Closed-form formulas for the Zhang–Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations

Discrete Applied Mathematics 198 (2016) 101–108

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Closed-form formulas for the Zhang–Zhang polynomials of benzenoid structures: Prolate rectangles and their generalizations Chien-Pin Chou a,∗ , Jin-Su Kang b , Henryk A. Witek a a

Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001, University Rd., Hsinchu City 30010, Taiwan, ROC b

Institute of Business Management, National Chiao Tung University, 118, Chung-Hsiao W. Rd., Taipei City 10044, Taiwan, ROC

article

info

Article history: Received 18 September 2014 Received in revised form 4 May 2015 Accepted 21 June 2015 Available online 10 July 2015 Keywords: Perfect matching Clar cover Clar structure Zhang–Zhang polynomial

abstract We show that the Zhang–Zhang (ZZ) polynomial of a benzenoid obtained by fusing a parallelogram M (m, n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. It seems possible to extend this important result also to cases where both fused structures are arbitrary Kekuléan benzenoids. Formal proofs of explicit forms of the ZZ polynomials for prolate rectangles Pr (m, n) and generalized prolate rectangles Pr ([m1 , m2 , . . . , mn ], n) follow as a straightforward application of the general  theory, giving ZZ (Pr (m, n), x) = (1 + (1 + x) · m)n and n ZZ (Pr ([m1 , m2 , . . . , mn ], n), x) = k=1 (1 + (1 + x) · mk ). © 2015 Elsevier B.V. All rights reserved.

1. Introduction Zhang–Zhang (ZZ) polynomials [20,22,18,21,12], sometimes referred to as Clar cover polynomials, are combinatorial objects used to enumerate conceivable Clar covers of benzenoid structures. They enable chemists to estimate the resonance energy of aromatic compounds and to predict plausible reaction sites using the concept of aromaticity that can be directly derived from ZZ polynomials [13,14,9,16]. For long time, the ZZ polynomials were known only for relatively small and structurally simple benzenoids owing to prohibitive cost of their determination. In our previous work [5], we reported an automatized algorithm and the resulting computer program (ZZCalculator) for computing ZZ polynomials of arbitrary pericondensed benzenoids with up to 500 vertices and arbitrary catacondensed benzenoids with up to 10,000 vertices. We have shown that ZZ polynomials computed for a number of consecutive members of a given family of benzenoids can be used to deduce closed-form formulas for the whole family. Such formulas for several classes of benzenoids have been reported. [5,3] The derived closed-form ZZ polynomial formulas are important for practical purposes but also can play an important role in understanding asymptotic behavior of large aromatic systems. However, the heuristic approach used by us for deriving closed-form ZZ polynomial formulas despite many advantages has some drawbacks. First, calculating ZZ polynomials of large pericondensed benzenoid systems can be costly. A good example here would be the application of our algorithm to determination of ZZ polynomials for the D6h hexagons O(n, n, n), for which treating systems larger than O(8, 8, 8) is practically infeasible as the total number of Clar covers easily exceeds 1040 . Second, the heuristic approach furnishes no formal proof for the discovered closed-form formulas, which consequently may fail to reproduce the ZZ polynomials for large

∗

Corresponding author. E-mail address: [email protected] (C.-P. Chou).

http://dx.doi.org/10.1016/j.dam.2015.06.020 0166-218X/© 2015 Elsevier B.V. All rights reserved.

