Abstract

A numeric parameter which studies the behaviour, structural, toxicological, experimental, and physicochemical properties of chemical compounds under several graphs’ isomorphism is known as topological index. In 2018, Ali and Trinajstić studied the first Zagreb connection index to evaluate the value of a molecule. This concept was first studied by Gutman and Trinajstić in 1972 to find the solution of -electron energy of alternant hydrocarbons. In this paper, the first Zagreb connection index and coindex are obtained in the form of exact formulae and upper bounds for the resultant graphs in terms of different indices of their factor graphs, where the resultant graphs are obtained by the product-related operations on graphs such as tensor product, strong product, symmetric difference, and disjunction. At the end, an analysis of the obtained results for the first Zagreb connection index and coindex on the aforesaid resultant graphs is interpreted with the help of numerical values and graphical depictions.

1. Introduction

Topological indices (TIs) are used in the study of cheminformatics which has key role in the formational structure of quantitative structures’ activity and property relationships to examine the chemical reactivity and experimental activity of a chemical compound in a molecular graph [1]. So, TIs predict both physical and chemical features that are defined in the molecular graphs such as surface tension, solubility, connectivity, freezing point, boiling point, melting point, critical temperature, polarizability, heat of evaporation, and formation [2]. In addition, the medical behaviours and a number of drug particles of different compounds have formed with the help of various TIs in the pharmaceutical networks (see [3]). In particular, the TIs called by connection number-based Zagreb indices are used to compute the correlation values among various octane isomers such as heat of evaporation, acentric factor, molecular weight, connectivity, critical temperature, stability, and density (see [4, 5]).

Product graphs play an essential part to develop new molecular graphs from simple graphs. For this purpose, Ashrafi et al. [6] defined the concept of coindices for several products on graphs. Das et al. [7] computed upper bounds for multiplicative Zagreb indices of operations such as join, corona, Cartesian, disjunction, and composition. The reformulated, multiplicative, hyper, first, second, and third Zagreb coindices with certain properties are defined in [8–13]. Relations between Zagreb coindices and some distance-based indices are computed in [14]. For more details, see [15–18]. For this purpose, we can define some operations such as tensor product, strong product, symmetric difference, and disjunction in the following definitions.

Definition 1. Tensor (or Kronecker or conjunctive or direct) product of two graphs and is a graph with vertex set and edge set , where andThe connection number of a vertex of is defined by . For more details, see Figure 1.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 1 

(a) , (b) , and (c) .

Definition 2. Strong product or normal product of two graphs and is obtained by taking the vertex set and edge set as and , where with conditionThe connection number of a vertex of is defined byif . For more details, see Figure 2.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 2 

(a) (b) , and (c) .

Definition 3. Symmetric difference of two graphs and is a graph with vertex set and edge set , where andbut not both hold at the same time, respectively.
The connection number of a vertex of is defined byFor more details, see Figure 3.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 3 

(a) , (b) , and (c) .

Definition 4. Disjunction of two graphs and is a graph with vertex set and edge set , where andThe connection number of a vertex of is defined byFor more details, see Figure 4.
The graph theory provides the significant tools in the field of modern chemistry that is exploited to develop several types of molecular graphs and also predicts their chemical properties. In 1972, Gutman and Trinajstić [19] defined the first degree-based (number of vertices at distance one) TI called the first Zagreb index to compute the total -electron energy of the molecules in molecular graphs. There are several TIs in literature, but degree-based TIs are studied more than others (see [20–29]).
Ali and Trinajstić [30] restudied the concept of connection number-based (number of vertices at distance two) TIs that were also defined by Gutman and Trinajstić in the same paper in 1972 to compute the total electron energy of the alternant hydrocarbons. They recalled them as Zagreb connection indices and reported that the chemical capability of the Zagreb connection indices is better than the classical Zagreb indices for the thirteen physicochemical properties of octane isomers. After two years, a few works have delivered on these connection number-based descriptors. Ali et al. [31] computed the analysis of Zagreb connection indices and coindices for some chemical structures of operations on graphs. For further studies and properties of the Zagreb connection indices, we refer to [32–40]. Recently, Gong et al. [41, 42] developed blocking lot-streaming flow shop scheduling problems and dynamic interval multiobjective optimization problems. These problems have been considered in various studies which have a close relation with the topic considered in this paper. For more details, see [43, 44].
In this paper, we compute the analysis for the first Zagreb connection index and coindex of the resultant graphs in the shape of exact formulae and upper bounds in terms of their factor graphs, where resultant graphs are obtained by operations such as tensor product, strong product, symmetric difference, and disjunction. Moreover, at the end, an analysis of the first Zagreb connection index and coindex on the aforesaid operations is included with the help of their numerical values and graphical depictions.
The rest of the paper is as follows: Section 2 represents the preliminary definitions and key results which are used in the main results, Section 3 contains the main results of product on graphs, and Section 4 includes the analysis and conclusions.

