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Journal of Molecular Structure (Theochem), 227 (1991) 79-86 Elsevier Science Publishers B.V., Amsterdam 79 THE RltDUCED GRAPH MODEL REVISITED* BORKA JERMAN-BLAiIC The Joief Stefan Institute, P.O. Box 199,610Ol Ljubljana, (Yugoslavia) SONJA NIKOLIC and NENAD TRINAJSTIC The Rugjer BoikoviE Institute, P.O. Box 1016,410Ol Zagreb, (Yugoslavia) (Received 2 October 1989) “It seems that the influence of physics and mathematics in chemistry will become even more important in the near future.” Rudolf Zahradnik (1988) [ 1 ] ABSTRACT The reduced graph model used for counting Kekul6 structures of benzenoid hydrocarbons is improved. The improvement is based on the Pascal recurrence algorithm for enumerating self- avoiding peak-to-valley paths and the John-Rempel-Sachs theorem. The improved version, un- like the original form, of the model, is applicable in principle to any benzenoid hydrocarbon. INTRODUCTION The reduced graph model has been introduced as an alternative method of representing benzenoid graphs [ 21. This model has been shown to be useful in combinatorial problems of benzenoid systems [ 2,3] such as the enumeration and generation of Kekule structures, the enumeration and generation of con- jugated circuits, the counting of benzenoid hydrocarbons, the construction of the sextet polynomial, etc. Nevertheless, the reduced graph model has mostly been used for Kekule-structure counts of benzenoid hydrocarbons. The crucial step in using the model is the count of vertical or self-avoiding peak-to-valley paths. This path count is related to the number of Kekule structures. Gordon and Davison [4] pointed out some time ago that there is a natural direct cor- respondence between self-avoiding peak-to-valley paths and the Kekule-struc- ture counts, but did not elaborate this idea further. In previous work we failed to solve two problems that arose in the application of the reduced graph model to Kekule-structure counts. These problems are *Dedicated to Professor Rudolph Zahradnrk, a friend and one of the most important contemporary chemical theoreticians in the Slavic world. 0166-1280/91/$03.50 0 1991- Elsevier Science Publishers B.V.

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Journal of Molecular Structure (Theochem), 227 (1991) 79-86 Elsevier Science Publishers B.V., Amsterdam

79

THE RltDUCED GRAPH MODEL REVISITED*

BORKA JERMAN-BLAiIC

The Joief Stefan Institute, P.O. Box 199,610Ol Ljubljana, (Yugoslavia)

SONJA NIKOLIC and NENAD TRINAJSTIC

The Rugjer BoikoviE Institute, P.O. Box 1016,410Ol Zagreb, (Yugoslavia)

(Received 2 October 1989)

“It seems that the influence of physics and mathematics in chemistry will become even more important in the near future.” Rudolf Zahradnik (1988) [ 1 ]

ABSTRACT

The reduced graph model used for counting Kekul6 structures of benzenoid hydrocarbons is improved. The improvement is based on the Pascal recurrence algorithm for enumerating self- avoiding peak-to-valley paths and the John-Rempel-Sachs theorem. The improved version, un- like the original form, of the model, is applicable in principle to any benzenoid hydrocarbon.

INTRODUCTION

The reduced graph model has been introduced as an alternative method of representing benzenoid graphs [ 21. This model has been shown to be useful in combinatorial problems of benzenoid systems [ 2,3] such as the enumeration and generation of Kekule structures, the enumeration and generation of con- jugated circuits, the counting of benzenoid hydrocarbons, the construction of the sextet polynomial, etc. Nevertheless, the reduced graph model has mostly been used for Kekule-structure counts of benzenoid hydrocarbons. The crucial step in using the model is the count of vertical or self-avoiding peak-to-valley paths. This path count is related to the number of Kekule structures. Gordon and Davison [4] pointed out some time ago that there is a natural direct cor- respondence between self-avoiding peak-to-valley paths and the Kekule-struc- ture counts, but did not elaborate this idea further.

In previous work we failed to solve two problems that arose in the application of the reduced graph model to Kekule-structure counts. These problems are

*Dedicated to Professor Rudolph Zahradnrk, a friend and one of the most important contemporary chemical theoreticians in the Slavic world.

0166-1280/91/$03.50 0 1991- Elsevier Science Publishers B.V.

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the efficient enumeration of self-avoiding peak-to-valley paths and elimina- tion of the uncoordinated paths beyond the brute-force approach. Here we report how these two deficiencies were removed by incorporating the Pascal recurrence algorithm [5] and the John-Rempel-Sachs theorem [6] into the framework of the reduced graph model.

