Zagreb Connection Indices of Molecular Graphs Based on Operations

Cao, Jinde; Ali, Usman; Javaid, Muhammad; Huang, Chuangxia

doi:https://doi.org/10.1155/2020/7385682

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Abstract Introduction Preliminaries Applications and Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 7385682 | https://doi.org/10.1155/2020/7385682

Zagreb Connection Indices of Molecular Graphs Based on Operations

Jinde Cao,¹Usman Ali,²Muhammad Javaid,²and Chuangxia Huang³

Academic Editor: Qingling Wang

Received15 Nov 2019

Accepted05 Feb 2020

Published30 Mar 2020

Abstract

Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.

1. Introduction

Graph theory has provided a variety of useful tools in which one of the best tools is a topological index (TI). Molecules and molecular compounds are often modeled by molecular graphs. The topological indices (TIs) predict hydrocarbon, physicochemical, and structural properties of the molecular graphs such as critical temperature, ZE-isomerism, chirality, solubility, molecular mass, and connectivity, see [1–4]. Medical behaviours of the drugs, crystallin materials, and nanomaterials which are very important for chemical and pharmaceutical industries are also studied by TIs, see [5–8]. Todeschini et al. [9] also reported that TIs are widely used in the study of quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs). These relationships play a vital role in the subject of cheminformatics, see [9–13].

TIs have been divided into different classes, but degree-based are studied more, see [1, 4, 7, 14, 15, 16]. Gutman and Trinajstić [17] investigated the correlation value between the total π-electron energy and the structure of a molecule using the first Zagreb index. Gutman et al. [18] developed their work and established another TI for molecular structures called the second Zagreb index. After that, many extended works have been appeared on these invariants. For more study, we refer to [9, 10, 19, 20]. Another TI was studied by Gutman and Trinajstić in the same paper [17], but there was not more attention on this index by other researchers up to 2017. Ali and Trinajstić [21] restudied this TI and renamed it as the modified first Zagreb connection index (ZCI). They also reported that it has more precise values of the correlation coefficients of various octane isomers. Du et al. [22], Ducoffe et al. [23], and Shao et al. [24] determined extremal alkanes and cycloalkanes under different conditions using this ZCI. Zhu et al. [25] established the lower bound by using the modified first ZCI of trees in terms of their order and maximum degree. Tang et al. [26] computed the first and second (ZCI) and modified first (ZCI) of the S-sum graphs.

A simple graph can be molded into a chemical structure by using some operations. First of all, Graovac and Pisanski [27] computed different results of the Wiener index using product based on operations. So, many chemical graphs can be generated by using simple graphs based on operations such as alkane is the corona product of , a type of cycloalkanes is cyclohexane that is the corona product of , a polynomial chain and nanotube are the Cartesian product of and , and a fence and closed fence are the lexicographic product of and . Up till now, many results of the various TIs have been presented under different molecular graphs based on operations, see [26, 28–38].

In this paper, we compute the second ZCI and modified first ZCI of the resultant graphs which are obtained by applying various operations of corona product, Cartesian product, and lexicographic product (composition) in the form of exact formulae and upper bounds. The rest of the paper is settled as Section 2 represents the preliminary definitions and results, Section 3 covers the general results of molecular graphs based on operations, and Section 4 includes the applications and conclusion.

2. Preliminaries

Let be a simple and connected molecular graph with a vertex set and an edge set . A null graph (N) has at least two vertices and no edge. It becomes a trivial graph if it has exactly one vertex and no edge. Todesehini et al. [9] defined for such that and are called the degree and connection number of the vertex . Now, throughout the paper, we assume that and are two connected graphs such that , , , and .

Alkanes are the simplest organic compounds containing a single bound between carbon atoms. They are also called hydrocarbon compounds. Its simple and Lewis structures are shown in Figure 1.

(a)

(b)

The some examples of alkanes are methane , ethane , and propane , and their Lewis structures are shown in Figure 2.

