
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)13240-9
10.1016/j.heliyon.2024.e37209
e37209
Research Article
Enumeration of the Hosoya index of pericondensed benzenoid system
Farooq Muhammad Talha muhammadtalharao1@gmail.com
ab
Almalki Norah norah@tu.edu.sa
c
Kaemawichanurat Pawaton pawaton.kae@kmutt.ac.th
ab⁎
a Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok, Thailand
b Mathematics and Statistics with Applications (MaSA), Thailand
c Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
⁎ Corresponding author at: Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi, Bangkok, Thailand. pawaton.kae@kmutt.ac.th
03 9 2024
15 9 2024
03 9 2024
10 17 e3720924 9 2023
26 8 2024
29 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
In chemical graph theory, topological indices play an important role and have various uses in quantitative structure-property relationships (QSPR) as well as quantitative structure-activity relationships (QSARs). The Hosoya index is one of them, performing a molecular descriptor in mathematical chemistry. Therefore, progressing with investigations requires computing the Hosoya index of distinct molecular graphs. This research presents a computational method for determining the Hosoya index of pericondensed benzenoid systems using the transfer matrix approach and the Hosoya vector.

MSC

05C92
Keywords

Topological index
Hosoya index
Benzenoid system
==== Body
pmc1 Preliminaries and introduction

In this paper, let G be a graph that is simple and connected. Let V(G) be the set of vertices and let E(G) be the set of edges of G. If we delete a vertex u from G and all its incident edges, we obtain a new graph which is expressed by G−u. If we delete an edge e from G, we obtain a new graph which is expressed by G−e. The collection of neighbors of a vertex v is represented by NG(v) which includes all the vertices that are adjacent to v. The closed neighbor set of v in G is NG[v]={v}∪NG(v). A matching of a graph G is a set of edges that do not share a common vertex. The size of the matching M is |M|. For a non-negative integer k, we let p(G,k) be the number of matchings of size k in a graph G. If we have a graph G, a matching of size one is a set of single edges. Thus, the number of matchings of size one is the total number of edges in the graph, implying that p(G,1)=|E(G)|. Further, the empty set is the only matching of size zero. Thus p(G,0)=1.

The topological index is the numerical descriptor that describes some physical properties of the molecules and these days numerous topological indices are introduced. It plays a crucial role in quantitative structure-activity relationships QSARs and QSPR analysis and holds significance in chemical graph theory. Topological indices can be used to predict the bioactivity of chemical compounds in the QSPR investigation. The forecasting power of degree-distance and distance-based topological indices is examined by Shirakol et al. [1] using QSPR analysis and, Luccic et al. [2] look for novel distance-related indices using the QSPR. In order to characterize the valuable topological indices based on predicting power, Sunilkumar et al. [3] investigated the QSPR analysis of degree-based topological indices. The QSPR/QSAR analysis between the indices and the physical characteristics of chemical compounds was developed by numerous studies [4], [5], [6], [7]. In 1971, a chemist Hauro Hosoya defined the Hosoya index [8], is Z(G)=∑k=0rp(G,k), where p(G,r)≠0 where r is the largest size of matchings of G, this index is also denoted as a Z index. The Hosoya index stands out as one of the most well-known topological indices in the field of mathematical chemistry. He also demonstrated how some of the chemical and physical characteristics of the Hosoya index in chemistry are closely linked to alkanes [8]. Numerous studies [9], [10], [11] demonstrated the relevance of Z-index in the theory of conjugated π-electron systems. This index is used to study the molecular topology of the molecules and how it relates to molecular characteristics. It also provides a numerical representation of the molecule's complexity and analyzes the molecular structures. Several studies on the Hosoya index of different molecular structures have been published [12], [13]. Matching theory is applied to topologically ascertain protein complexes. He also introduced the Wiener index [8]. This index has been used to describe the properties of chemical compounds in terms of their molecular structure. These indices are widely utilized and analyzed as molecular descriptors for evaluating various physicochemical characteristics of the corresponding chemical compounds based on their molecular structure. Dobrynin [14], [15] works on molecular graphs of unbranched cata-condensed benzenoid hydrocarbons, which are included in the Wiener index for hexagonal chains and some special families of hexagonal chains [16]. In 2016, Michael et al. examined the polycyclic benzenoid network using several distance-based topological indices, including the Wiener, Szeged, and Schultz indices [17].