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members of a given family not used in the parameterization process. To resolve these problems, we have recently developed [6,8] a graphical ZZDecomposer toolkit applicable for formal recursive decompositions [12] of ZZ polynomials for arbitrary benzenoid structures. The correctness of most of the heuristically derived closed-form ZZ polynomial formulas reported in our previous work [5,3] has been confirmed formally with ZZDecomposer [6,8,4,7,17]. We anticipate that the appearance of ZZDecomposer is an important step forward in the development and further refinement of the theory of ZZ polynomials and cube polynomials, which seem to be intimately related [19,1]. In the preceding paper [7] we proved (Theorem 7 of [7] given again here as Theorem 1) that the ZZ polynomial of a benzenoid M (m, n)∥M (m′ , n′ ), obtained by fusing two parallelograms [11] M (m, n) and M (m′ , n′ ), is equal to the product of the ZZ polynomials of the individual components. This important result has been used in formal derivation of the closed-form formulas of the ZZ polynomials of chevrons and generalized chevrons (Eqs. (16) and (21) of [7], respectively). In the current paper, we extend this result to structures M (m, n)∥ABC , in which only one of components is a parallelogram M (m, n), while the other one (ABC ) is a completely arbitrary (Kekuléan or non-Kekuléan) benzenoid structure. This extension (Theorem 2) gives us a formal basis for deriving ZZ polynomials for prolate rectangles Pr (m, n) and generalized prolate rectangles Pr ([m1 , m2 , . . . , mn ], n). The possibility of extending this result to a benzenoid obtained by fusing two arbitrary Kekuléan structures is discussed (Conjecture 5). The practical application of the formulated theory is a formal proof of the explicit form of ZZ polynomials of prolate rectangles Pr (m, n) and generalized prolate rectangles Pr ([m1 , m2 , . . . , mn ], n) (Theorems 3 and 4, respectively), derived by us previously in a heuristic fashion. 2. Properties of Zhang–Zhang polynomials In this section, we review those properties of ZZ polynomials that are relevant in the context of the current study. More details about ZZ polynomials and their basic properties can be found elsewhere [20,22,18,21,12,5,3,10]. The Zhang–Zhang polynomial ZZ (S , x) of a benzenoid S is defined as ZZ (S , x) =

Cl 

C i xi ,

(1)

i=0

where Cl is the Clar number of a given benzenoid S and Ci is the number of conceivable Clar covers of order i. A Clar cover of order i contains exactly i aromatic sextets. The ZZ polynomial can be computed recursively [12] using the following three important properties: Property 1. Let the vertices A and B of a graph S, both either of order 2 or 3, be connected by an edge AB. If the edge AB does not belong to any hexagonal ring, then the ZZ polynomial of the graph S can be expressed as a sum of ZZ polynomials of two simpler subgraphs, S − AB and S − A − B, i.e., the graph S with the edge AB deleted and the graph S with the vertices A and B removed [20,12,5,7]. Property 2. Let the vertices A and B of a graph S, both either of order 2 or 3, be connected by an edge AB. If the edge AB belongs to a single hexagonal ring s, then the ZZ polynomial of the graph S can be expressed as a sum of ZZ (S − AB, x), ZZ (S − A − B, x), and x · ZZ (S − s, x), where S − AB and S − A − B have the same meaning like in Property 1 and S − s denotes the graph S with the six vertices belonging to the ring s removed [20,12,5,7]. Property 3. Let the graph S consist of two disconnected fragments, S1 and S2. Then, the ZZ polynomial of S can be expressed as the product of the ZZ polynomials of the disconnected fragments, S1 and S2 [20,12,5,7]. These three properties are essential for understanding and proving Theorems 1 and 2, which state that the ZZ polynomials of the fused structures M (m, n)∥M (m′ , n′ ) and M (m, n)∥ABC , respectively, can be expressed as products of the ZZ polynomials of their disconnected components. 3. ZZ polynomial of two fused parallelograms M (m, n)∥M (m′ , n′ ) A fused benzenoid M (m, n)∥M (m′ , n′ ) can be obtained from two parallelograms M (m, n) and M (m′ , n′ ) placed in the same hexagonal lattice and connected by one or more additional edges without adding any additional vertices. Two of many possible ways of fusing the parallelogram M (3, 8) with M (2, 5) are illustrated in Fig. 1. For convenience, we follow the same convention and notation as in [15]. The vertices in M (m, n) which are located in the first layer and may be used for drawing additional edges connecting both parallelograms are labeled as k1 , k2 , k3 , . . . , kn , and the vertices located on the second layer are labeled as l0 , l1 , l2 , l3 , . . . , ln ; details are shown in Fig. 2. When performing graph decomposition of the fused benzenoid M (m, n)∥M (m′ , n′ ) with respect to the additional edges connecting both parallelograms, the substructures (subgraphs) can be recursively generated (Property 1 or 2) by one of the following operations: (i) the edge between the vertices ki and k′i is deleted, leaving both vertices untouched, or (ii) the vertices ki and k′i are removed together with the edge connecting them, or (iii) the vertices ki , li , ki+1 and k′i , l′i , k′i+1 are removed together with the edges ki k′i and ki+1 k′i+1 ; the last operation is permissible only when both edges, ki k′i and ki+1 k′i+1 , are present in the fused benzenoid at the given stage of the decomposition process.

