# Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms

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## Abstract

**:**

## 1. Introduction

## 2. Counting Clar Covers

- Kekulé count $\mathbf{K}={\mathbf{c}}_{\mathbf{0}}=\mathrm{ZZ}(\mathbf{B},\mathbf{0})$,
- Clar count $\mathbf{C}={\mathbf{c}}_{\mathbf{0}}+\dots +{\mathbf{c}}_{\mathbf{Cl}}=\mathrm{ZZ}(\mathbf{B},\mathbf{1})$,
- Clar number $Cl=\mathrm{degree}\left(\mathrm{ZZ}\right(\mathbf{B},x\left)\right)$,
- the number of Clar formulas ${c}_{Cl}$,
- and the first Herndon number ${h}_{1}={c}_{1}$ [22].

## 3. ZZ Polynomials of Complex Benzenoid Structures

## 4. Results

#### 4.1. Ribbons $Rb\left(\right)open="("\; close=")">{n}_{1},{n}_{2}-{n}_{1},{m}_{2},{m}_{1}-{m}_{2}$, $Rb\left(\right)open="("\; close=")">{n}_{1},{n}_{2}-{n}_{1},{m}_{2},{m}_{1}-\mu $, and $Rb\left(\right)open="("\; close=")">{n}_{1},{n}_{2}-{n}_{1},{m}_{2},{m}_{1}$

#### 4.2. Two Vertically Overlapping Parallelograms $MvM({m}_{2},{n}_{2},{m}_{1},{n}_{1},\mu ,\nu )$

- We introduce in $MvM$ a system of parallel horizontal lines referred to as the elementary cuts ${I}_{k}$ in the way shown in Figure 5. Each such elementary cut is perpendicular to some vertical edges of $MvM$ and dissects them into halves. The number of elementary cuts introduced in this way is ${n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu -1$. For convenience, we augment this system with two additional elementary cuts, ${I}_{0}$ and ${I}_{{n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu}$ in the way shown in Figure 5.
- The set of vertical edges of $MvM$ intersected by the elementary cut ${I}_{k}$ is referred to as the interface ${i}_{k}$. Each edge belonging to the interface ${i}_{k}$ is referred to as an interface bond. Simple geometrical considerations allow establishing that the number of interfaces in $MvM$ is ${n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu -1$. It is beneficial to augment this set again with two additional empty interfaces ${i}_{0}$ and ${i}_{{n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu}$ in the way shown in Figure 5.
- The set of edges of $MvM$ located at least partially between the elementary cuts ${I}_{k-1}$ and ${I}_{k}$ is referred to as the fragment ${f}_{k}$. $MvM$ has ${n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu $ fragments.
- For $k=1,\dots ,{n}_{2}+{m}_{2}+{n}_{1}+{m}_{1}-\mu -\nu $, ${i}_{k-1}$ is the upper interface of ${f}_{k}$ and ${i}_{k}$ is the lower interface of ${f}_{k}$.
- Interface bonds in each fragment ${f}_{k}$ are numbered from left to right. The leftmost interface edge in ${f}_{k}$ is referred to as ${e}_{\mathrm{first}}\equiv {e}_{0}$ and the the rightmost interface edge in ${f}_{k}$, as ${e}_{\mathrm{last}}$.
- Each fragment ${f}_{k}$ can be assigned an attribute of shape ($\mathtt{W}\equiv \mathrm{wider}$, $\mathtt{N}\equiv \mathrm{narrower}$, $\mathtt{R}\equiv \mathrm{to}-\mathrm{the}-\mathrm{right}$, and $\mathtt{L}\equiv \mathrm{to}-\mathrm{the}-\mathrm{left}$), which is defined in the following way$$\mathrm{shape}\left(\right)open="("\; close=")">{f}_{k}$$Following this convention, it is possible to assign the attribute of shape to the whole structure $MvM$, simply by listing the shape of each fragment from the top to the bottom. For example, the shape sequences for the two structures $MvM$ in Figure 5 are: (a) $\mathtt{WWWWWRRNLWWWLLLNNNNNNN}$ and (b) $\mathtt{WWWWRRRWWWRRNNRRRRNNNNN}$.
- Let us now consider an arbitrary Clar cover $\mathbf{C}$ of $MvM$. For every interface edge e in $MvM$, we define a covering order function $\mathrm{ord}$, which assumes the following values$$\mathrm{ord}\phantom{\rule{4pt}{0ex}}\left(e\right)=\left(\right)open="\{"\; close>\begin{array}{cc}1\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}\exists {K}_{2}\in \mathbf{C}:e\in E\left({K}_{2}\right)\hfill \\ \frac{1}{2}\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}\exists {C}_{6}\in \mathbf{C}:e\in E\left({C}_{6}\right)\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}$$
- The concept of covering order (or briefly: order) can be naturally extended to interfaces. We define the covering order of the interface i as$$\mathrm{ord}\phantom{\rule{4pt}{0ex}}\left(i\right)=\sum _{e\in i}\mathrm{ord}\phantom{\rule{4pt}{0ex}}\left(e\right)$$
- Since the interface ${i}_{0}$ is empty, we naturally have $\mathrm{ord}\phantom{\rule{4pt}{0ex}}\left(\right)open="("\; close=")">{i}_{0}$. The orders of the remaining interfaces can be recursively computed from the First rule of interface theory [51,52], which for an arbitrary Clar cover $\mathbf{C}$ relates the covering order of the interface ${i}_{k}$ to the covering order of the interface ${i}_{k-1}$ and the shape of the fragment ${f}_{k}$ in the following way
- (a)
- If ${f}_{k}$ has the shape $\mathtt{W}$, then $\mathrm{ord}\left({i}_{k}\right)=\mathrm{ord}\left({i}_{k-1}\right)+1$.
- (b)
- If ${f}_{k}$ has the shape $\mathtt{N}$, then $\mathrm{ord}\left({i}_{k}\right)=\mathrm{ord}\left({i}_{k-1}\right)-1$.
- (c)
- If ${f}_{k}$ has the shape $\mathtt{R}$ or $\mathtt{L}$, then $\mathrm{ord}\left({i}_{k}\right)=\mathrm{ord}\left({i}_{k-1}\right)$.

