# How to Find the Fries Structures for Benzenoid Hydrocarbons

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Method and Applications

**A**, the elements of which, a

_{ij}, represent CC bonds. The adjacency matrix is schematically presented in Figure 4a.

**A**(Figure 4b) contains the Pauling bond orders [30]. The Hadamard product [31,32] (for definition, see Eq. 1) of matrices

^{−1}**A**and

**A**, gives matrix

^{−1}**P**, which contains exclusively Pauling bond orders, p

_{1}_{rs}.

**C**=

**A**○

**B**c

_{ij}= a

_{ij}·b

_{ij}

_{ij}

_{,}b

_{ij}, and c

_{ij}are elements of (n×m) matrices

**A**,

**B**, and

**C**, respectively.

_{1}, k

_{2}, or k

_{3}, matrices

**K**(1),

**K**(2), and

**K**(3), which contain full information about the positions of double bonds. These matrices are shown in Figure 6.

**K**(1),

**K**(2), and

**K**(3), are so-called self-inverse matrices with the property

**K**matrix representing a canonical structure, $\sqrt{\left|\mathrm{det}K\right|}=1$ because, for one structure, K = 1.

**K**matrix that represents a Fries canonical-structure, we construct from matrices

**A**and

**P**, as presented in Figure 4, a recurrence function, denoted as a Fries structure generating function (FGF). This function is defined in Equation 3.

**A**stands for an adjacency matrix, and

**P**for the matrix of Pauling bond orders. The multiplication follows the Hadamard rule, and n is a number of steps in the recurrence procedure [31,32]. In Figure 7, a graphical illustration is presented of the procedure as applied to the naphthalene molecule.

**K**(2), which represent the Fries structure for naphthalene (Figure 4).

_{n}| = 1. Note that, in subsequent iterations, because of the property given in Equation 2, the

**P**matrix is closer to the

_{n}**K**matrix. The determinant of

**P**is a useful measure of the FGF convergence.

**K**(F), of its Fries structure

**P**

_{n}that is a superposition of matrices

**K**(F

_{1}) and

**K**(F

_{2}) (see Equation 6 and Figure 9 and Figure 10).

_{64}H

_{26}. These structures can be easily transformed into Clar structures with the maximum number of sextets, as shown in Figure 12.

## 3. Conclusions

## Acknowledgements

## References and Notes

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**Figure 1.**Dibenzo[bc,kl]coronene molecule.

**A)**Fries structure. Stars denote Kekulé-type six-membered rings.

**B)**Clar structure with maximal number of isolated sextets.

**Figure 4.**

**A)**The adjacency matrix,

**A**, for naphthalene (1 and 0 are represented by black boxes and white boxes, respectively).

**B)**The inverse matrix

**A**(light-gray boxes and dark-gray boxes correspond to the values of 1/3 and 2/3, respectively; white boxes correspond to negative or zero values).

^{−1}**C)**The Hadamard product of matrices

**A**and

**A**(light-gray boxes and dark-gray boxes correspond to the values of 1/3 and 2/3, respectively; white boxes correspond to values of zero).

^{−1}**Figure 6.**Matrices

**K**(1),

**K**(2), and

**K**(3), which correspond to Kekulé structures k

_{1}, k

_{2}, and k

_{3}. Double CC bonds (represented in a graphical way below each matrix) are denoted by the entry 1.

**Figure 7.**Graphical illustration of the procedure for obtaining the Fries structure of naphthalene in four cycles of iteration. The elements of matrices are shown in a symbolic way as varying degrees of grayness.

**Figure 10.**Illustration of superposition of k

_{1}and k

_{2}with matrices below,

**K**(F

_{1}),

**P**

_{n}, and

**K**(F

_{2}). Solid lines and black boxes visualize p

_{rs}= 1, dashed lines and gray boxes indicate p

_{rs}= 0.5.

**Figure 11.**Fries structures for phenanthrene (A), triphenylene (B), benz[a]pyrene (C), coronene (D), and one large benzenoid hydrocarbon (E) obtained form FGF procedure in 10 cycles of interaction.

**Figure 12.**Clar structure with the maximal number of sextets deduced from the Fries structure for phenanthrene (A), triphenylene (B), benz[a]pyrene (C), coronene (D), and one large benzenoid hydrocarbon (E).

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**MDPI and ACS Style**

Ciesielski, A.; Krygowski, T.M.; Cyrański, M.K.
How to Find the Fries Structures for Benzenoid Hydrocarbons. *Symmetry* **2010**, *2*, 1390-1400.
https://doi.org/10.3390/sym2031390

**AMA Style**

Ciesielski A, Krygowski TM, Cyrański MK.
How to Find the Fries Structures for Benzenoid Hydrocarbons. *Symmetry*. 2010; 2(3):1390-1400.
https://doi.org/10.3390/sym2031390

**Chicago/Turabian Style**

Ciesielski, Arkadiusz, Tadeusz M. Krygowski, and Michał K. Cyrański.
2010. "How to Find the Fries Structures for Benzenoid Hydrocarbons" *Symmetry* 2, no. 3: 1390-1400.
https://doi.org/10.3390/sym2031390