The operator technique and the transfer matrix technique play an important role in solving various combinatorial enumeration problems, like matchings. Many studies use these techniques [18], [19] and Randić et al. introduced an algorithm to obtain matching polynomials of benzenoid chains by using the transfer matrix technique [20]. Cruz et al. [21] also introduced a method for calculating the Hosoya index of a catacondensed hexagonal system. In this approach, they proposed a Hosoya vector and utilized the transfer matrix technique. In this study, we work on the pericondensed benzenoid system by utilizing the transfer matrix technique and Hosoya vector. The researchers are working on the Hosoya index and Merrifield Simmons index and different topological indexes of chemical structure. The following works are the motivation for the current study. In these studies, Sinan [22], [23], [24], [25] computed the number of matchings in catacondensed benzenoid systems, the Hosoya index of primitive coronoid systems, and special double benzenoid chains. Hosoya and Ohkami [18] employed an operator technique to derive recurrence equations for the characteristic, matching, and Z-counting polynomials associated with linear, kinked, and zigzag configurations of polyacenes. There are few studies on benzenoid (hexagonal) systems in [26], [27], [28], [29], and some extremal hexagonal chain characteristics were investigated [30], [31], [32]. In his study, Aleksander [33] introduced linear algorithms to facilitate the calculation of the Hosoya Index and the Hosoya matrix for a tree graph. Various methods have been employed to analyze different topological indices like spectral techniques [34], [35]. The transfer matrix technique has proven effective in solving numerous combinatorial and enumeration problems in mathematical chemistry. The Hosoya index is highly responsive to a molecule's structural characteristics. It records significant data regarding the configuration of bonds and atoms, offering insights into the structure of molecules. It has adaptability to a variety of chemical configurations demonstrating its versatility. Its broad applicability in chemical research is attributed to its ability to be applied to various kinds of molecules. It has been calculated using an analytical method, such as the transfer matrix technique. This improves its computing efficiency and makes it easier to incorporate into other research approaches.

In theoretical chemistry, benzenoid hydrocarbons are represented by hexagonal graph structures, the graphs whose inner faces are hexagons. To represent a benzenoid hydrocarbon molecule as a graph structure, a finite 2-connected graph is utilized, with hexagons representing the regions corresponding to the molecule's rings, and excluding the exterior region that is a hexagon. If a hexagonal system contains at least one vertex that is part of three different hexagons, it is said to be pericondensed; otherwise, it is said to be catacondensed. Here let us use Xn and Yn to distinguish between two different pericondensed benzenoid systems in Figure 1, Figure 2.Figure 1 Benzenoid system Xn.

Figure 1

Figure 2 Benzenoid system Yn.

Figure 2

2 Methodology

In this section, a novel approach is introduced for the computation of the Hosoya index of two different pericondensed benzenoid systems Xn and Yn that are presented in Figure 1, Figure 2. Two important Recurrence relations (1) and (2) are used to drive the transfer matrices and the entries of these matrices are calculated by the Python algorithm. By using the Hosoya vector and these calculated matrices of benzenoid systems in MATLAB (the algorithms are embedded in the link https://github.com/Muhammadtalhafarooq/matching-of-graph), the Hosoya index is calculated for the required cells of the system. In Xn, the path graph with vertices {aj+n,uj+n,ij+n,sj+n,bj+n}, n≥0 is highlighted in red, representing the merging and intersection graph of two cells of benzenoid systems and this path graph is also referred as a graph Q. Here the subgraph Q and the subgraph of five hexagons given in explicit form merge along a path, and this merging path appears as P5 in Figure 3, Figure 4 and P4 in Figure 9, Figure 10 respectively. Similarly, in Yn, the path graph with vertices {aj+n,uj+n,ij+n,sj+n,bj+n}, n≥0 serves as the merging path P5 for two subgraphs, Q and five hexagons.Xn:aj≡dj+n,ij≡gj+n,bj≡cj+nYn:aj≡dj+n,ij≡gj+n,fj≡qj+n,bj≡kj+n

The transfer matrix technique is just one method among various approaches for calculating the Hosoya index. Different methods come with distinct advantages and disadvantages. This method does not apply to carbon nanotubes, trees, and bipartite graphs. This is suitable for linear and random chains of benzenoid systems. Using this method, we cannot calculate the k-matching of the molecular structure, it only provides the total number of matchings. In the future, we plan to explore methods for calculating the k-matching. The Hosoya index is more sensitive as compared to other topological indices because it focuses on the connections between atoms rather than capturing the degrees of the atoms in the molecules.Figure 3 Benzenoid system Xn.

Figure 3

Figure 4 Benzenoid system Yn.