- The interface orders obtained in this way are actually independent of the choice of the Clar cover $\mathbf{C}$, as they are completely determined by the condition $\mathrm{ord}\left(\right)open="("\; close=")">{i}_{0}$ and the shape sequence $\mathtt{WWWWWRRNLWWWLLLNNNNNNN}$. Therefore, the interface orders are identical for every Clar cover $\mathbf{C}$ of $MvM$ and can be treated as an inherent property of $MvM$ allowing enumerating and constructing the set of Clar covers of $MvM$. The interface covering orders computed in this way are listed in red for the two structures $MvM$ shown in Figure 5.
- The number of interface bonds in every non-empty interface ${i}_{k}$ of $MvM$ is larger by 1 from the order of this interface, $\mathrm{ord}\left({i}_{k}\right)$, as can be easily seen from Figure 5. This property holds for a general structure of this type, as both the interface orders and the numbers of interface bonds in consecutive interfaces depend in the same manner on the shape of the fragment between the interfaces, except for the first and the last fragment.
- An explicit formula for the interface order $\mathrm{ord}\left({i}_{k}\right)$ as a function of the interface number k is somewhat cumbersome. It can be shown that$$\mathrm{ord}\left({i}_{k}\right)={p}_{1}\left(k\right)+{p}_{2}\left(k\right)-{p}_{12}\left(k\right),$$$$\begin{array}{ccc}\hfill {p}_{1}\left(k\right)& =& \left(\right)open="\{"\; close>\begin{array}{cc}{\displaystyle 0}\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k{m}_{2}+{n}_{2}+1-\mu -\nu \hfill \\ {\displaystyle k-{m}_{2}-{n}_{2}+\mu +\nu}\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}{m}_{2}+{n}_{2}+1-\mu -\nu \le k{m}_{2}+{n}_{2}-\mu -\nu +\mathrm{min}({m}_{1},{n}_{1})\hfill \\ {\displaystyle \mathrm{min}({m}_{1},{n}_{1})}\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}{m}_{2}+{n}_{2}-\mu -\nu +\mathrm{min}({m}_{1},{n}_{1})\le k\le {m}_{2}+{n}_{2}-\mu -\nu +\mathrm{max}({m}_{1},{n}_{1})\hfill \\ {\displaystyle {m}_{2}+{n}_{2}+{m}_{1}+{n}_{1}-\mu -\nu -k}\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k{m}_{2}+{n}_{2}-\mu -\nu +\mathrm{max}({m}_{1},{n}_{1})\hfill \end{array}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {p}_{2}\left(k\right)& =& \left(\right)open="\{"\; close>\begin{array}{cc}k\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k\mathrm{min}({m}_{2},{n}_{2})\hfill \\ \mathrm{min}({m}_{2},{n}_{2})\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}\mathrm{min}({m}_{2},{n}_{2})\le k\mathrm{max}({m}_{2},{n}_{2})\hfill \\ {m}_{2}+{n}_{2}-k\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}\mathrm{max}({m}_{2},{n}_{2})\le k\le {m}_{2}+{n}_{2}-1\hfill \\ 0\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k{m}_{2}+{n}_{2}-1\hfill \end{array}\hfill \end{array}$$$$\begin{array}{ccc}\hfill {p}_{12}\left(k\right)& =& \left(\right)open="\{"\; close>\begin{array}{cc}0\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k{m}_{2}+{n}_{2}+1-\mu -\nu \hfill \\ k-{m}_{2}-{n}_{2}+\mu +\nu \hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}{m}_{2}+{n}_{2}+1-\mu -\nu \le k{m}_{2}+{n}_{2}-\mu \hfill \\ \mathrm{min}(\mu ,\nu )\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}{m}_{2}+{n}_{2}-\mu \le k{m}_{2}+{n}_{2}-\nu \hfill \\ {m}_{2}+{n}_{2}-k\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}{m}_{2}+{n}_{2}-\nu \le k\le {m}_{2}+{n}_{2}-1\hfill \\ 0\hfill & \mathrm{when}\phantom{\rule{4pt}{0ex}}k{m}_{2}+{n}_{2}-1\hfill \end{array}\hfill \end{array}$$