(a)

(b)

(c)

Cycloalkanes are cyclic organic compounds containing the closed chain of carbon atoms. In other words, a cycloalkane is arranged into a chemical structure obtained a single ring (sometime side chains may be attached), and all of the carbon-carbon bonds are single. These are classified into two classes as homocyclic and heterocyclic compounds. If cyclo-organic compounds containing between carbons [1 to 5], [6 to 10], and [11 to on wards] are called small, mediam, and large cyclo organic compounds, respectively. Cycloalkanes are the isomers of alkene, e.g., is used as the same chemical formula of cyclopropane as well as propene. The general formula of cycloalkanes is . The sets of are examples of cycloalkanes that are classified by homocyclic and heterocyclic compounds, respectively. Furthermore, the following Lewis structures of these cycloalkanes are shown in Figure 3.

(a)

(b)

Definition 1. For a graph Q, the first Zagreb index , second Zagreb index , and their coindices are defined asGutman et al. [17, 18, 39] defined the different degree-based TIs which are frequently used in the studies of QSPR and QSAR [40–43]. Corresponding to these degree-based TIs, the connection-based TIs are defined in Definition 2. For further studies of connection-based TIs, see [21, 22, 44].

Definition 2. For a graph Q, the first Zagreb connection index , second Zagreb connection index , and modified first Zagreb connection index are defined as

Definition 3. The corona product of two graphs and is obtained by taking one copy of and copies of (i.e., ) and then by joining each vertex of the copy of to the vertex of one copy of , where . Also, a number of vertex set and edge set are defined as: and . For more detail, see Figure 4.

(a)

(b)

(c)

Definition 4. The Cartesian product and lexicographic product or composition of two graphs and are obtained by taking the vertex set and the edge set , where with conditions(i)either or ,(ii)either or , respectively.For more detail, see Figures 5 and 6.

(a)

(b)

(c)

(a)

(b)

(c)

Lemma 1 (see [45]). Let Q be a connected graph and be its complement. Then, (i) , (ii) , and (iii) , where and .

Lemma 2 (see [46]). Let Q be a connected graph with n vertices and e edges. Then, , where equality holds if and only if Q is a free graph.

Lemma 3 (see [26]). Let Q be a connected and free graph with n vertices and e edges. Then,

3. Main Results

This section consists on the main results.

Theorem 1. Let and be two connected and free graphs. Then, and of the corona product of and are as follows:

Proof. (a)If for any either or , where and(i)Case I: if , then .(ii)Case II: if , then Take Also take Similarly, Consequently, (b) Take Also take Similarly, Consequently,

Theorem 2. Let and be two connected and free graphs. Then, and of the Cartesian product of and are as follows:

Proof. (a)For , , and , we have Take Similarly, Consequently, (b)Consider Take Similarly, Consequently,

Theorem 3. Let and be two connected graphs. Then, and of the composition (or lexicographic product) of and are as follows:

Proof. (a)For , , and , we have Take Also take Let’s suppose that Similarly, Consequently, (b) Take Also take Similarly, Consequently,

4. Applications and Conclusion

In this section, we present some applications of the obtained results for particular chemical structures such as alkanes (see Figure 7), cycloalkanes (see Figure 8), linear polynomial chain (see Figure 9), carbon nanotubes (see Figure 10), fence (see Figure 11), and closed fence (see Figure 12). We also give the both exact and computed values of the obtained results for the aforesaid particular chemical structures to develop an easy understanding. Let be a null graph, , , and be three particular alkanes called by paths, and , , and be cycles.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

4.1. Corona Product

Example 1. Alkane ().(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of

Example 2. Cyclohexane ().(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of

4.2. Cartesian Product

Example 3. Polynomial chain.(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of

Example 4. Carbon nanotube ().(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of

4.3. Lexicographic Product

Example 5. Fence ().(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of

Example 6. Closed fence ().(i)Exact value of (ii)Exact value of (iii)Computed value of (iv)Computed value of In this paper, we have computed the general results related to the second ZCI and modified first ZCI of the resultant graphs which are obtained with the help of various operations of product on graphs such as corona product, Cartesian product, and lexicographic product (composition). The obtained results also illustrated with the help of particular class of molecular graphs. However, the problem is still open to compute the ZCI of the molecular graphs under the different operations of subdivision, addition and product, etc.

Data Availability

All the data are included within this paper. However, the reader may contact the corresponding author for more details of the data.

Conflicts of Interest

The authors have no conflicts of interest.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (grant no 61833005) and in part by the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence (grant no BM2017002).

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