Zhang-Zhang Polynomials of Ribbons

We report a closed-form formula for the Zhang-Zhang polynomial (aka ZZ polynomial or Clar covering polynomial) of an important class of elementary pericondensed benzenoids $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$ usually referred to as ribbons. A straightforward derivation is based on the recently developed interface theory of benzenoids [Langner and Witek, MATCH Commun. Math. Comput. Chem. 84, 143--176 (2020)]. The discovered formula provides compact expressions for various topological invariants of $Rb\left(n_{1},n_{2},m_{1},m_{2}\right)$: the number of Kekul\'e structures, the number of Clar covers, its Clar number, and the number of Clar structures. The last two classes of elementary benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes $O\left(k,m,n\right)$ and oblate rectangles $Or\left(m,n\right)$.


Introduction
Consider a regular pericondensed benzenoid depicted in Fig. 1, which can be fully characterized by specifying four structural parameters: n 1 , n 2 , m 1 , and m 2 . Structures of this type constitute an important family of elementary pericondensed benzenoids and occupy a pronounced position in the general theory of Clar covers among other, highly symmetric structures such as parallelograms M (m, n) [1,2,3,4], parallelogram chains [5], hexagons O (k, m, n) [1,6,7,8,9,10], oblate and prolate rectangles Or (m, n) [11,8,9] and P r (m, n) [12,8,13], chevrons Ch (k, m, n) [1,14,8,4], and generalized chevrons Ch (k, m, n 1 , n 2 ) [4]. In situations when m 1 = n 1 , the structure shown in Fig. 1 is traditionally referred to as a ribbon or a V-shaped benzenoid and has been symbolically denoted by V (k, m, n) [14,15,16], where k = m 1 = n 1 , m = m 1 + m 2 , and n = n 1 + n 2 . Here, we consider a wider class of ribbons, allowing the parameters m 1 and n 1 to be different. We still refer to these structures as ribbons, but we represent them by a new symbol Rb (n 1 , n 2 , m 1 , m 2 ) capable of accommodating the extra new parameter in contrast to V (k, m, n). The Clar theory of generalized ribbons Rb (n 1 , n 2 , m 1 , m 2 )-which we attempt to construct in the current paper-clearly encompasses the Clar theory of regular ribbons V (k, m, n), similarly as before we were able to show [4] that the Clar theory of generalized chevrons Ch (k, m, n 1 , n 2 ) includes as a special case the Clar theory of regular chevrons Ch (k, m, n). The questions we want to answer in the current paper are: (i) How many Kekulé structures exist for Rb (n 1 , n 2 , m 1 , m 2 )? (ii) How many Clar covers can be constructed for Rb (n 1 , n 2 , m 1 , m 2 )? (iii) What is the Clar number of Rb (n 1 , n 2 , m 1 , m 2 )?
(iv) How many Clar structures can be constructed for Rb (n 1 , n 2 , m 1 , m 2 )? (v) What is the ZZ polynomial of Rb (n 1 , n 2 , m 1 , m 2 )? Note that the solution to the last problem is sufficient for answering all the posed here questions, justifying the title and the scope of the current paper.

Preliminaries
A benzenoid is a planar hydrocarbon B consisting of fused benzene rings. From a graph theoretical point, B is defined as a 2-connected finite plane graph such that every interior face is a regular hexagon [17]. A Kekulé structure K is a resonance structure of B constructed using only double bonds [18]. A Clar cover C is a resonance structure of B constructed using double bonds and aromatic Clar sextets [19]. From a graph theoretical point, a Kekulé structure K is a spanning subgraph of B all of whose components are K 2 and a Clar cover C is a spanning subgraph of B whose components are either K 2 or hexagons C 6 . Note that most difficulties originating from this double terminology can be circumvented if one establishes two correspondences: a complete graph on 2 vertices K 2 ≡ double bond and a cycle graph on 6 vertices C 6 ≡ aromatic Clar sextet. The maximal number of hexagons C 6 that can be accommodated in C is referred to as the Clar number Cl of B [19,20]. The Clar covers with Cl aromatic sextets C 6 are referred to as the Clar structures of B [19,20]. The Clar covers with k aromatic sextets C 6 are referred to as the Clar covers of order k. If we represent the number of Clar covers of order k for B by c k , we can define a combinatorial polynomial usually referred to as the Clar covering polynomial of B or the Zhang-Zhang polynomial of B or, shortly, the ZZ polynomial of B. [21,22,23,12] Clearly, the ZZ polynomial of B has the following inviting properties: • The number of Kekulé structures of B is given by K {B} = c 0 = ZZ (B, 0).
• The Clar number of B is given by Cl = deg (ZZ (B, x)).