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Fig. 1. Two of many possible ways of fusing two parallelograms M (3, 8) and M (2, 5).

Fig. 2. Schematic classification of peripheral vertices in the structure M (m, n).

Theorem 1 ([7]). Let the structure M (m, n)∥M (m′ , n′ ) be a benzenoid obtained by fusing two aligned parallelograms M (m, n) and M (m′ , n′ ) with a certain number p of new edges connecting pairs of vertically aligned vertices (ki , k′i ). Then, the ZZ polynomial of the fused structure M (m, n)∥M (m′ , n′ ) is equal to the product of the ZZ polynomials of its original components ZZ (M (m, n)∥M (m′ , n′ ), x) = ZZ (M (m, n), x) · ZZ (M (m′ , n′ ), x).

(2)

The complete proof of Theorem 1, presented in [15], is quite technical and is based on a number of auxiliary lemmas. Here, only an outline of the proof is given in order to facilitate the proof of Theorem 2 presented in Section 4. Recursive decomposition of the structure M (m, n)∥M (m′ , n′ ) with respect to the additional edges connecting both parallelograms, which leads to a complete separation of both parallelogram-based substructures, produces a large number of disconnected ˜ j (m, n) · M ˜ j (m′ , n′ ). Each of such substructures consists of two disconnected fragments, M ˜ j (m, n) and benzenoid structures M

˜ j (m′ , n′ ). Using Properties 1–3 allows us to express the ZZ polynomial of the structure M (m, n)∥M (m′ , n′ ) as a sum M ZZ (M (m, n)∥M (m′ , n′ ), x) =



˜ j (m, n), x) · ZZ (M ˜ j (m′ , n′ ), x) xsj · ZZ (M

(3)

j

where the summation runs over all substructures generated in the recursive decomposition process and sj denotes the

˜ j (m, n) · M ˜ j (m′ , n′ ). The structure M ˜ j (m, n) number of times the operation (iii) was selected in the process of generating M ˜ j (m′ , n′ )) is obtained from M (m, n) (or M (m′ , n′ )) by removing a certain number of vertices ki (or k′i ) and a certain (or M number of vertices li (or l′i ) associated with the sequence of operations (i)–(iii) in the decomposition path leading to the ˜ j (m, n) · M ˜ j m′ , n′ . Note that only in one case, namely when the sequence of operations (i, i, i, . . . , i) was substructure M 



˜ j (m, n) · M ˜ j m′ , n′ , the resulting disconnected fragments M ˜ j (m, n) and M ˜ j m ′ , n′ selected to generate the substructure M 







˜ j (m, n) are not defective. In all other cases at least one vertex was removed from M (m, n) (and M m′ , n ) to generate M 

 ′

˜ j m′ , n′ ). Since the ZZ polynomial of a defective parallelogram M ˜ is necessarily zero by Theorem 6 of [7], all the (and M terms containing the defective parallelograms in the decomposition tree vanish, leaving on the right hand side of Eq. (3) only a single term x0 · ZZ (M (m, n)) · ZZ (M (m′ , n′ )), which proves Theorem 1. 



4. ZZ polynomial of a parallelogram M (m, n) fused with an arbitrary benzenoid structure ABC In this Section, we extend Theorem 1 to a more general case in which the parallelogram M (m, n) is fused not with another parallelogram M (m′ , n′ ) but with an arbitrary benzenoid structure ABC . Fig. 3 shows two examples of such fused benzenoid structures, a parallelogram M (3, 6) fused with a hexagon O(3, 3, 3) and a parallelogram M (3, 7) fused with a multiple zigzag chains Z (4, 4).

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Fig. 3. Two examples of fused benzenoid structures: M (3, 6)∥O(3, 3, 3) (left) and M (3, 7)∥Z (4, 4) (right).