Figure 4

3 Enumeration of the Hosoya index of pericondensed benzenoid systems

In [36], we can find two practical recurrence relations that can be used to compute the Hosoya index for a graph G.(1) Z(G)=∏i=1kZ(Gi)

Where connected components of G are denoted by G1, G2, G3,...,Gk.(2) Z(G)=Z(G−xy)+Z(G−x−y)

for a graph G with edge e=xy. Definition 1 Consider a graph G and a, u, i, v and b be the five vertices of a path in G and we represent the Hosoya vector of graph G at the path P7 with the vertices v, d, o, g, p, c and r by Zdgc(G), Zvop(G) and Zopr(G) and is defined as:

Zdgc(G)=[Z(G),Z(G−d),Z(G−g),Z(G−c),Z(G−d−g),Z(G−g−c),Z(G−d−c),Z(G−d−g−c)]T

Zvop(G)=[Z(G),Z(G−v),Z(G−o),Z(G−p),Z(G−v−o),Z(G−v−p),Z(G−o−p),Z(G−v−o−p)]T

Zopr(G)=[Z(G),Z(G−o),Z(G−p),Z(G−r),Z(G−o−p),Z(G−o−r),Z(G−r−p),Z(G−o−p−r)]T

To clarify the following theorems, graphs are presented in Figure 5, Figure 6, Figure 7, which illustrate different aspects of the concepts discussed.Figure 5 Graph G from Definition 1 used in Theorem 1.

Figure 5

Figure 6 Graph G from Definition 1 used in Theorem 2.

Figure 6

Figure 7 Graph G from Definition 1 used in Theorem 3.

Figure 7

3.1 Calculating the Hosoya index of benzenoid system Xn

Theorem 1 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the pathP5, which has the endpointsaandbof the graph Q see inFig. 5. Then

Zdgc(G)=A⋅Zaib(Q)

whereA=[120857461257426227026211257427429812829614014060612298324144298144144645742962981402741401286026212814464140687232262140144721286864322701401446814072683211260643260323216]

Proof Using formulae (1) and (2), we obtain values for the following expressions: Z(G),Z(G−d),Z(G−g),Z(G−c),Z(G−d−g),Z(G−g−c),Z(G−d−c),Z(G−d−g−c). Let us remove the appropriate combinations of edges ae, ij, and fb from each matching subgraph.Z(G)=Z(G−ae−ij−fb)+Z(G−ae−ij−f−b)+Z(G−ae−i−j−fb)+Z(G−ae−i−j−f−b)+Z(G−a−e−ij−fb)+Z(G−a−e−ij−f−b)+Z(G−a−e−i−j−fb)+Z(G−a−e−i−j−f−b)=1208Z(Q)+574Z(Q−b)+612Z(Q−i)+262Z(Q−i−b)+574Z(Q−a)+270Z(Q−a−b)+262Z(Q−a−i)+112Z(Q−a−i−b)=(1208,574,612,262,574,270,262,112)⋅Zaib(Q)Z(G−d)=Z(G−d−ae−ij−fb)+Z(G−d−ae−ij−f−b)+Z(G−d−ae−i−j−fb)+Z(G−d−ae−i−j−f−b)+Z(G−d−a−e−ij−fb)+Z(G−d−a−e−ij−f−b)+Z(G−d−a−e−i−j−fb)+Z(G−d−a−e−i−j−f−b)=574Z(Q)+274Z(Q−b)+298Z(Q−i)+128Z(Q−i−b)+296Z(Q−a)+140Z(Q−a−b)+140Z(Q−a−i)+60Z(Q−a−i−b)=(574,274,298,128,296,140,140,60)⋅Zaib(Q)Z(G−g)=Z(G−g−ae−ij−fb)+Z(G−g−ae−ij−f−b)+Z(G−g−ae−i−j−fb)+Z(G−g−ae−i−j−f−b)+Z(G−g−a−e−ij−fb)+Z(G−g−a−e−ij−f−b)+Z(G−g−a−e−i−j−fb)+Z(G−g−a−e−i−j−f−b)=612Z(Q)+298Z(Q−b)+324Z(Q−i)+144Z(Q−i−b)+298Z(Q−a)+144Z(Q−a−b)+144Z(Q−a−i)+64Z(Q−a−i−b)=(612,298,324,144,298,144,144,64)⋅Zaib(Q)Z(G−c)=Z(G−c−ae−ij−fb)+Z(G−c−ae−ij−f−b)+Z(G−c−ae−i−j−fb)+Z(G−c−ae−i−j−f−b)+Z(G−c−a−e−ij−fb)+Z(G−c−a−e−ij−f−b)+Z(G−c−a−e−i−j−fb)+Z(G−c−a−e−i−j−f−b)=574Z(Q)+296Z(Q−b)+298Z(Q−i)+140Z(Q−i−b)+274Z(Q−a)+140Z(Q−a−b)+128Z(Q−a−i)+60Z(Q−a−i−b)=(574,296,298,140,274,140,128,60)⋅Zaib(Q)Z(G−d−g)=Z(G−d−g−ae−ij−fb)+Z(G−d−g−ae−ij−f−b)+Z(G−d−g−ae−i−j−fb)+Z(G−d−g−ae−i−j−f−b)+Z(G−d−g−a−e−ij−fb)+Z(G−d−g−a−e−ij−f−b)+Z(G−d−g−a−e−i−j−fb)+Z(G−d−g−a−e−i−j−f−b)=262Z(Q)+128Z(Q−b)+144Z(Q−i)+64Z(Q−i−b)+140Z(Q−a)+68Z(Q−a−b)+72Z(Q−a−i)+32Z(Q−a−i−b)=(262,128,144,64,140,68,72,32)⋅Zaib(Q)