(a)

(b)

(c)

(a)
(b)
(c)

Figure 4 

(a) , (b) , and (c) .

2. Preliminaries

Let be a simple graph such that the order and size of are and . The distance between any two vertices and of a graph is equal to the length of a shortest path connecting them. For and a positive integer , denotes the open -neighborhood of in a graph , where is called -distance degree of a vertex in any graph . The degree of a vertex in a graph is the number of edges incident on it, and it is denoted by . In particular: If , (number of vertices at distance one from ) If ,  connection number of  b (number of vertices at distance two from )

The complement of a graph is denoted by . It is also simple with the same vertex set as of , but edge set is defined as . Thus, , where is a complete graph of order and size . Moreover, if , then and , where and are the degrees of the vertex in and , respectively. Now, throughout the paper, for two graphs and , we assume that , , , and . Finally, it is important to note that Zagreb connection coindices of are not Zagreb connection indices of because the connection number works according to . For any terminology or notion which are not mentioned here, we refer to [45, 46].

Definition 5. For a graph , the first Zagreb index and second Zagreb index are defined asThese degree-based indices are defined by Gutman, Trinajstić, and Ruscic (see [19, 47]) which are frequently used to predict better outcomes of the various parameters related to the molecular networks such as chirality, complexity, entropy, heat energy, ZE-isomerism, heat capacity, absolute value of correlation coefficient, chromatographic, retention times in chromatographic, pH, and molar ratio [19, 48]. Symmetrical to these degree-based TIs, the connection-based TIs are discussed in Definitions 6 and 7.

Definition 6. For a graph , the first Zagreb connection index is defined as

Definition 7. For a graph , the modified first Zagreb connection index and second Zagreb connection index are defined as

Definition 8. For a graph , the first Zagreb coindex and second Zagreb coindex are defined asThese coindices associated with the degree-based classical Zagreb indices are defined by Ashrafi et al. (see [6]). The coindices associated with the connection-based Zagreb indices are defined in Definition 9.

Definition 9. For a graph , the first Zagreb connection coindex and second Zagreb connection coindex are defined asThe modified coindices associated with the connection number-based Zagreb indices are defined in Definition 10. These modified coindices associated with the connection number-based Zagreb indices are defined by Ali et al. (see [49]).

Definition 10. For a graph , the modified first Zagreb connection coindex and the modified second Zagreb connection coindex are defined asThe degree/connection-based coindices defined in Definitions 8–10 study the various physicochemical and isomer properties of molecules on the basis of the adjacency and nonadjacency pairs of vertices in the molecular networks. For more details, see [6, 30, 31, 36].
Now, we present some important results which are used in the main results.

Lemma 1 (see [50]). Let be a connected graph and be its complement with vertices and edges. Then,(a)(b)

Lemma 2 (see [36]). Let be a connected and -free graph with vertices and edges. Then,(a)(b)

3. Main Results

This section contains the main results for the first Zagreb connection index and first Zagreb connection coindex of the product on graphs obtained under the operations of tensor product, strong product, symmetric difference, and disjunction.

Theorem 1. Let and be two connected and -free graphs. Then, and of the tensor product are

Proof. Since , where , , and ,Using Lemma 2 (a) and (b),

Theorem 2. Let and be two connected and -free graphs. Then, and of the strong product are

Proof. Since , where , , and ,Using Lemma 2 (a) and (b),We takeWe know thatAlso, we takeAgain, we takeConsequently,