Theorem 2. Let the structure M (m, n)∥ABC be a benzenoid obtained by fusing a parallelogram M (m, n) with an arbitrary benzenoid structure ABC using a certain number p of additional edges connecting some of the vertices ki of M (m, n) and the corresponding vertices in the aligned structure ABC . (For examples, see Fig. 3.) The ZZ polynomial of such fused benzenoid structure M (m, n)∥ABC is equal to the product of the ZZ polynomials of its original components ZZ ((M (m, n)∥ABC ), x) = ZZ (M (m, n), x) · ZZ (ABC , x).

(4)

Proof. The proof of this theorem follows very closely that of Theorem 1. The recursive decomposition of the structure M (m, n)∥ABC with respect to the edges connecting the parallelogram M (m, n) and the arbitrary benzenoid ABC after at ˜ j (m, n) · ABCj′ . Each of such substructures consists of two most p steps yields a large number of disconnected structures M

˜ j (m, n) and ABCj′ . Using Properties 1–3 allows us to express the ZZ polynomial of the structure disconnected fragments, M M (m, n)∥ABC as a sum ZZ (M (m, n)∥ABC , x) =



˜ j (m, n), x) · ZZ (ABCj′ , x), xsj · ZZ (M

(5)

j

where the summation runs over all substructures generated in the recursive decomposition process and sj denotes the

˜ j (m, n) · ABCj′ . The structure M ˜ j (m, n) is number of times the operation (iii) was selected in the process of generating M obtained from M (m, n) by removing a certain number of vertices ki and a certain number of vertices li as stipulated by the ˜ j (m, n) is necessarily defective if at operations (i)–(iii) selected in the decomposition process. We note that the structure M least one of the operations (ii) or (iii) has been selected in the decomposition process. We have shown previously (Theorem ˜ (m, n), x) vanishes. Therefore, all summands but one in 6 of [7]) that the ZZ polynomial of a defective parallelogram ZZ (M ˜ Eq. (5) vanish because of the multiplicative term ZZ (Mj (m, n), x), reducing the sum to a single surviving decomposition path in which the operation (i) has been selected at every decomposition step. In this decomposition path no vertex has been ˜ j (m, n) = M (m, n) and ABCj′ = ABC for this particular removed from neither of the structures M (m, n) and ABC , giving M decomposition path j. Since the operation (i) has been selected all the time, sj = 0, and we have ZZ (M (m, n)∥ABC , x) = x0 · ZZ (M (m, n), x) · ZZ (ABC , x),

(6)

which justifies the thesis of our theorem. Note that the proof presented here does not assume any particular form of the arbitrary structure ABC , which can be Kekuléan or non-Kekuléan.  5. ZZ polynomials of prolate rectangles Pr (m, n) and their generalizations In previous work [3], we have reported closed-form formulas for ZZ polynomials of prolate rectangle Pr (m, n) and their generalizations Pr ([m1 , m2 , . . . , mn ], n). Intuitively, these benzenoid structures can be conceived as multiply fused polyacenes and therefore their ZZ polynomials should be closely related to the ZZ polynomials of polyacenes. Theorems 3 and 4 give a formalization of these intuitive concepts, providing us with formally correct expressions for the ZZ polynomials of these two important classes of benzenoids. Theorem 3. ZZ polynomial of a prolate rectangle Pr (m, n) is given by ZZ (Pr (m, n), x) = (1 + (1 + x) · m)n .

(7)

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105

Proof (By induction on n). In the case n = 1, Pr (m, 1) reduces to the polyacene L(m), for which the ZZ polynomial is given by m(1 + x) + 1 [20], satisfying the base case. Let us now assume that ZZ (Pr (m, k), x) = (m(1 + x) + 1)k for all k ≤ n. We notice that Pr (m, n + 1) can be considered as Pr (m, 1)∥Pr (m, n) and since Pr (m, 1) = M (m, 1) = L(m), we have by Theorem 2 ZZ (Pr (m, n + 1), x) = ZZ (Pr (m, 1)∥Pr (m, n), x) = ZZ (M (m, 1)∥Pr (m, n), x) = ZZ (M (m, 1), x) · ZZ (Pr (m, n), x)

= (m(1 + x) + 1)n+1 which proves the inductive step.

