Abstract
Numerous studies based on mathematical models and tools indicate that there is a strong inherent relationship between the chemical properties of the chemical compounds and drugs with their molecular structures. In the last two decades, the graph-theoretic techniques are frequently used to analyse the various physicochemical and structural properties of the molecular graphs which play a vital role in chemical engineering and pharmaceutical industry. In this paper, we compute Zagreb indices of the generalized sum graphs in the form of the different indices of their factor graphs, where generalized sum graphs are obtained under the operations of subdivision and strong product of graphs. Moreover, the obtained results are illustrated with the help of particular classes of graphs and analysed to find the efficient subclass with dominant indices.
1. Introduction
In many fields (chemistry, physics, computer science, and electrical networks) various physicochemical and structural properties such as melting point, boiling point, chemical bonds, bond energy, solubility, surface tension, critical temperature, connectivity, stability, density, and polarizability are studied with the help of various TIs (degree-based, distance-based, and polynomial-based). Moreover, degree-based TIs have been used as a powerful approach to discover many new drugs, such as anineoplastics, anticonvulsants, antiallergics, antimalarials, and silico generation (see [1]). Therefore, this practice has proven that the TIs and the quantitative structure-activity (or structure-property) relationships (QSAR or QSPR) have presented a foundation stone in chemical engineering and pharmaceutical industry for the process of the drug design and discovery (see [2, 3]).
Let be a collection of (molecular) graphs in which each graph is considered as a simple graph without multiedges and loops. A topological index (TI) is a function that assigns a real number to each element (graph) of , where is a set of real numbers. Moreover, for two graphs and , if and only if is isomorphic to . Mostly, TIs are computed for the hydrogen-suppressed molecular graphs in which the atoms are represented by nodes and bonds between them by edges. In 1947, Wiener index (path number), first distance-based TI, is utilized in the study of paraffin’s boiling point [4]. Gutman and Trinajstic [5] calculated total -electrons energy of the molecules through a degree-based TI called as the first Zagreb index (FZI). They also studied the various properties of the second Zagreb index (SZI) in the same paper. In chemical graph theory, many more TIs are introduced in [6], but degree-based TIs are prominent than others. For more details, we refer to [7, 8].
On the other hand, operations on graphs (addition, complement, deletion, switching, subdivision, union, intersection, and product) also play a very important role in the construction of new graphs and structures. Yan et al. [9] introduced the four operations on the subdivision of a graph and obtained the Wiener index of these resultant graphs . For , Taeri and Eliasi [10] defined the -sum graphs using the Cartesian product on graphs and , where are connected graphs. They also studied the Wiener index of these -sum graphs. Liu et al. [11] constructed -sum graphs with the help of the Cartesian product on the graphs and and calculated the first general Zagreb index for these graphs, i.e., . Liu et al. [12] introduced the generalized -sum (-sum) graphs with the help of the Cartesian product on the graphs and , where represents some integral value. Moreover, they calculated the mathematical expressions of the Zagreb indices for these graphs, i.e., and . Furthermore, Awais et al. [13, 14] computed the forgotten topological and hyper-Zagreb indices of generalized F-sum graphs based on Cartesian product in terms of its factor graphs. Recently, Awais et al. [15] computed the first general Zagreb index of -sum graphs in terms of TIs of their factor graphs.
In the current study, we study the generalized -sum graphs which are obtained under the operation of strong product on the graphs and , where and is some counting number. Mainly, we compute the Zagreb indices of these generalized -sum graphs based on strong product such as and . Moreover, a comparison is also organized of the generalized -sum graphs , , , and with respect to both the Zagreb indices ( and ). The rest of the paper is settled as follows: Section 2 covers basic notions, Section 3 predicated on main results, and conclusively Section 4 included the application and conclusion.”
