Hosoya and Harary Polynomials of Hourglass and Rhombic Benzenoid Systems

Teaching Department of Public Basic Course, Anhui International Studies University, Hefei 231201, China Department of Mathematics and Statistics, University of Lahore, Lahore 54000, Pakistan Department of Mathematics, Lahore Leads University, Lahore, Pakistan Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan Department of Mathematics, Government College University, Lahore, Pakistan


Introduction
Cheminformatics is a new branch of science which relates chemistry, mathematics, and computer sciences. Quantitative structure-activity (QSAR) and structure-property relationships (QSPR) are the main components of cheminformatics which are helpful to study physicochemical properties of chemical compounds [1][2][3].
A topological index is a numeric quantity associated with a graph which characterizes the topology of graph and is invariant under graph automorphism [4][5][6][7][8]. ere are numerous applications of graph theory in the field of structural chemistry. e first well-known use of a topological index in chemistry was by Wiener in the study of paraffin boiling points [9]. After that, in order to explain physicochemical properties, various topological indices have been introduced. e Hosoya polynomial of a graph is a generating function about distance distribution, introduced by Haruo Hosoya in 1988 [10]. is polynomial has many chemical applications [11]; in particular, Wiener index can be directly obtained from the polynomial and studied extensively [12][13][14]. e Wiener index was first introduced by Harold Wiener in 1947 to study the boiling points of paraffin [9]. It plays an important role in the so-called inverse structure-property relationship problems [15]. For more details about this topological polynomial and index, see the paper series and the references therein [16][17][18][19][20]. In this report, we study Hosoya polynomials, Wiener index, and hyper-Wiener index of hourglass and rhombic benzenoid systems.
Definition 2 (Hosoya polynomial [10]). e Hosoya polynomial of a connected graph G is denoted by H(G, x) and defined as follows: (G) x d (v,u) , (1) where d (u, v) denotes the distance between vertices u and v.
Definition 3 (Wiener index [9]). e Wiener index of a connected graph G is denoted by W(G) and defined as the sum of distances between all pairs of vertices in G, i.e., it can be formulated as follows: Note that the first derivative of the Hosoya polynomial at x � 1 is equal to the Wiener index: Definition 4 (modified Wiener index). e modified Wiener index of a connected graph G is denoted by W λ (G) and defined as the sum of λ power distances between all pairs of vertices in G, where λ � 1, 2, 3, 4, . . . i.e., it can be formulated as follows: For detailed survey about this index, see [21][22][23].

Methodology
To compute the Hosoya polynomial of a graph G, we need to compute number of pairs of vertices at distance 1, 2, 3, For this purpose, we use mathematical induction. Here, the dia(X n ) � 4n − 1 and dia(R n ) � 2n + 1. e general view of Hosoya polynomial is as below, where d is the diameter of graph.

Computational Results
Benzenoid hydrocarbons play a vital role in our environment and in the food and chemical industries. Benzenoid molecular graphs are systems with deleted hydrogens. It is a connected geometric figure obtained by arranging congruent regular hexagons in a plane, so that two hexagons are either disjoint or have a common edge.
is figure divides the plane into one infinite (external) region and a number of finite (internal) regions. All internal regions must be regular hexagons. Benzenoid systems are of considerable importance in theoretical chemistry because they are the natural graph representation of benzenoid hydrocarbons. A vertex of a hexagonal system belongs to, at most, three hexagons. A vertex shared by three hexagons is called an internal vertex.
Definition 10 (benzenoid hourglass system). Let X n denotes the benzenoid hourglass, which is obtained from two copies of a triangular benzenoid T n by overlapping their external hexagons ( Figure 1).