(8)



Theorem 4. ZZ polynomial of a generalized prolate rectangle Pr ([m1 , m2 , . . . , mn ], n) is given by ZZ (Pr ([m1 , m2 , . . . , mn ], n), x) =

n 

(mj (1 + x) + 1).

(9)

j =1

Proof (By induction on n). In the case n = 1, Pr ([m1 ], 1) reduces to the polyacene L(m1 ) for which the ZZ polynomial is given k by m1 (1 + x)+ 1, [20] satisfying the base case. Let us now assume that ZZ (Pr ([m1 , m2 , . . . , mk ], k), x) = j=1 (mj (1 + x)+ 1) for all k ≤ n. We notice that Pr ([m1 , m2 , . . . , mn , mn+1 ], n + 1) can be considered as Pr ([mn+1 ], 1)∥Pr ([m1 , m2 , . . . , mn ], n) and since Pr ([mn+1 ], 1) = M (mn+1 , 1) = L(mn+1 ), we have by Theorem 2 ZZ (Pr ([m1 , m2 , . . . , mn , mn+1 ], n + 1), x) = ZZ (Pr ([mn+1 ], 1)∥Pr ([m1 , m2 , . . . , mn ], n), x) = ZZ (M (mn+1 , 1)∥Pr ([m1 , m2 , . . . , mn ], n), x) = ZZ (M (mn+1 , 1), x) · ZZ (Pr ([m1 , m2 , . . . , mn ], n), x)

=

n +1  

mj (1 + x) + 1



(10)

j=1

which proves the inductive step.



Theorems 3 and 4 provide formal proofs of Eqs. (62) and (63) of [3], respectively, which were derived in our previous work using heuristic reasoning. Note finally that Eq. (7) appeared originally in [21] (in the text of proof of Theorem 10) but no formal demonstration of its (intuitively obvious) validity was provided. Eq. (7) can be conveniently transformed to additive form giving explicit information about the number ck of Clar covers with exactly k aromatic sextets ZZ (Pr (m, n), x) = (m(1 + x) + 1)n n    n mk (1 + x)k = k k=0

=

n    n

k

k=0



(1 + m)n−k xk .  

(11)

ck

Analogous additive formulas for Pr ([m1 , m2 , . . . , mn ], n) are somewhat more complicated and involve the theory of symmetric polynomials (see for example [15]). We first transform Eq. (9) in the following way ZZ (Pr ([m1 , m2 , . . . , mn ], n), x) =

n 

(mj (1 + x) + 1)

j =1

=

n 

(mj + 1) ·

j =1





 n   mj x+1 . mj + 1 j =1

(12)



M

We can express now Eq. (9) in additive form; we have ZZ (Pr ([m1 , m2 , . . . , mn ], n), x) =

n 

ek (m1 , m2 , . . . , mn )(1 + x)k

k=0

=

n 

 Mek

k=0



m1

,

m2

m1 + 1 m2 + 1

 ck

,...,

mn



mn + 1



xk

(13)

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a

b

Fig. 4. Two examples of prolate benzenoid structures: (a) prolate rectangle Pr (5, 4) and (b) a generalized prolate benzenoid structure Pr ([6, 8, 8, 9], 4).

where M has been defined in Eq. (12) and ek (z1 , z2 , . . . zn ) are the elementary symmetric functions, defined as ek (z1 , z2 , . . . , zn ) =



zj1 · zj2 · · · · · zjk

(14)

1≤j1
for which the expressions appearing on the left hand-side of Eq. (12) are simply generating functions. Two examples of prolate structures, Pr (5, 4) and Pr ([6, 8, 8, 9], 4), are presented in Fig. 4 for quick, visual reference. By Theorems 3 and 4, the ZZ polynomial of these structures can be computed as ZZ (Pr (5, 4), x) = (5x + 6)4

= 1 + 20(1 + x) + 150(1 + x)2 + 500(1 + x)3 + 625(1 + x)4 = 1296 + 4320x + 5400x2 + 3000x3 + 625x4

(15)

ZZ (Pr ([6, 8, 8, 9], 4), x) = (6x + 7)(8x + 9) (9x + 10) 2

= 1 + 31(1 + x) + 358(1 + x)2 + 1824(1 + x)3 + 3456(1 + x)4 = 5670 + 20043x + 26566x2 + 15648x3 + 3456x4 .