2. Preliminaries
A graph is a structure consisting of two finite sets of vertices and edges in which pairs of vertices are connected by edges. In particular, a graph will refer to a simple undirected graph if each edge connects two distinct vertices and there are no parallel edges. Throughout the paper, the order of is , and the size of a is . Given two vertices and , if , then and are said to be adjacent. The strength of edges which are incident on any node is known as its degree [16]. Here, we defined few topological indices.
Definition 1. Let be a simple undirected graph. The first Zagreb index and second Zagreb index are:In 1972, Trinajsti and Gutman [5] introduced these two TIs which are used in study of structure-based properties of (molecular) graphs (see [17–19]). In 1960, Sabidussi [6] introduced the strong product for two graphs and with vertex set as Cartesian product such that and will be adjacent in iff and is adjacent to or and is adjacent to or is adjacent to and is adjacent to . Strong product is union of tensor product and Cartesian product.
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Definition 2. The four generalized operations related to the subdivision of graphs defined in [15] are given as follows:(i)k-subdivision operation can be made by adding new node in each of , where is an integral value(ii)k-semitotal-point graph with node set and link set (iii)k-semitotal line graph with node set and link set .(iv)k-total point graph with node set and link set (for more details, see Figure 1).
Definition 3. Let be two graphs, is an operation, and is obtained after applying on having edge-set and node set . The generalized -sum graph is a graph having node set:such that two nodes of are adjacent iff or or [], where is a positive number. We noticed that the generalized -sum graphs contain copies of graphs that are labeled with the nodes of . For more details, see Figures 2 and 3.
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3. Main Results
Now, we will prove the key results of and in terms of its factor graphs and . We assume that be two simple, undirected, and connected graphs with order and size, and respectively.
Theorem 1. Let and be two connected graphs such that , . For ,
Proof. Let be the degree of a vertex in the graph .ConsiderSince in this case , we haveSince in this case , we haveConsequently, we getConsiderTherefore,
Theorem 2. Let and be two connected graphs such that , . For ,
Proof. Let be the degree of a vertex in the graph .Hence,ConsiderConsequently, we have
Theorem 3. Let and be two connected graphs such that , . For ,
Proof. Let be the degree of a vertex in the graph .Now, we split this sum into two parts for the vertices, and , where . Assume that , where cover the edges of which are in the same edges of and of in two different adjacent edges of .Now, we split this sum into two parts for the vertices, and , where . Assume that , where cover the edges of which are in the same edges of and of in two different adjacent edges of .Consequently, we haveNext,ConsiderNow, we split this sum into two parts for the vertices, and , where . Assume that which are in the same edges of and of in two different adjacent edges of .where is the added vertex in the edge and is added vertex in the edges of :where is the number of neighbors which are common vertices of and in .Now, we split this sum into two parts for the vertices, and , where . Assume that which are in the same edges of and of in two different adjacent edges of .where is the added vertex in the edge and is added vertex in the edges of :where is the number of neighbors which are common vertices of and in .
Consequently, we have
Theorem 4. Let and be two connected graphs such that , . For ,
4. Applications and Conclusion
In this section, we have computed the first and second Zagreb indices of generalized -sum graphs based on strong product as application of Theorem 1 to Theorem 4 for given as follows. We also find the subclass with better Zagreb indices as presented in Tables 1 and 2 and Figure 4(i)-sum:(ii)-sum:(iii)-sum:(iv)-sum:
Now, we close our discussion with the conclusion that both the Zagreb indices of the generalized T-sum graph are dominant among the Zagreb indices of all the generalized sum graphs as shown in Figure 4. We also conclude that, in the generalized T-sum graph, the number of vertices (atoms) and edges (bonds) between them are more than the other graphs of this family for each integral values of . Thus, the role of Zagreb indices for generalized T-sum graph remains dominant for each integral values of . However, the problem is still open to find different degree- and distance-based TIs for the generalized sum graphs obtained under the various operations of product of graphs.
Data Availability
The data used to support the findings of this study are included within the article. Additional data can be obtained from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
Acknowledgments
This work was supported by University of Tabuk, Tabuk, Saudi Arabia.