Abstract
The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer and a network , the networks and are its derived networks called as generalized subdivided and generalized semitotal point networks, where and are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks, and , and are -sum networks which are obtained by the lexicographic product of and , respectively, where . In this paper, for the integral value , we find exact values of the first and second Zagreb indices for generalized -sum networks. Furthermore, the obtained findings are general extensions of some known results for only . At the end, a comparison among the different generalized -sum networks with respect to first and second Zagreb indices is also included.
1. Introduction
Topological index (TI) being a molecular descriptor is a mathematical measure that associates a molecular network with a real number and predicts the underlying molecular network’s biological, chemical, and structural properties. Molecular descriptors were used by Wiener [1] and Trinjastic and Gutman and Trinajstić [2] to determine the boiling point of paraffin and the total -electron energy of the molecules, respectively. TIs are also used in the study of cheminformatics to classify molecules in terms of quantitative structure behavior and property relationships. Most notably, all TIs are invariants under the networks’ isomorphism parameter [3, 4]. Many TIs exist in the literature for networks. Degree-based TIs, distance-based TIs, and polynomial-based TIs are the three major types of these. The degrees-based TIs are more familiar than the others [5].
In the field of chemical network theory, networks operations are frequently used to discover the new families of networks. Yan et al. [6] explained the subdivision and semitotal point operations on a molecular network and attain the Wiener indices of the consequent networks and . After that, Eliasi and Taeri [7] explained the -sum network by using Cartesian product of and , where . Deng et al. [8] defined the -sum network with the help of lexicographic product and also calculated the first and second Zagreb indices, and Akhter and Imran [9] also calculated the forgotten index of the -sum networks. For more studies of the TIs under the operations of networks, see [10, 11].
Recently, for some integer , Liu et al. [12] defined the generalized subdivision and generalized semitotal point operations. They also computed the 1st and 2nd Zagreb indices for T-sum networks with the help of Cartesian product, i.e., for [13, 14]. In this study, we find the exact values of the Zagreb indices for the generalized -sum networks which are constructed with the help of generalized subdivision, generalized semitotal point operations, and lexicographic product. In the remaining paper, Section 1 has some previous knowledge related to our work and Section 2 has some basic definition. The key findings are presented in Section 3, and the conclusion and applications of these indices are presented in Section 4.
2. Preliminaries
We consider undirected, connected, and simple networks, where is a vertex set and is an edge set. In addition, is order and is size of G. Each vertices of a molecular network is referred to as an atom, and edges reflect the bonding between atoms.
The number of incident edges is called its degree. The networks and and , and are called path , cycle , and complete , respectively [3].
Definition 1. Let and be two networks with their vertex sets as and , respectively. The Cartesian product of these networks is defined as follows:with conditions either in and or and in .
Definition 2. Let and be two networks with their vertex sets as and , respectively. Then, the lexicographic product of these networks is defined aswith conditions either. in and or and in .
For integer , Liu et al. [12] defined the following graphs using the generalized subdivision and semitotal point operations:(i) network is found out by inserting -vertices in each edge of (ii) is found out from by connecting the old vertices which are adjacent in Eliasi and Taeri [7] proposed the generalized -sum networks under the operation of two connected networks and based on the Cartesian product. Deng et al. [8] defined the generalized -sum networks under the operation of two connected networks and based on the Lexicographic product as follows.
Definition 3. Let and be two networks and and be a network with vertex set and edge set . Then, the generalized -sum network is a graph with the vertex set:with conditions either in and or and in .
For more explanation, see Figures 1–4 .
(a)
(b)
(c)
3. Main Results
In this part, we will learn about the Zagreb indices of T-sum networks obtained by the operations of generalized subdivision and generalized semitotal point and lexicographic product.
Theorem 1. Let and be two networks; then, for some integer k, the first Zagreb index of -sum network is
Proof. Let be the degree of a vertex in the network :Now, first, we calculateSo, the result is
Theorem 2. Let and be two networks; then, for some integer k, the second Zagreb index of -sum network is
Proof. Let be the degree of a vertex in the graph :Now, first, we calculateSo, the final result is
Theorem 3. Let and be two networks; then, for some integer k, the first Zagreb index of -sum network is
Proof. Let be the degree of a vertex in the graph :Now, first, we calculateSo, the final result is
Theorem 4. Let and be two networks; then, for some integer k, the second Zagreb index of -sum network is
Proof. Let be the degree of a vertex in the network :Now, first, we calculateSo, the final result is
4. Conclusion
Let and be two networks; then, we have , , , , , , , and .
Applying these values on the above derived results in Theorems 1–4, we obtain Tables 1–4 . The graphical representations of the abovementioned results in tables are shown in Figures 5–8 . In Figure 9, we summarize the results of generalized -sum networks from k = 1 to k = 4.
For ,(a)(b)(c)(d)
For ,(a)(b)(c)(d)
For ,(a)(b)(c)(d)
For ,(a)(b)(c)(d)
In this paper, we proved the result of 1st and 2nd Zagreb indices of networks , by using generalized subdivision and generalized semitotal point operation and lexicographic product. The first and second Zagreb indices are in the form , where (i = 1, 2). We plot the network of these derived exact values individually and after that plot a combine network for to compare these results. In this result, it is clearly shown that the 2nd Zagreb indices shows better result as compared to others’ exact values.
Data Availability
All the data are included within this paper. However, more details of the data can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors have no conflicts of interest.
Acknowledgments
This work was supported by the Humanities and School science project of Anhui provincial Education Department: research on social crisis management in Anhui cities under the background of the data (subject no: SK2016A0233).