Z(G−g−c)=Z(G−g−c−ae−ij−fb)+Z(G−g−c−ae−ij−f−b)+Z(G−g−c−ae−i−j−fb)+Z(G−g−c−ae−i−j−f−b)+Z(G−g−c−a−e−ij−fb)+Z(G−g−c−a−e−ij−f−b)+Z(G−g−c−a−e−i−j−fb)+Z(G−g−c−a−e−i−j−f−b)=262Z(Q)+140Z(Q−b)+144Z(Q−i)+72Z(Q−i−b)+128Z(Q−a)+68Z(Q−a−b)+64Z(Q−a−i)+32Z(Q−a−i−b)=(262,140,144,72,128,68,64,32)⋅Zaib(Q)Z(G−d−c)=Z(G−d−c−ae−ij−fb)+Z(G−d−c−ae−ij−f−b)+Z(G−d−c−ae−i−j−fb)+Z(G−d−c−ae−i−j−f−b)+Z(G−d−c−a−e−ij−fb)+Z(G−d−c−a−e−ij−f−b)+Z(G−d−c−a−e−i−j−fb)+Z(G−d−c−a−e−i−j−f−b)=270Z(Q)+140Z(Q−b)+144Z(Q−i)+68Z(Q−i−b)+140Z(Q−a)+72Z(Q−a−b)+68Z(Q−a−i)+32Z(Q−a−i−b)=(270,140,144,68,140,72,68,32)⋅Zaib(Q)Z(G−d−g−c)=Z(G−d−g−c−ae−ij−fb)+Z(G−d−g−c−ae−ij−f−b)+Z(G−d−g−c−ae−i−j−fb)+Z(G−d−g−c−ae−i−j−f−b)+Z(G−d−g−c−a−e−ij−fb)+Z(G−d−g−c−a−e−ij−f−b)+Z(G−d−g−c−a−e−i−j−fb)+Z(G−d−g−c−a−e−i−j−f−b)=112Z(Q)+60Z(Q−b)+64Z(Q−i)+32Z(Q−i−b)+60Z(Q−a)+32Z(Q−a−b)+32Z(Q−a−i)+16Z(Q−a−i−b)=(112,60,64,32,60,32,32,16)⋅Zaib(Q)

Therefore, we can conclude that Zdgc(G) can be expressed as A⋅Zaib(Q), where Q is a path graph P5 with vertices a,u,i,v,b as given in Theorem 1, Zdgc(G)=A⋅Zaib(Q) and Zaib(Q)=[8,5,4,5,2,3,2,1]T is a Hosoya vector of that path P5 and Zdgc(G)=[19822,9146,9782,9248,4292,4352,4440,1908]T. So, the total number of matching of a graph in Fig. 5 is 19822. □

Corollary 1 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the path P5 , which has the endpoints aandb of the graph Q see in Fig. 5 . Then Zvop(G)=Aˆ⋅Zaib(Q)

where Aˆ=[12085746125742622702621125302512622131121009641442207208195889078334421952082077890883316979786533302611191848876333332121446364632427249552424219983]

Proof Utilizing the Definition 1, it becomes evident that the rows of the vector Zdgc(G) have been computed in Theorem 1. Thus, we can infer that Zvop(G) is equivalent to the product of the matrix Aˆ and the vector Zaib(Q), where Q is P5 with vertices a,u,i,v,b as given in Theorem 1. Zvop(G)=Aˆ⋅Zaib(Q) and Zaib(Q)=[8,5,4,5,2,3,2,1]T is a Hosoya vector of P5 and Zvop(G)=[19822,8365,7013,7013,2603,2921,2224,825]T. So the total number of matchings of a graph in Fig. 5, is 19822. □