The continuous interest in Kekule structures [ 7-91 is related to their use in . simple classical pictures [lo] of delocalized chemical bonding, pictures that carry over to modern quantum-chemical theories of benzenoid hydrocarbons [ 111. Kekule structures are a basis for several forms of valence-bond (VB ) resonance-theoretic models [ 121 such as the Pauling-Wheland model [ 131, the Simpson-Herndon model [ 14,151, the conjugated-circuit model [ 161, the VB model using only significant VB structures [ 171, the bond-orbital-reso- nance-theory [ 181, the unified VB theory [ 191, etc. Besides, Kekulk structures are important in understanding the mathematical basis of the intimate con- nection between the resonance-theoretic model of Pauling and the simple mo- lecular orbital model of Hiickel [ 201 and in the development of modern chem- istry in which they play a significant role [ 13b,c, 211.

OUTLINE OF THE REDUCED GRAPH MODEL

We depict the o carbon skeletons of benzenoid hydrocarbons by benzenoid graphs [ 221. The benzenoid graph G represents a connected subsection of an infinite hexagonal lattice 2% The infinite hexagonal lattice H is a planar bi- partite infinite 3-regular graph [ 221. Three disjunctive sets of parallel edges arranged in rows are present in the lattice H. We can arbitrarily choose one of these three sets and call it vertical and the remaining two we can denote, in two different ways, as left and right diagonal. The lattice H with edges labelled as vertical, left diagonal and right diagonal is called the oriented hexagonal lattice. Horizontal rows of hexagons in the oriented lattice Hare called levels of lattice H.

The oriented hexagonal lattice H may be transformed into the trigonal planar lattice T according to the following conversion rules:

V(T) = {vertical edges of H} (1)

E(T) = { (u l,u 2 ) 1 either pair u 1 ,LJ 2 belongs to the same ring in H

or u 1 ,u 2 are connected by a diagonal edge} (21

where the symbols V and E stand for vertices and edges, respectively, in T. By applying these transformation rules each G in H may be converted into a graph G’ that is a part of T and is called a reduced graph. In Fig. 1 we illustrate the transformation of a hexagonal lattice H into a trigonal lattice T, simultane- ously changing the representation of a benzenoid hydrocarbon from the ben- zenoid graph G to the reduced graph G’ .

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formation

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Fig. 1. The transformation of the hexagonal lattice H into the trigonal lattice T and the simulta- neous conversion from the benzenoid graph (G) to the reduced graph (G’) representation of benzocoronene.

KEKUL$-STRUCTURE COUNT

Kekuld structures of benzenoid hydrocarbons may be depicted by Kekule graphs [24]. Kekule graphs are isomorphic to l-factors [23]. l-Factor of G is a graph F such that: (i ) F is a spanning disconnected subgraph of G; and (ii) the components of F are only K2 graphs, i.e. F is a l-regular graph.

We investigate the l-factors of G in the oriented lattice H. It is evident that whenever we determine which of the vertical edges belong to the l-factors of G and which do not, this completely determines the assignment of other edges. We need, therefore, work only with the vertices of G’ . Note that the number of l-factors is invariant on the orientation of G. For convenience we orient the benzenoid graphs under consideration north-south. Peaks and valleys are ver- tices of valency two on the northern and southern parts of the periphery of G [6a]. To retain the concept of peaks and valleys, we can add purely formally above and/or below each last horizontal edge in G’ a dummy vertex and thus create an upper and/or lower triangle. This form of the reduced graph is named the complete reduced graph and is denoted as 8. We will consider only this type of the reduced graph in the present report. In G, the set of added vertices to G’ to form upper triangles is denoted by 9 and these vertices are called peaks. Similarly, the set of added vertices to G’ to form lower triangles is de- noted by Y and these vertices are called valleys. As an illustrative example of G, we give the complete reduced graph of benzocoronene in Fig. 2.

A sequence of adjacent vertices from a peak to a valley in G, connected only with diagonal edges is called vertical or self-avoiding peak-to-valley path [ 51. No other types of paths are required in G. For our purpose two theorems are important.

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Fig. 2. The complete reduced graph 6 of benzocoronene. Peaks and valleys are denoted by black dots.

Theorem 1: G has a l-factor if, and only if, there exists a l-l mapping f: 9-V such that for each pe 9 there exists a path to valley (a peak-to-valley path), f(p) E V, and these paths are pairwise disjunctive (self-avoiding paths), i.e. no two paths have a common vertex.

The proof of this theorem is given elsewhere [ 3e]. This is a very basic theo- rem because it gives the necessary and sufficient condition for a graph G to possess a l-factor. A related statement can be found in the seminal work by Gordon and Davison [4] supported only by a pictorial representation for the case of a single peak and a single valley.