- Let us assume that a $1+\dots +1+0$ partition was selected for ${i}_{{n}_{2}+{m}_{2}-\mu}$ with the single bond in position $k\in [0,\dots ,{o}_{\mu}]$. Somewhat involved geometric considerations show that the range of indices of non-fixed interface bonds in ${i}_{{n}_{2}+{m}_{2}-\nu}$ associated with this choice is given by $[\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k)]$.
- Consequently, a single bond in interface ${i}_{{n}_{2}+{m}_{2}-\mu}$ in position k permits placing a single bond in interface ${i}_{{n}_{2}+{m}_{2}-\nu}$ in position $l\in [\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k)]$ or permits placing an aromatic ring in hexagon $l\in [1+\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k)]$.
- Let us assume now that a $1+\dots +1+\frac{1}{2}+\frac{1}{2}$ partition was selected for ${i}_{{n}_{2}+{m}_{2}-\mu}$ with the aromatic ring in hexagon $k\in [1,\dots ,{o}_{\mu}]$. Again, geometric considerations show that the range of indices of non-fixed interface bonds in ${i}_{{n}_{2}+{m}_{2}-\nu}$ associated with this choice of covering for ${i}_{{n}_{2}+{m}_{2}-\mu}$ is given by $[\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k+1)]$.
- Consequently, an aromatic ring in hexagon k of interface ${i}_{{n}_{2}+{m}_{2}-\mu}$ permits placing a single bond in interface ${i}_{{n}_{2}+{m}_{2}-\nu}$ in position $l\in [\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k+1)]$ or permits placing an aromatic ring in hexagon $l\in [1+\mathrm{max}(k+{o}_{\nu}-\mu ,0),{o}_{\nu}-\mathrm{max}(0,\nu -k+1)]$.

`for computing the ZZ polynomial of an arbitrary structure $MvM({m}_{2},{n}_{2},{m}_{1},{n}_{1},\mu ,\nu )$ together with several $MvM$ structures and their ZZ polynomials computed in a twofold way: black text reports the ZZ polynomials computed in a brute force manner using ZZDecomposer and the blue text in red frames reports the ZZ polynomials computed using the provided MAPLE procedure. Note that the computational time needed to evaluate the ZZ polynomials of larger structures using recursive decompositions with ZZDecomposer exceeds a few minutes, while the MAPLE procedure produces identical results instantaneously.`

**MvM**#### 4.3. Two Horizontally Overlapping Parallelograms $MhM({m}_{1},{n}_{1},{m}_{2},{n}_{2},\mu ,\nu )$

#### 4.4. Two Intersecting Parallelograms $MxM({m}_{2},{n}_{2},{m}_{1},{n}_{1},\mu ,\nu )$

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**25 Clar covers of the parallelogram $M(2,3)$ can be constructed by a recursive assignment of single (no covering, symbol S), double (${K}_{2}$ covering, symbol D), or aromatic ring (${C}_{6}$ covering, symbol R) character to the edges of $M(2,3)$ marked with a red circle. Each Clar cover of order 0 (i.e., Kekulé structure, depicted in cyan frames) contributes a term 1 to $\mathrm{ZZ}\left(M\right(2,3),x)$, each Clar cover of order 1 (purple frames) contributes a term x to $\mathrm{ZZ}\left(M\right(2,3),x)$, and each Clar cover of order 2 (orange frames) contributes a term ${x}^{2}$ to $\mathrm{ZZ}\left(M\right(2,3),x)$. The resulting ZZ polynomial of $M(2,3)$ is thus equal to $10+12x+3{x}^{2}$.