Corollary 2 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the pathP5, which has the endpointsaandbof the graph Q see inFig. 5. ThenZopr(G)=A˜⋅Zaib(Q)

whereA˜=[12085746125742622702621125302512622131121009641442207208195889078335302132622519610011241169797865333026111696578792630331119176888432333312552124248993]

Proof Utilizing the Definition 1, it becomes evident that the rows of the vector Zdgc(G) have been computed in Theorem 1. Thus, we can infer that Zopr(G) is equivalent to the product of the matrix A˜ and the vector Zaib(Q), where Q is P5 with vertices a,u,i,v,b as given in Theorem 1. Zopr(G)=A˜⋅Zaib(Q) and Zaib(Q)=[8,5,4,5,2,3,2,1]T is a Hosoya vector of P5 and Zopr(G)=[19822,8365,7013,8365,2603,2603,2921,825]T. So the total number of matchings of a graph in Fig. 5 is 19822. □

Corollary 3 LetXnbe a pericondensed system with n Anthracene as shown inFig. 1. ThenZdgc(Xn)=An−1⋅[1208,574,612,574,262,270,262,112]T

Proof According to Theorem 1, we know that Zdgc(G)=A⋅Zxyz(Q). If we apply the Theorem 1 to Xn for n−1 times, we get Zdgc(Xn)=An−1⋅Zxyz(Q′) where Q′ is the joined pair of three hexagons and is an Anthracene. Therefore, we can conclude that Zxyz(Q′) equals to [1208,574,612,574,262,270,262,112]T. Hence, the expression for Zdgc(Xn) becomes Zdgqk(Xn)=An−1⋅[1208,574,612,574,262,270,262,112]T. □

Theorem 2 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the pathP4, which has the endpointsi,uandbof the graph Q see inFig. 6. Then

Zvdg(G)=B⋅Ziub(Q)

whereB=[17828448741208612574374262743351358530262251153112870414438574298274188128910442468612324298208144509242254361184172109793941911982801441368864402196216262144128966423611512616690815640]

Proof By utilizing formulae (1) and (2), we get the values of Z(G),Z(G−v),Z(G−d),Z(G−g),Z(G−v−d),Z(G−v−g),Z(G−d−g),Z(G−v−d−g). Let us remove the appropriate combinations of edges ia, uf, and bj from each matching subgraphsZ(G)=782,844,874,1208,612,574,374,262⋅Ziub(Q)Z(G−v)=743,351,358,530,262,251,153,112⋅Ziub(Q)Z(G−d)=870,414,438,574,298,274,188,128⋅Ziub(Q)Z(G−g)=910,442,468,612,324,298,208,144⋅Ziub(Q)Z(G−v−d)=509,242,254,361,184,172,109,79⋅Ziub(Q)Z(G−v−g)=394,191,198,280,144,136,88,64⋅Ziub(Q)Z(G−d−g)=402,196,216,262,144,128,96,64⋅Ziub(Q)Z(G−v−d−g)=236,115,126,166,90,81,56,40⋅Ziub(Q)

Therefore, we can conclude that Zvdg(G) can be expressed as B⋅Ziub(Q), where Q is P4 with vertices i,u,b as given in Theorem 2, Zvdg(G)=B⋅Ziub(Q) and Ziub(Q)=[5,3,2,3,2,2,1,1]T is a Hosoya vector of P4. □

Theorem 3 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the pathP4, which has the endpointsa,uandiof the graph Q see inFig. 7. Then

Zvdg(G)=C⋅Zaui(Q)

whereC=[17821208874844374574612262781530374313137213262968485744264362002962981409106124684422082983241445333612632181001481847041628020816978114144543902622082081041401447224716613010452709036]

Proof By utilizing formulae (1) and (2), we get the values of Z(G),Z(G−v),Z(G−d),Z(G−g),Z(G−v−d),Z(G−v−g),Z(G−d−g),Z(G−v−d−g). Let us remove the appropriate combinations of edges af, uj, and ib from each matching subgraphsZ(G)=1782,1208,874,844,374,574,612,262⋅Zaui(Q)Z(G−v)=781,530,374,313,137,213,262,96⋅Zaui(Q)Z(G−d)=848,574,426,436,200,296,298,140⋅Zaui(Q)Z(G−g)=910,612,468,442,208,298,324,144⋅Zaui(Q)Z(G−v−d)=533,361,263,218,100,148,184,70⋅Zaui(Q)Z(G−v−g)=416,280,208,169,78,114,144,54⋅Zaui(Q)Z(G−d−g)=390,262,208,208,104,140,144,72⋅Zaui(Q)Z(G−v−d−g)=247,166,130,104,52,70,90,36⋅Zaui(Q)