Theorem 1. For the benzenoid hourglass system X n , we have
Proof. To prove this theorem, we need to compute |a m (n)| where m � 1, 2, 3, . . . , 4n − 1. It is easy to verify that a 0 (n) � |V| � 2 n 2 + 4n − 2 , e remaining proof is divided into six parts which are according to the parity of m.
Using a similar fashion, we have It implies that In a similar fashion, we infer In terms of mathematical induction, we yield It can be observed from Figure 1 that Now, one can conclude that By means of the same trick, we obtain a 6 (1) � 0, a 6 (2) � 0, a 6 (3) � 84, a 6 (4) � 174, a 6 (5) � 300, which reveals that In light of the similar approach, we get Hence, by mathematical induction, we have It can be observed from Figure 1 that Now, one can conclude that Using a similar fashion, we have It implies that 4 Journal of Chemistry In a similar fashion, we infer In terms of mathematical induction, we yield It can be observed from Figure 1 that Now, one can conclude that a 2n+m (n) � (n − 1) 3 3 Using a similar fashion, we have It implies that In a similar fashion, we infer In terms of mathematical induction, we yield Case 5. m � 2n + 1. It can be observed from Figure 1 that Journal of Chemistry Now, one can conclude that It can be observed from Figure 1 that Now, one can conclude that By what have been mentioned above, we get our required result.
From the above theorem, we get the following results immediately.

Benzenoid Rhombus
System. Consider a benzenoid system in which hexagons are arranged to form a rhombic shape, say, R n , where n represents number of hexagons along each boundary of the rhombic as given in Figure 2.
It can be observed from Figure 2 that In terms of mathematical induction, we yield (52) It can be observed from Figure 2 that Now, one can conclude that By means of the same trick, we obtain which reveals that a 6 (n) � 18(n − 2) 2 + 36 * (n − 2).
In light of the similar approach, we get (57) Hence, by mathematical induction, we have a m (n) � 3mn 2 − 2m n + 7m 3 24

Journal of Chemistry
Now for m � 2n + 2 to m � 4n − 1, we will generalize in this way. By observing Table 1, values in italics show the distances from 2n + 2 to 4n − 1, but the values in the table are in descending order, so first we generalized this in ascending order and then reverse its order as required, let p i be the values in ascending order as follows (from Table 1). (59) Hence, one can conclude that For m ≡ 1mod(2), 1 ≤ m ≤ 2n − 3. So to reverse its order put i � (− m + 2n − 2), we get a m (n) � (− m + 2n − 2) 3 24 (61) Hence, one can conclude that For m ≡ 0mod(2), 2 ≤ m ≤ 2n − 2. So to reverse its order put i � (− m + 2n), we get a m (n) � (− m + 2n) 3 24 By what has been mentioned above, we get our desired result.

Conclusions
Wiener demonstrated that the Wiener index is firmly connected to the boiling point of alkane. Later work on quantitative structure-activity connections demonstrated that it is additionally corresponded to different amounts including the parameters of its basic point the thickness, surface strain, and consistency of its fluid stage and the van der Waals surface territory of the molecules. Wiener index is a valuable topological index in the structureproperty relationship since it is the measurement of compactness of particle regarding its basic characteristics, for example, spreading and cyclicity. Utilizations of benzene follow a long history. In the nineteenth and midtwentieth centuries, benzene was utilized as an aftershave lotion due to its wonderful smell. Before the 1920s, benzene was as often as possible utilized as a modern dissolvable, particularly to degrease metal. As its lethality wound up self-evident, benzene was displaced by different solvents, particularly toluene (methylbenzene), which has comparable physical properties yet is not as cancer-causing. In 1903, Ludwig Roselius promoted the utilization of benzene to decaffeinate espresso. is disclosure prompted the creation of Sanka. is procedure was later ended. Benzene was generally utilized as a noteworthy part in numerous shopper items, for example, Liquid Wrench, a few paint strippers, elastic concretes, spot removers, and different items. Produce of a portion of these benzene-containing details stopped in around 1950, albeit Liquid Wrench kept on containing critical measures of benzene until the late 1970s. In this present paper, we computed Hosoya polynomial, Wiener index, and hyper-Wiener index of zigzag and benzenoid rhombus systems. It is an interesting problem to find out distance-based topological indices for the families of graphs studied in [42][43][44][45][46][47].

Data Availability
All the data are included within this paper.

Conflicts of Interest
e authors of this paper declare that they have no conflicts of interest.