The Second Hyper-Zagreb Coindex of Chemical Graphs and Some Applications

&e second hyper-Zagreb coindex is an efficient topological index that enables us to describe a molecule from its molecular graph. In this current study, we shall evaluate the second hyper-Zagreb coindex of some chemical graphs. In this study, we compute the value of the second hyper-Zagreb coindex of some chemical graph structures such as sildenafil, aspirin, and nicotine. We also present explicit formulas of the second hyper-Zagreb coindex of any graph that results from some interesting graphical operations such as tensor product, Cartesian product, composition, and strong product, and apply them on a q-multiwalled nanotorus.


Introduction
A graph can be identified by a corresponding numerical value, a sequence of numbers, or a special polynomial or a matrix. Special attention is directed to chemical graphs which constitute a wonderful topic in graph theory because of the abundance of applications in chemistry or in medical science [1,2]. Topological index and coindex are invariant under graph automorphism. e computation of these numerical quantities is useful and well-proven in medical information of new drugs without resorting to chemical experiments [3,4]. All graphs in this study are finite and simple, let G be a finite simple graph on V(G) � n, vertices, and E(G) � m, edges, and the degree of a vertex v is the number of edges event to v, denoted by δ G (v). e complement of G, denoted by G, is a simple graph on the same set of vertices V(G), in which two vertices u and v are adjacent by an edge uv, if and only if they are not adjacent in G. Hence, uv ∈ E(G) if and only if uv ∉ E(G). Obviously, we have E(G)⋃ E(G) � E(K n ), so m � E(G) � n 2 − m, and the degree of a vertex u in G is given by Gutman and Trinajestić [5] introduced the first and second Zagreb indices as follows: In 2008, Došlić defined Zagreb coindices [6], which are given as follows: Later in 2010, Ashrafi et al. have established the following nice formulas for the precise relationship between the first and second Zagreb indices and their coindices [7]: In 2013, Shirdel et al. [8] introduced degree-based Zagreb indices named hyper-Zagreb index which is defined as In 2013, Ranjini et al. introduced and defined the third Zagreb index of a graph as [9] Furtula and Gutman in 2015 introduced the forgotten index (F-index) [10], which is defined as In 2016, De et al. introduced forgotten coindex as follows: In 2016, Veylaki et al. [11] introduced hyper-Zagreb coindex as follows: In 2016, Wei et al. [12] defined new version of Zagreb topological indices. It is called the hyper-Zagreb index that is defined as above. en, the second hyper-Zagreb index of a graph G is defined as the sum of the weights (δ G (u)δ G (v)) 2 and is equal to In 2020, Alameri et al. [13,14] defined a new degreebased of Zagreb indices named Y-index and Y-coindex as where Here, we define a new version of Zagreb topological indices, based on the hyper-Zagreb index that is defined as above. It is called the second hyper-Zagreb index of a graph G and defined as the sum of the weights (δ G (u)δ G (v)) 2 , such that uv ∉ E(G) and is equal to Eventhough, there are several research reports contributing to the computation of topological indices of chemical graphs. However, the studies on the computation of topological coindices of octane isomers are very limited.
is study focused on one of the important topological coindices named the second hyper-Zagreb coindex. Some chemical graphs were obtained by this parameter. Moreover, the second hyper-Zagreb coindex of graph operations was computed and gave some of their applications such as a qmultiwalled nanotorus.

Preliminaries
is section is devoted to some preparatory results that will play a prominent role in our study.
Definition 2.1 (see [15,16]). Suppose that G 1 and G 2 are two connected graphs, then and xy ∈ E(G 2 ) or ab ∈ E(G 1 ) and x � y. (iii) e composition G 1 [G 2 ] of G 1 and G 2 with disjoint vertex sets V(G 1 ) and V(G 2 ) and edge sets E(G 1 ) and E(G 2 ) is the graph with vertex set V (G 1 ) × V (G 2 ) and any two vertices u � ( , and any two vertices (u 1 , v 1 ) and (u 2 , v 2 ) are adjacent if and only if Lemma 2 (see [17,18]). Let G 1 and G 2 be graphs with [17,18]). Let G 1 , G 2 be two graphs with n 1 , n 2 vertices and m 1 , m 2 edges, respectively, then.