• The number of Clar structures of B is given by
. These relations demonstrate the claim made at the end of Section 1, where we have written that determination of the ZZ polynomial of B answers most graph-theoretically relevant questions about B. Zhang and Zhang were able to demonstrate that the ZZ polynomials possess a rich structure of recursive decomposition properties [21,22,12], which enable their fast and robust computations in practical applications. (See for example Properties 1-7 in [3].) Consequently, the ZZ polynomial of an arbitrary benzenoid B can be efficiently computed using recursive decomposition algorithms [11,3,24] or determined using interface theory of benzenoids [25,26,27,28]. A useful practical tool for determination of ZZ polynomials is ZZDecomposer [24,9]. With this freely downloadable [29,30] software, one can conveniently define a graph representation corresponding to a given benzenoid B using a mouse drawing pad and subsequently use it to find the ZZ polynomial of B, generate the set of Clar covers of B, and determine its structural similarity to other, related benzenoids.
In the most typical depth-decomposition mode of ZZDecomposer, used below in Fig. 2 to prove Eqs. (2) and (3), ZZDecomposer generates a recurrence relation for the analyzed benzenoid structure, which relates its ZZ polynomial to the ZZ polynomials of structurally related benzenoids and often allows for determination of a closed-form formulas for the whole family of structurally similar benzenoids. Another useful feature of ZZDecomposer is generating vector graphics that can be easily incorporated in publications.

Heuristic determination of the ZZ polynomial from recurrence relations
Before presenting a formal derivation of the ZZ polynomial for the ribbon Rb (n 1 , n 2 , m 1 , m 2 ) in Section 4, first we discuss here a heuristic reasoning suggesting how such formulas can be discovered for a general benzenoid B. This goal can be readily achieved using ZZDecomposer described in Section 2 by considering the first two members of this family of structures, Rb (1, n 2 , m 1 , m 2 ) and Rb (2, n 2 , m 1 , m 2 ), and performing their multi-step recursive decompositions with respect to the covering character of the vertical edges depicted in blue (with a black dot) in Fig. 2. The process of assigning to these edges single bond covering S , double bond covering D , or aromatic ring covering R , as it is demonstrated in Fig. 2, allows us to partition the set of Clar covers of Rb (1, n 2 , m 1 , m 2 ) and Rb (2, n 2 , m 1 , m 2 ) into three and five, respectively, subsets, each of them consisting of a region of fixed bonds separating parallelogram-shaped regions with not fixed bonds. Consequently, the ZZ polynomials of Rb (1, n 2 , m 1 , m 2 ) and Rb (2, n 2 , m 1 , m 2 ) can be expressed by summing over the subsets ZZ polynomials. Moreover, the formulas can be written compactly in terms of the ZZ polynomials of the parallelograms M (m, n) as It is easy to notice regularities and patterns in these formulas. Remembering that ZZ (M (m, 0) , x) = 1, we can rewrite Eqs. (2) and (3) for situations when n 1 ≤ m 1 in the following form The final generalization applies to situations when n 1 > m 1 . It is clear that the ZZ polynomials of Rb (n 1 , n 2 , m 1 , m 2 ) and Rb (m 1 , m 2 , n 1 , n 2 ) should be identical as there is a clear (horizontal mirror reflection) isomorphism between the sets of Clar covers of both structures. Indeed, Eq. (4) reflects this symmetry except for the upper summations limits; it is easy to see that the appropriate change relies on replacing n 1 by min (n 1 , m 1 ). Consequently, the general formula for the ZZ polynomial of Rb (n 1 , n 2 , m 1 , m 2 ) must have the following symmetric form This formula can be further simplified by substituting an explicit form of the ZZ polynomial for the parallelogram M (m, n) Introducing this formula into Eq. (5), we obtain Numerical experiments performed with ZZDecomposer for various values of the parameters n 1 , n 2 , m 1 , and m 2 show that formulas given by Eqs. (9) and (10) are indeed correct. Formal demonstration of correctness of Eqs. (9) and (10) is presented in the next Section; the proof is based on the recently developed interface theory of benzenoids [27,28].
As we mentioned earlier, the ribbon Rb (n 1 , n 2 , m 1 , m 2 ) has been also denoted in the earlier literature by the symbol V (k, m, n), where k = m 1 = n 1 , m = m 1 + m 2 , and n = n 1 + n 2 or by V (k 1 , k 2 , m, n), where k 1 = n 1 , k 2 = m 1 , m = m 1 + m 2 , and n = n 1 + n 2 . Therefore, for consistency, we also give explicit formulas for the ZZ polynomials of V (k, m, n) and V (k 1 , k 2 , m, n) using their structural parameters in the formulas. We have ZZ (V (k, m, n) , x) = (11) Note that analogous formulas for the number of Kekulé structures given by Cyvin and Gutman (as Eq. (19) of [15]) have both of the inner summations evaluated to a closed binomial form. This is indeed possible for Kekulé structures, for which x = 0, where the following binomial identity (Eq. (5.22) of [31]) can be used.
In the case of the ZZ polynomial, Eq. (13) takes on the following form which renders Eqs. (11) and (12) in the hypergeometric form analogous to Eq. (9). These hypergeometric functions reduce to the obvious polynomial form or to Jacobi polynomials, and no other functional identities exist that would allow to express them as some wellknown functions.