**Figure 2.**Brute-force computation of the ZZ polynomial of the $M(10,10)$ parallelogram using ZZDecomposer takes about 14 s on a desktop computer.

**Figure 3.**All types of Kekuléan composite benzenoids that can be formed by overlapping two identically oriented parallelogram-shaped benzenoids, $M({m}_{1},{n}_{1})$ and $M({m}_{2},{n}_{2})$. The parallelograms are depicted using blue and red shading and the overlapping region is depicted in purple.

**Figure 4.**All types of non-Kekuléan composite benzenoids that can be formed by overlapping two identically oriented parallelogram-shaped benzenoids, $M({m}_{1},{n}_{1})$ and $M({m}_{2},{n}_{2})$. The parallelograms are depicted using blue and red shading and the overlapping region is depicted in purple.

**Figure 5.**Elementary cuts (in gray), interface orders (in red), and the numbers of interface bonds (in blue) for each interface ${i}_{k}$ of (

**a**) two vertically overlapping parallelograms $M(7,5)$ and $M(7,10)$ with the intersection region corresponding to the parallelogram $M(4,3)$ and (

**b**) two vertically overlapping parallelograms $M(12,4)$ and $M(13,15)$ with the intersection region corresponding to the parallelogram $M(9,2)$. The interface bonds are defined as the bonds intersected by the elementary cuts. The interface ${i}_{k}$ is defined as the set of interface bonds intersected by the same elementary cut ${I}_{k}$. The depicted-in-blue interface cuts ${I}_{8}$ and ${I}_{9}$ (left panel) and ${I}_{7}$ and ${I}_{14}$ (right panel) are used to partition the studied structures into smaller substructures, which allows us to derive closed-form formulas of their ZZ polynomials. For details, see text.

**Figure 6.**Assigning bond coverings to the interface ${i}_{8}$ of the structure (a) from Figure 5 results in nine different systems of fixed bonds (depicted in gray). Five of these systems correspond to four double bonds and one single bond (upper row, partitions $1+\dots +1+0$) and four of these systems correspond to one aromatic ring and three double bonds (lower row, partitions $1+\dots +1+\frac{1}{2}+\frac{1}{2}$). The non-fixed components of each graph (depicted in black) correspond to two disconnected subgraphs of $MvM$, here both with the shape of a parallelogram. The shapes of the parallelograms are determined by the position of the single bond or by the position of the aromatic ring.

**Figure 7.**Four examples of partially covered structures $MvM(11,6,11,9,9,3)$ show that the non-covered regions consists of three parallelograms with their shapes determined by the location of the single and/or aromatic bonds in the both covered interfaces ${i}_{{n}_{2}+{m}_{2}-\mu}$ and ${i}_{{n}_{2}+{m}_{2}-\nu}$.

**Figure 8.**Maple procedure for computing the ZZ polynomial of an arbitrary structure $MvM\equiv MvM({m}_{2},{n}_{2},{m}_{1},{n}_{1},\mu ,\nu )$ according to Equations (20)–(29). As a numerical verification of correctness of the derived formula, several typical structures $MvM$ is shown together with their ZZ polynomials evaluated in a twofold manner: with ZZDecomposer (in black) and with the provided MAPLE routine (in blue in red frames).

**Figure 9.**The composite benzenoid $MxM({m}_{2},{n}_{2},{m}_{1},{n}_{1},\mu ,\nu )$ shown originally in Figure 3 can be formally treated as a superposition of two parallelograms, $M\left(\right)open="("\; close=")">{m}_{1},{n}_{1}$ (red shading) and $M\left(\right)open="("\; close=")">{m}_{2},{n}_{2}$ (blue shading), or two ribbons, $Rb\left(\right)open="("\; close=")">{n}_{1},\nu ,{m}_{2},\mu $ (red shading) and $Rb\left(\right)open="("\; close=")">{n}_{1},{n}_{2}-{n}_{1}-\nu ,{m}_{2},{m}_{1}-{m}_{2}-\mu $ (blue shading).

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## Share and Cite

**MDPI and ACS Style**

Witek, H.A.; Langner, J.
Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms. *Symmetry* **2020**, *12*, 1599.
https://doi.org/10.3390/sym12101599

**AMA Style**

Witek HA, Langner J.
Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms. *Symmetry*. 2020; 12(10):1599.
https://doi.org/10.3390/sym12101599

**Chicago/Turabian Style**

Witek, Henryk A., and Johanna Langner.
2020. "Clar Covers of Overlapping Benzenoids: Case of Two Identically-Oriented Parallelograms" *Symmetry* 12, no. 10: 1599.
https://doi.org/10.3390/sym12101599