Abstract
The 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.
1. 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. The 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 vertices, and edges, and the degree of a vertex is the number of edges event to , denoted by . The complement of , denoted by , is a simple graph on the same set of vertices , in which two vertices and are adjacent by an edge , if and only if they are not adjacent in G. Hence, if and only if Obviously, we have , so , and the degree of a vertex u in 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. Then, the second hyper-Zagreb index of a graph G is defined as the sum of the weights and is equal to
In 2020, Alameri et al. [13, 14] defined a new degree-based of Zagreb indices named Y-index and Y-coindex aswhere
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 , such that 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. This 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 q-multiwalled nanotorus.
2. Preliminaries
This section is devoted to some preparatory results that will play a prominent role in our study.
Definition 2.1. (see [15, 16]). Suppose that and are two connected graphs, then(i)The tensor product of and is the graph with and (ii)The Cartesian product of and has the vertex set , and is an edge of if a = b and or and x = y.(iii)The composition of and with disjoint vertex sets and and edge sets and is the graph with vertex set and any two vertices is adjacent with whenever ( is adjacent with ) or ( = and is adjacent with ).(iv)The strong product of and is a graph with , and any two vertices and are adjacent if and only if and} or and}.
Lemma 2 (see [17, 18]). Let and be graphs with , , , and . Then,(i)(ii)(iii) (iv) (v)(vi) (a)(b)(c)
Lemma 2.3. (see [17, 18]). Let be two graphs with vertices and edges, respectively, then.(i) (ii) (iii)(iv)
Lemma 2.4. (see [17, 18]). Let be two simple graphs with vertices and edges, respectively, then(i) (ii) (iii) (iv)
Lemma 2.5. (see [17, 18]). Let be two simple graphs with vertices and edges, respectively, then(i)(ii)(iii)(iv)
3. Main Results
In the following section, we study the second hyper-Zagreb coindex of some chemical graph structures, exactly sildenafil, aspirin, and nicotine.
Proposition 3.1. Let be a graph with n vertices and m edges. Then,
Proof. For the proof (Theorem 3.2), we refer to [10].
Proposition 3.2. Let be a graph with n vertices and m edges. Then,
Proof. By definition of the second hyper-Zagreb coindex and using a similar method, as above in Proposition 3.1, thenSildenafil (C22H30N6O4S) is a drug used for pulmonary arterial hypertension. It is taken by mouth or injection into a vein (Figure 1) [19].
Proposition 3.3. The second hyper-Zagreb coindex of sildenafil.
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
Also, by Table 2 and definition of the second hyper-Zagreb index, we have
Using Proposition 3.2, we have
Aspirin (C9H8O4) is known as acetylsalicylic acid (ASA). 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. The 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
Using Proposition 3.2, we have
Nicotine (C10H14N2) 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 [22–24].
By Table 3 and definitions of the first Zagreb index and the Y-index, we have
Proposition 3.5. The second hyper-Zagreb coindex of nicotine.
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
4. 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 , the Cartesian product , the composition , and the strong product . Also, we apply this coindex on a q-multiwalled nanotorus.
Theorem 4.1. The second hyper-Zagreb coindex of is given by
Proof. We have , given in Proposition 3.2, and by replacing each by , which yields , and by using (Lemma 2.2–Lemma 2.4), we obtain the required.
All proofs in Theorems 4.2–4.4 are given as Theorem 4.1.
Theorem 4.2. The second hyper-Zagreb coindex of is given by
Theorem 4.3. The second hyper-Zagreb coindex of is given by
Theorem 4.4. The second hyper-Zagreb coindex of is given by
Proof.
In [19, 25–27], 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.
Corollary 4.5. Let T = T[p; q] be the molecular graph of a nanotorus. Then, the first Zagreb index of a q-multiwalled nanotorus is
Proof. The proof of the above corollary is given by Gao et al. in [3]. Obviously,
Corollary 4.6. Let T = T[p; q] be the molecular graph of a nanotorus. Then, the Y-index of a q-multiwalled nanotorus is
Proof. We have by Lemma 4.2,Then,Therefore,
Corollary 4.7. Let T = T[p; q] be the molecular graph of a nanotorus. Then, the second hyper-Zagreb index of a q-multiwalled nanotorus is
Proof. We have by Lemma 2.5,As proof in Corollary 4.6, we havewhereNow, we apply the second hyper-Zagreb coindex on a q-multiwalled nanotorus using Cartesian product operation.
Corollary 4.8. Let T = T[p; q] be the molecular graph of a nanotorus. Then, the second hyper-Zagreb coindex of a q-multiwalled nanotorus is
Proof. We have by Proposition 3.2,Then,By using Corollaries 4.5–4.7, we obtain
5. 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. The 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
The authors declare that they have no conflicts of interest.