Theorem 2: If (P) ij is the number of self-avoiding peak-to-valley paths in G starting at pi (pie 9’ ) and ending at ui (Uie V), then

K(G)=detlP] (3)

where K(G) is the number of l-factors (Kekule structures) of G whose com- plete reduced graph is G, whilst P is a matrix containing as elements the self- avoiding peak-to-valley path counts.

The above theorem was introduced and proved by John, Rempel and Sachs [ 61 for benzenoid graphs and the same type of proof applies to the correspond- ing complete reduced graphs.

The count of self-avoiding peak-to-valley paths is a time-consuming com- binatorial problem. One way to handle it is by using the Pascal recurrence algorithm [ 5,251. The Pascal recurrence algorithm simply makes counts (P) ij of the uncoordinated paths such that at the i-th vertex of G one writes in the sum of the numbers at immediately preceding vertices above it. The initiating value at the peak is 1.

APPLICATION

The procedure, based on the reduced graph model, for the Kekule-structure count for benzenoid hydrocarbons consists of the following steps.

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(a) Represent a benzenoid hydrocarbon by the corresponding benzenoid graph G and orient it north-south.

(b) Transform G into-a complete reduced graph 6. (c)Make n copies of G with n being the number of peaks. (d) Count for each copy the number of self-avoiding peak-to-valley paths

using the Pascal recurrence algorithm. (e) Apply the John-Rempel-Sachs counting theorem to obtain the number

of l-factors (Kekule structures) of G (benzenoid hydrocarbon). We take here benzocoronene as an illustrative example for the above pro-

cedure. The benzenoid graph corresponding to benzocoronene is given in Fig. 1. The complete reduced graph G of benzocoronene is given in Fig. 2 and in Fig. 3 we give the count of self-avoiding peak-to-valley paths in two copies of G using the Pascal recurrence algorithm. We made two copies of G because there are two peaks in G.

Since there are two peaks and two valleys, the count of paths leads to 2 x 2 P matrix and the following number of l-factors (Kekule structures) of G:

K(G) =det I I 1 i =54-20~34

Because a-set of self-avoiding peak-to-valley uncoordinated paths corre- sponds to one l-factor these paths can be easily converted into Kekule struc- tures. The size of the set is determined by the number of peaks i.e. the number of copies of G. The generation procedure is based on the following simple rule: the vertices in the set of self-avoiding peak-to-valley paths in G correspond to single bonds in a given Kekule structure of the related benzenoid hydrocarbon. A set of self-avoiding peak-to-valley paths may be transformed into a corre- sponding Kekule structure in three steps: (i) give the graphical representation of the set of self-avoiding peak-to-valley uncoordinated paths over G; (ii) transform G with given paths into a hexagonal structure with allocated single bonds corresponding to the positions of vertices in the uncoordinated paths; and (iii) convert this structure into a corresponding Kekuld structure by add- ing double bonds in the appropriate positions.

Fig. 3. The counts of self-avoiding peak-to-valley paths in two copies of the complete reduced graph corresponding to henzocoronene. The relevant peak and valleys in each copy are indicated by black dots.

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ti) (ii) (iii)

Fig. 4. The conversion of two arbitrarily selected self-avoiding peak-to-valley uncoordinated paths over G into the corresponding Kekul6 structure of benzocoronene in three steps as described in the text.

As an explanatory example we constructed one Kekuld structure of benzo- coronene from the corresponding two self-avoiding peak-to-valley uncoordi- nated paths (see Fig. 4).

The above approach may be extended to coronoid hydrocarbons. However, the extended procedure is not as simple and attractive as the procedure de- scribed for benzenoid hydrocarbons.

CONCLUDING REMARKS

The reduced graph model is improved and can be used in the present form to enumerate efficiently the Kekule structures of benzenoid hydrocarbons of moderate size. The improvement in the model is related to the introduction of the potent Pascal recurrence algorithm for counting self-avoiding peak-to-val- ley paths. The conceptual advantage of the reduced graph (trigonal) represen- tation over the hexagonal representation of benzenoid hydrocarbons is that the self-avoiding peak-to-valley paths can now be traced in the true graph- theoretical sense, unlike when the hexagonal network is used. One way of prov- ing the Pascal recurrence algorithm is to use a square-planar lattice [ 51 which is a sub-lattice of the trigonal lattice used in this work.

The other improvement in the model is connected with the use of the pow- erful John-Rempel-Sachs theorem by which the Kekule-structure count is linked in an elegant manner with the counts of self-avoiding peak-to-valley paths.

ACKNOWLEDGEMENT

This research was supported in part by the Federal Committee for Devel- opment, Grant No. P-339.

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