed [139], such that Sez(G) = ln(C(G)), had already been mentioned (Theorem 4.6 3.4). The graph C(G) is of some relevance in Clar s aromatic sextet theory. The name Ciar graph has been proposed for C(G). 

Babic D (1993) Isospectral Benzenoid Graphs with an Odd Number of Vertices. J Math Chem 12 137... 

Cioslowski J (1991) A Conjecture on Benzenoid Graphs. J Math Chem 6 111... 

Explore if the complexity algorithm mentioned above holds for selected classes of molecules, such as (i) acyclic alkane graphs, (ii) cata-condensed benzenoid graphs, or (iii) smaller polycylic graphs. 

Try to characterize peri-condensed benzenoid graphs that have a Hamiltonian path. Problem 17... 

Figure 9 Three examples of benzenoid graphs exhibiting excessive degeneracies, discussed by Wild, Keller, and GUnthard ...

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