Formal derivation of the ZZ polynomial from the interface theory of benzenoids
Consider the ribbon B ≡ Rb (n 1 , n 2 , m 1 , m 2 ) in the orientation shown in Fig. 1. We introduce a system of m 1 + n 2 + m 2 + n 1 − 1 elementary cuts I k intersecting the vertical edges of B in the way shown in Fig. 3. The set of vertical edges intersected by the elementary cut I k is referred to as the interface i k of B. It is convenient to augment the set {i 1 , . . . , i m 1 +n 2 +m 2 +n 1 −1 } of interfaces by two empty interfaces, i 0 and i m 1 +n 2 +m 2 +n 1 , located, respectively, above and below B.
Let us further refer to all the edges and vertices of B located (at least partially) between the elementary cuts I k−1 and I k as the fragment f k of B. We augment the set of fragments with two additional fragments: f 1 including all edges and vertices located (at least partially) above the elementary cut I 1 and f m 1 +n 2 +m 2 +n 1 including all edges and vertices located (at least partially) below the elementary cut I m 1 +n 2 +m 2 +n 1 −1 . It is clear from these definitions that for 1 ≤ k ≤ m 1 + n 2 + m 2 + n 1 Consider now a fragment f k ⊂ B together with its upper interface i k−1 and its lower interface i k . Denote by e first the left-most vertical edge of f k , and by e last , the right-most vertical edge of f k . We can now define the function shape as follows W when e first ∈ i k and e last ∈ i k N when e first ∈ i k−1 and e last ∈ i k−1 R when e first ∈ i k−1 and e last ∈ i k L when e first ∈ i k and e last ∈ i k−1 (14) The symbols W, N, R, and L describe geometrically the shape (respectively: wider, narrower, to-the-right, and to-the-left) of each fragment. Using this terminology, it is possible to apply the function shape to B = (f 1 , f 2 , . . . , f m 1 +n 2 +m 2 +n 1 ), simply by mapping it to the sequence of its fragments. For example, the shape of the structure shown in Fig. 1 is specified by the following sequence shape (Rb (3, 6, 5, 4)) = WWWWWLLLLNNRRRRNNN Let us now consider an arbitrary Clar cover C of B. For every edge e of B , we define a covering order function ord (e) as follows This definition can be naturally extended to covering order of interfaces by defining the order of the interface ord (i) as The interfaces i 0 and i m 1 +n 2 +m 2 +n 1 are empty, thus ord (i 0 ) = 0 = ord (i m 1 +n 2 +m 2 +n 1 ) .
It turns out that the orders of the remaining interfaces can be conveniently computed in an iterative fashion using Theorem 1. (First rule of interface theory: interface order criterion) [27,28] Let C be some Clar cover of a benzenoid B. Let f k be a fragment of B, and let i k−1 and i k be the upper and lower interfaces of f k , respectively. The following conditions are always satisfied.
(a) If f k has the shape W, then ord(i k ) = ord(i k−1 ) + 1.