Main Results
In the following section, we study the second hyper-Zagreb coindex of some chemical graph structures, exactly sildenafil, aspirin, and nicotine.
Proof. For the proof ( eorem 3.2), we refer to [10]. □ Proposition 3.2. Let G be a graph with n vertices and m edges. en, Proof. By definition of the second hyper-Zagreb coindex and using a similar method, as above in Proposition 3.1, then Sildenafil (C 22 H 30 N 6 O 4 S) is a drug used for pulmonary arterial hypertension. It is taken by mouth or injection into a vein (Figure 1) [19]. From the graph structure of sildenafil (Figure 1), it is easy to obtain the dataset in Tables 1 and 2.
By Table 1 and definitions of the first Zagreb index and the Y-index, we have   Aspirin has many medicinal uses as it is a drug that is used to reduce fever or inflammation, also given after a heart attack to reduce the risk of death. Aspirin is also used as a nonsteroidal anti-inflammatory drug because it has an antiplatelet effect by inhibiting its normal functioning. Also, a lot of evidence indicates that aspirin is considered a chemical agent that may limit and reduce the incidence of general cancers (Figure 2) [20,21].

Proposition 3.4. e second hyper-Zagreb coindex of aspirin.
From the graph structure of aspirin (Figure 2), it is easy to obtain the dataset in Tables 2 and 3.
Also, by Table 4 and definition of the second hyper-Zagreb index, we have Nicotine (C 10 H 14 N 2 ) is an alkaloid that is widely used as an anxiolytic. Nicotine is used as a drug to quit smoking, and if it is not used well, it can lead to addiction. Many types of research conducted on animals indicate that some inhibitors found in tobacco smoke, such as monoamine oxidase, may enhance some of the addictive properties of nicotine (Figure 3) [21,22]. Any unexplained terminology is standard, typically as in [22][23][24].
By Table 3 and definitions of the first Zagreb index and the Y-index, we have From the graph structure of nicotine (Figure 3), it is easy to obtain the dataset in Tables 5 and 6.
By Table 5 and definitions of the first Zagreb index and the Y-index, we have Also, by Table 6 and definition of the second hyper-Zagreb index, we have Using Proposition 3.2, we have

Applications
In the following section, we provide the exact value of the second hyper-Zagreb coindex of graphs that are arisen from mathematical operations such as the tensor product G 1 ⊗ G 2 , the Cartesian product G 1 × G 2 , the composition , and the strong product G 1 * G 2 . Also, we apply this coindex on a q-multiwalled nanotorus.
Theorem 4.1. e second hyper-Zagreb coindex of G 1 × G 2 is given by Proof. We have HM 2 (G) � 1/2M 1 2 (G) − 1/2Y(G) − H M 2 (G), given in Proposition 3.2, and by replacing each G by e second hyper-Zagreb coindex of G 1 * G 2 is given by   e second hyper-Zagreb coindex of G 1 ⊗ G 2 is given by In [19,[25][26][27], authors computed some topological indices of molecular graph of a nanotorus (Figure 4). In this section, we compute the second hyper-Zagreb coindex of a molecular graph of a nanotorus. Proof.
e proof of the above corollary is given by Gao et al. in [3]. Obviously, en, erefore, Proof. We have by Lemma 2.5, As proof in Corollary 4.6, we have where HM 2 P n � (16n − 40), HM 2 (T) � 243 2 pq. (37) Now, we apply the second hyper-Zagreb coindex on a qmultiwalled nanotorus using Cartesian product operation.

Conclusion
In this study, we obtained the value of the second hyper-Zagreb coindex of some chemical graphs, and we computed some explicit formulas for their numbers under several graph operations. Also, we applied the second hyper-Zagreb coindex on a q-multiwalled nanotorus. e results of this work may be used as a predictor, especially in the chemical graph theory. For example, in quantitative structure-activity relationships (QSAR) modelling, the predictors consist of theoretical molecular descriptors of chemicals.

Data Availability
No data were used to support this study, except for the references that were mentioned.

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