Therefore, we can conclude that Zvdg(G) can be expressed as C⋅Zaui(Q), here Q is P4 with vertices a,u,i as given in Theorem 3. Zvdg(G)=C⋅Zaui(Q) and Zaui(Q)=[5,3,2,3,1,2,2,1]T is a Hosoya vector of P4. □

Theorem 4 Consider a graph G formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the path P4 , which has the endpoints i,uandb of the graph Q see in Fig. 6 . Then Zgcr(G)=D⋅Ziub(Q)

where D=[17828448741208612574374262910442468612324298208144848436426574298296200140781313374530262213137963902082082621441401047241616920828014411478545332182633611841481007024710413016690705236]

Proof By utilizing formulae (1) and (2), we get the values of Z(G),Z(G−g),Z(G−c),Z(G−r),Z(G−g−c),Z(G−g−r),Z(G−c−r),Z(G−g−c−r). Let us remove the appropriate combinations of edges af, uj, and ib from each matching subgraphsZ(G)=1782,844,874,1208,612,574,374,262⋅Ziub(Q)Z(G−g)=910,442,468,612,324,298,208,144⋅Ziub(Q)Z(G−c)=848,436,426,574,298,296,200,140⋅Ziub(Q)Z(G−r)=781,313,374,530,262,213,137,96⋅Ziub(Q)Z(G−g−c)=390,208,208,262,144,140,104,72⋅Ziub(Q)Z(G−g−r)=416,169,208,280,144,114,78,54⋅Ziub(Q)Z(G−c−r)=533,218,263,361,184,148,100,70⋅Ziub(Q)Z(G−g−c−r)=247,104,130,166,90,70,52,36⋅Ziub(Q)

Therefore, we can conclude that Zgcr(G) can be expressed as D⋅Ziub(Q), here Q is P4 with vertices i,u,b as given in Theorem 4, Zgcr(G)=D⋅Ziub(Q) and Ziub(Q)=[5,3,2,3,2,2,1,1]T is a Hosoya vector of P4. □

Example 1 Let P(A,A,A,A) be a four cells of system Xn as shown in Fig. 8:Figure 8 Benzenoid system Xn.

Figure 8

Let us calculate the Hosoya index of P(A,A,A,A) using the Hosoya vector associated with the path P5 containing vertices d,gande. We can compute an appropriate product involving some of the matrices A,A,A,A and the vector (8,5,4,5,2,3,2,1)T to determine pwvu(P5), which represents the Hosoya index of P(A,A,A,A), as follows:Zdge(P(A,A,A,A))=A⋅Zaib(QI)=A⋅A⋅Zhkl(QII)=A⋅A⋅A⋅Zprs(QIII)=A⋅A⋅A⋅A⋅Zwvu(P5)Zdge(P(A,A,A,A))=A⋅A⋅A⋅A⋅(8,5,4,5,2,3,2,1)T

Hence Zdge((P(A,A,A,A))=(221071150978800,103098889673392,110241587500320,103928450304496,48265433402432,48753116691136,49792870585280,21328417625664)T where QI, QII, QIII are corresponding subgraphs in P(A,A,A,A). As a result, we get a Hosoya index of Z(P(A,A,A,A))=221071150978800. In conclusion, we can enumerate the Hosoya index of an n-Anthracene chain by performing a specific matrix-vector product involving the matrices A,A,A,A, and the vector (8,5,4,5,2,3,2,1)T.

Example 2 Let P(B,B,B,B) be a four cells of system Zn as shown in Fig. 9:Figure 9 Benzenoid system Zn.

Figure 9

Let us calculate the Hosoya index of P(B,B,B,B) by using the Hosoya vector associated with the path P4 containing vertices v,dandg. We can compute an appropriate product involving some of the matrices B,B,B,B and the vector (5,3,2,3,2,2,1,1)T to determine pwvu(P4), which represents the Hosoya index of P(B,B,B,B), as follows:Zvdg(P(B,B,B,B))=B⋅Zaib(QI)=B⋅B⋅Zhkl(QII)=B⋅B⋅B⋅Zprs(QIII)=B⋅B⋅B⋅B⋅Zwvu(P4)Zvdg(P(B,B,B,B))=B⋅B⋅B⋅B⋅(5,3,2,3,2,2,1,1)T

Hence Zvdg((P(B,B,B,B))=(937654889535374394692700169907457205064574574484631806543706272055045285311211842333825315214331287993564128169636514237)T where QI, QII, QIII are corresponding subgraphs in P(B,B,B,B). As a result, we get a Hosoya index of Z(P(B,B,B,B))=937654889535374. In conclusion, we can enumerate the Hosoya index of n-Anthracene chain by performing a specific matrix-vector product involving the matrices B,B,B,B, and the vector (5,3,2,3,2,2,1,1)T.

