Fibonacci graphs

In this review we will focus on polyhex graphs, caterpillar trees, Clar graphs and several related polyomino gra s [19] In addition sets of graphs obeying certain types of recursive relations, called "Fibonacci Graphs" will be discussed particularly from the point of view of their computational importance ... 

In this part we review three types of equivalencies viz. Graphs possessing equivalent alternative models, homologous series of graphs and a recently introduced [36] class of equivalent graphs of computational importance called Fibonacci graphs-... 

A very special type of homologous set of graphs of specific topological construction which was recently discovered [36] is called Fibonacci graphs. The simplest two such classes are being paths and cycles. More general constructions are own in Fig. 15... 

Fig- 15 External (top) and inteinal (bottom) construction of Fibonacci graphs-... 

Gutman, I. and El-Basil, S. (1986). Fibonacci Graphs. MATCH (Comm.Math. Comp. Chem.), 20, 81-94. 

These graphs could be of great computational importance as is illustrated in Fig 16 where the nonadjacent numbers of 18-annuleno-18-annulene are hand-computed using the concept of Fibonacci grains ... 

Hosoya found that for linear graphs the Z indices are the Fibonacci numbers, i.e. Z = Ea) where A is the number of atoms in the molecular graph therefore, for a linear graph, it is closely related to the -> Merrifield-Simmons index [Gutman et ai, 1992 Randic et ai, 1996b]. 

Moreover, again for linear graphs (i.e. n-alkanes), the Hosoya Z index coincides with the Fibonacci number F , and thus Hosoya and Merrifield-Simmons indices are both closely and directly related. For monocyclic graphs C and iso-path graphs i - L the following relationships between Fibonacci numbers and the Merrifield-Simmons index hold ... 

Table S-3. Fibonacci numbers (F ) and corresponding Merrifield-Simmons indices for linear (Z- ), cyclic (C), and iso-path graphs...

Balaban, A.T. and Tomescu, T. (1985). Chemical Graphs. XLI. Numbers of Conjugated Circuits and Kekule Structures for Zigzag Catafusenes and (j, k)-hexes Generalized Fibonacci Numbers. MATCH (Comm.Math.Comp.Chem.), 17, 91-120. 

Balaban, A.T. and Tomescu, 1. (1984) Chemical graphs. XL. Three relations between the Fibonacci sequence and the numbers of Kekule structures for non-branched cata-condensed polycyclic aromatic hydrocarbons. Croat. Chem. Acta, 57, 391-404. 

Balaban, A.T. and Tomescu, 1. (1985) Chemical graphs. XLl. Numbers of conjugated circuits and Kekule structures for zigzag catafiisenes and (J, k)-hexes generahzed Fibonacci numbers. MATCH Commun. Math. Comput. Chem., 17, 91—120. 

Figure 12-13. Density graph P2/(zx) of the thick twofold layers, with spacings ah -plane, 1.48 A, feg -plane, 0.92 A, fcg -plane, 1.48 A, afc -plane. The plateau of the graph defines the terminations. The width of the support of the plateau equals exactly /2 and defines the Fibonacci sequence of twofold...

S. El-Basil and D. J. Klein, Fibonacci numbers in the topological theory of benzenoid hydrocarbons and related graphs, J. Math. Chem. 3 (1989) 1-23. 

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