(b)
If f k has the shape N, then ord(i k ) = ord(i k−1 ) − 1.
(c) If f k has the shape R or L , then ord(i k ) = ord(i k−1 ).
The interface orders determined this way starting with ord (i 0 ) = 0 depend only on the shape of B and are independent of the choice of C. Therefore, the interface orders are identical for every Clar cover C. It is straightforward to show that the interface orders of Rb (3, 6, 5, 4) shown in Fig. 1 are specified by the following sequence (ord (i 0 ) , . . . , ord (i m 1 +n 2 +m 2 +n 1 )) = (0, 1, 2, 3, 4, 5, 5, 5, 5, 5, 4, 3, 3, 3, 3, 3, 2, 1, 0) and the interface orders of Rb (5,9,4,6) shown in Fig. 3 are specified by the following sequence (ord (i 0 ) , . . . , ord (i m 1 +n 2 +m 2 +n 1 )) = 0, 1, 2, 3, 4, 5 Let us now consider in detail the interface i m 1 +n 2 of Rb (n 1 , n 2 , m 1 , m 2 ). Application of Theorem 1 to i m 1 +n 2 shows that ord(i m 1 +n 2 ) = N ≡ min (m 1 , n 1 ). At the same time simple geometrical considerations show that i m 1 +n 2 consists of N + 1 vertical edges e 0 , . . . , e N , where the numbering proceeds from right to left. Since and since each ord (e k ) can take on only three values: 0, 1 2 , and 1, the interface order ord(i m 1 +n 2 ) = N can be created from the interface edges orders ord (e k ) only in two possible ways N = 1 + . . .   Fig. 4 shows that the systems of double bonds in the interface i m 1 +n 2 propagates down and up in B, uniquely deciding the covering orders for a large portion of this structure. Each Clar cover belonging to this class shares this region of fixed bonds. However, there remain two disconnected regions in B, each in the shape of a parallelogram, for which the covering characters are not determined by the covering of the interface i m 1 +n 2 . The sizes of these two parallelograms are determined by the structural parameters n 1 , n 2 , m 1 , and m 2 and the location k of the single bond. It is easy to see that the upper parallelogram is M (m 1 − k, n 2 + k) and the lower one is M (m 2 + k, n 1 − k). The product of the ZZ polynomials of these two parallelograms Figure  Let us now consider a class of Clar covers of B corresponding to Eq. (21) with a hexagon C 6 located at the positions e k−1 and e k of the interface i m 1 +n 2 and double bonds in its remaining positions. Fig. 5 shows that the covered bonds in the interface i m 1 +n 2 induce a system of double bonds in the interfaces i m 1 +n 2 −1 and i m 1 +n 2 +1 , which propagate down and up in B, again uniquely deciding the covering orders for a large portion of this structure. Each Clar cover belonging to this class shares this region of fixed bonds. Again, there remain two disconnected regions in B, each in the shape of a parallelogram, for which the covering characters are not determined by the covering of the interface i m 1 +n 2 . The sizes of these two parallelograms are determined by the structural parameters n 1 , n 2 , m 1 , and m 2 and the location k of the Clar sextet. It is again easy to see that the upper parallelogram is M (m 1 − k, n 2 − 1 + k) and the lower one is M (m 2 − 1 + k, n 1 − k). The product of the ZZ polynomials of these two parallelograms ZZ (M (m 1 − k, n 2 − 1 + k) , x) · ZZ (M (m 2 − 1 + k, n 1 − k) , x) describes the contribution of this class of Clar covers to the ZZ polynomial of B. Sum of these contributions for k ∈ {1, . . . , N } reproduces Eq. (7) and concludes the proof of Eq. (5).

Discussion and conclusion
We have derived a closed-form formula for the ZZ polynomial of ribbons B ≡ Rb (n 1 , n 2 , m 1 , m 2 ), an important class of elementary pericondensed benzenoids. The formal demonstration of its correctness is based on the recently developed interface theory of benzenoids. The discovered formula uniquely determines the most important topological invariants of Rb (n 1 , n 2 , m 1 , m 2 ): • the number of Kekulé structures Interestingly, it is straightforward to obtain the Clar number of Rb (n 1 , n 2 , m 1 , m 2 ) and the number of Clar structures of Rb (n 1 , n 2 , m 1 , m 2 ) directly from the ZZ polynomial, but it seems to be a formidable task to extract these two quantities directly from the structural constants n 1 , n 2 , m 1 , and m 2 . For example, at the moment, the most compact formula for Cl in terms of the structural constants n 1 , n 2 , m 1 , and m 2 that we are aware of is given by the following expression Cl = max (Cl s , Cl r ) , where Cl s = max k∈{0,...,min(m 1 ,n 1 )} (min (m 1 − k, n 2 + k) + min (n 1 − k, m 2 + k)) Cl r = max k∈{1,...,min(m 1 ,n 1 )} (1 + min (m 1 − k, n 2 − 1 + k) + min (n 1 − k, m 2 − 1 + k)) (24) Both of these terms are needed, as the following examples show. For Rb (2, 2, 1, 1), we have Cl s = 3 and Cl r = 2, so Cl = Cl s . For Rb (3, 2, 1, 2), we have Cl s = 3 and Cl r = 4, so Cl = Cl r . We believe that Eqs. (22)-(24) cannot be simplified much. Therefore, it should be very instructive to see this expression for researchers who try to determine Clar numbers of simple benzenoids directly from geometrical considerations. The structural complexity of this formula suggests that transforming relatively easy geometrical constructs into an algebraic expression can be cumbersome. The last two classes of elementary pericondensed benzenoids, for which closed-form ZZ polynomial formulas remain to be found, are hexagonal flakes O (k, m, n) and oblate rectangles Or (m, n). We hope that our results will stimulate mathematicians and mathematicallyoriented chemists to discover these two last missing formulas.