Example 3 Let P(B,C,B,D) be a four cells of system Wn as shown in Fig. 10:Figure 10 Benzenoid system Wn.

Figure 10

Let us calculate the Hosoya index of P(B,C,B,D) by using the Hosoya vector associated with the path P4 containing vertices v,dandg. We can compute an appropriate product involving some of the matrices B,C,B,D, and the vector (5,3,2,3,2,2,1,1)T to determine pwvu(P4), which represents the Hosoya index of P(B,C,B,D), as follows:Zvdg(P(B,C,B,D))=B⋅Zaib(QI)=B⋅C⋅Zhkl(QII)=B⋅C⋅D⋅Zprs(QIII)=B⋅C⋅D⋅B⋅Zwvu(P4)Zvdg(P(B,C,B,D))=B⋅C⋅D⋅B⋅(5,3,2,3,2,2,1,1)T

Hence Zvdg((P(B,C,B,D))=(939798679239988,395608001242043,458239316749354,485738726336802,272682196746724,212334730171881,214812195763020,128463392982209)T where QI, QII, QIII are corresponding subgraphs in P(B,C,B,D). As a result, we get a Hosoya index of Z(P(B,C,B,D))=939798679239988.

In conclusion, we can enumerate the Hosoya index of n-Anthracene chain by performing a specific matrix-vector product involving the matrices B,C,B,D, and the vector (5,3,2,3,2,2,1,1)T.

3.2 Calculating the Hosoya index of benzenoid system Yn

Definition 2 Let G be a graph formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the path P5, which has the endpoints a,i,fandb of the graph Q and we represent the Hosoya vector of graph G at the path P5 with vertices a, i, f and b by Zdgqk(G) and is defined as:

Zdgqk(G)=[Z(G)Z(G−d)Z(G−g)Z(G−q)Z(G−k)Z(G−d−g)Z(G−d−q)Z(G−d−k)Z(G−g−q)Z(G−g−k)Z(G−q−k)Z(G−d−g−q)Z(G−d−g−k)Z(G−g−q−k)Z(G−d−q−k)Z(G−d−g−q−k)]

Theorem 5 Let G be a graph formed by merging the edges of a graph Q with a hexagonal system that consists of five hexagons located at the path P5 , which has the endpoints a,i,fandb of the graph Q. Then

Zdgqk(G)=A⋅Zaifb(Q)

A=[88051945245251928026830620826828012916616612980413244208192254128129150871141295876805436403237212220233132123137105130131637876653940323322021223713113013710512313265767863394132541922082441291141508712912854807658362771631441321708889100637889425255392619711410593121626570455463303639271819011787871175854723654582436362416168979595975755565055573033333018197121931051146354704565622739363018227170132144163897810063898839555242261166765577138404130333920222418121358363638342395127394218262618121167157656739334130403818242220121358363638342395127394218262618128049393949262430182426121616128]

Proof By using the Definition 2 of the Hosoya vector, we need to compute these Z(G),Z(G−d),Z(G−g),…,Z(G−d−g−q−k) by deleting independent edges ap, ic, fe and bh by using formulae (1) and (2), from each matching subgraph:Z(G)=880,519,452,452,519,280,268,306,208,268,280,129,166,166,129,80⋅Zaifb(Q)Z(G−d)=413,244,208,192,254,128,129,150,87,114,129,58,76,80,54,36⋅Zaifb(Q)Z(G−g)=403,237,212,220,233,132,123,137,105,130,131,63,78,76,65,39⋅Zaifb(Q)Z(G−q)=403,233,220,212,237,131,130,137,105,123,132,65,76,78,63,39⋅Zaifb(Q)Z(G−k)=413,254,192,208,244,129,114,150,87,129,128,54,80,76,58,36⋅Zaifb(Q)Z(G−d−g)=277,163,144,132,170,88,89,100,63,78,89,42,52,55,39,26⋅Zaifb(Q)Z(G−d−q)=197,114,105,93,121,62,65,70,45,54,63,30,36,39,27,18⋅Zaifb(Q)Z(G−d−k)=190,117,87,87,117,58,54,72,36,54,58,24,36,36,24,16⋅Zaifb(Q)Z(G−g−q)=168,97,95,95,97,57,55,56,50,55,57,30,33,33,30,18⋅Zaifb(Q)Z(G−g−k)=197,121,93,105,114,63,54,70,45,65,62,27,39,36,30,18⋅Zaifb(Q)Z(G−q−k)=227,170,132,144,163,89,78,100,63,89,88,39,55,52,42,26⋅Zaifb(Q)Z(G−d−g−q)=116,67,65,57,71,38,40,41,30,33,39,20,22,24,18,12⋅Zaifb(Q)Z(G−d−g−k)=135,83,63,63,83,42,39,51,27,39,42,18,26,26,18,12⋅Zaifb(Q)Z(G−g−q−k)=116,71,57,65,67,39,33,41,30,40,38,18,24,22,20,12⋅Zaifb(Q)Z(G−d−q−k)=135,83,63,63,83,42,39,51,27,39,42,18,26,26,18,12⋅Zaifb(Q)Z(G−d−g−q−k)=80,49,39,39,49,26,24,30,18,24,26,12,16,16,12,8⋅Zaifb(Q)