(16)

Results obtained with equivalent formulas (Eq. (11) for Pr (5, 4) and Eqs. (12) and (13) for Pr ([6, 8, 8, 9], 4) are consistent and reproduce the brute force calculations performed with the ZZ Calculator [5]. 6. ZZ polynomials of structures obtained by fusing two arbitrary benzenoids We have further noticed, mainly on the basis of extensive numerical experiments, that Theorems 1 and 2 can be further generalized to structures obtained by fusing two arbitrary Kekuléan benzenoids, say A and B. This observation can be formalized as the following. Conjecture 5. Let A and B be two arbitrary Kekuléan benzenoids. Let the structure A∥B be obtained by fusing A and B, i.e., first arranging them in a way that a certain border of A is parallel to a certain border of B and subsequently drawing a number of additional edges between the adjacent, vertically aligned pairs of vertices ki and k′i , where ki ∈ A and k′i ∈ B. (Some exemplary structures obtained in that way are shown in Fig. 5. For additional details on fusing two benzenoids, see Section 3 and Fig. 2.) Then, the ZZ polynomial of such a fused benzenoid structure A∥B is equal to the product of the ZZ polynomials of its original components ZZ (A∥B, x) = ZZ (A, x) · ZZ (B, x).

(17)

No formal proof of this conjecture can be offered here but, nevertheless, we communicate it to stimulate other researcher in the field to look for the proof of this important result, which seems to be generally valid. Note that another way of expressing Conjecture 5 is saying that the structure A∥B is essentially disconnected and the edges added in the process of fusing must necessarily be fixed. It is important to discuss the assumptions of this conjecture. It is essential that at least one of the structures A or B is Kekuléan, since it is easy to find a counterexample of a composite structure which is Kekuléan and has a non-vanishing ZZ polynomial but its disconnected components are non-Kekuléan and have vanishing ZZ polynomials; some examples are shown in Fig. 6. It is not clear if both of the disconnected fragments need to be Kekuléan in order for the conjecture to hold. It also seems important that the fusing of the structures A and B is performed only with respect to a single border of A and a single border of B as we have observed that Conjecture 5 does not generalize to situations when the additional edges responsible for fusing the structures A and B are not parallel to each other (for an example of such a structure, see Fig. 7). Other issue which requires further studies is the ‘‘simple connectedness’’ of the resulting fused structure, as it seems that Conjecture 5 does not hold in situations when the fusing introduces a hole in the structure A∥B. The term ‘‘hole’’ refers here to missing vertices and not to missing edges; missing edges seem not to cause any complications. The result (Conjecture 5) anticipated by us from

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107

Fig. 5. Example of benzenoid structures obtained by fusing two arbitrary Kekuléan structures.

a

b

c

Fig. 6. Three examples of fusing two non-Kekuléan benzenoid structures to produce Kekuléan benzenoids. (a) Oblate rectangle Or (1, 2) obtained by fusing two well-known non-Kekuléan triangulenes. (b) A benzenoid obtained by fusing two non-Kekuléan (odd number of atoms) defective parallelograms. (c) A benzenoid obtained by fusing two non-Kekuléan G structures studied by D. Chen et al. [2].

a

b

c

Fig. 7. Benzenoid structures obtained by fusing two multiple polyacenes L(5, 2) and L(3, 2) in three different ways. The ZZ polynomials of (a) and (b) can be decomposed into the ZZ polynomial of their separated components using Conjecture 5. However, (c) is an example that Conjecture 5 does not hold, when the extra fusing edges are not parallel.

numerical experiments may become a mile stone in further development of the theory of ZZ polynomials. Therefore, we decided to include it in this work in its current incomplete form, hoping that other researchers will be luckier and will have more insight to provide a rigorous statement of Conjecture 5 and an appropriate proof satisfying all formal requirements. 7. Conclusion We formally demonstrate (Theorem 2) that the ZZ polynomial of a benzenoid obtained by fusing a parallelogram M (m, n) with an arbitrary benzenoid structure ABC can be simply computed as a product of the ZZ polynomials of both fragments. This

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