Therefore, we can conclude that Zdgqk(G) can be expressed as S⋅Zaifb(Q), where Q is P5 with vertices a,i,f,b as given in Theorem 5. Zdgqk(G)=S⋅Zaifb(Q) and Zaifb(Q)=[8,5,3,3,5,3,2,3,1,2,3,1,2,2,1,1]T is a Hosoya vector of P5. So the number of matchings of a graph in Fig. 11, is 19822, Zdgqk(G)=[19822,9248,9156,9156,9248,6258,4438,4236,3874,4438,5858,2656,3028,2656,3028,1820]T. □Figure 11 Graph G from Definition 2 used in Theorem 5.

Figure 11

Corollary 4 Let Yn be a pericondensed system with n Anthracene as shown in Fig. 2 . Then Zdgqk(Yn)=An⋅[148,65,70,70,65,44,32,28,30,32,44,19,20,32,19,12]T

Proof According to Theorem 5, we know that Zdgqk(G)=A⋅Zwxyz(Q). If we apply this Theorem 5 to Yn for n times, we get Zdgqk(Yn)=An⋅Zwxyz(Q′) where Q′ is the joined pair of two hexagons and is naphthalene. Therefore, we can conclude that Zwxyz(Q′) equals to [148,65,70,70,65,44,32,28,30,32,44,19,20,32,19,12]T. Hence, the expression for Zdgqk(Yn) becomes Zdgqk(Yn)=An⋅[148,65,70,70,65,44,32,28,30,32,44,19,20,32,19,12]T. □

4 Conclusion

In this work, we calculate the Hosoya index of pericondensed benzenoid systems Xn and Yn and provide a method based on two important recurrence relations, the Hosoya vector, the transfer matrix technique, and algorithms. The Hosoya index of the linear pericondensed benzenoid systems Xn and Yn, as well as the random and regular pericondensed benzenoid systems Zn and Wn, is calculated by utilizing these transfer matrices and the Hosoya vector in MATLAB in no time. The Hosoya index is used to study the molecular topology of the molecules and how it relates to molecular characteristics. It provides a numerical representation of the molecule's complexity analyzes molecular structures and is also helpful in the QSPR and QSAR analysis of the chemical structures. Hence, this comparable approach could be employed for calculating the Hosoya index of various benzenoid systems.

Funding statement

Muhammad Talha Farooq is supported by Petchra Pra Jom Klao Ph.D. Research Scholarship from King Mongkut's University of Technology Thonburi (25/2565 ). Norah Almalki's support is provided by the Deanship of Scientific Research at 10.13039/501100006261 Taif University . Pawaton Kaemawichanurat has been funded by National Research Council of Thailand (NRCT) and King Mongkut's University of Technology Thonburi (N42A660926 ).

CRediT authorship contribution statement

Muhammad Talha Farooq: Writing – original draft, Methodology, Investigation, Formal analysis. Norah Almalki: Writing – review & editing, Investigation, Formal analysis. Pawaton Kaemawichanurat: Supervision, Project administration, Conceptualization.

Declaration of Competing Interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Muhammad Talha Farooq reports financial support was provided by 10.13039/501100004705 King Mongkut's University of Technology Thonburi . Pawaton Kaemawichanurat reports financial support was provided by 10.13039/501100004705 King Mongkut's University of Technology Thonburi . Norah Almalki reports financial support was provided by 10.13039/501100006261 Taif University . If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

All data used for the research are contained in the article.
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