Abstract
A topological index can be focused on uprising of a chemical structure into a real number. The degree-based topological indices have an active place among all topological indices. These topological descriptors intentionally associate certain physicochemical assets of the corresponding chemical compounds. Graph theory plays a very useful role in such type of research directions. The hex-derived networks have vast applications in computer science, physical sciences, and medical science, and these networks are constructed by hexagonal mesh networks. In this paper, we determined the exact values of vertex-edge-degree-based topological descriptors for hex-derived networks and , which are generated by the hexagonal network of dimension .
1. Introduction
The applications of topological descriptors of chemical structure are nowadays a normal process in the education of structure-property relations, specifically in QSPR/QSAR studies. Topological indices play a dynamic part in the QSPR/QSAR study. They associate certain physicochemical assets of chemical compounds. Graph theory has provided pharmacists with an assortment of suitable apparatuses, such as topological indices. Chemicals and chemical compounds are frequently displayed by chemical graphs. A chemical graph is an illustration of the structural formula of a chemical compound in terms of graph theory, in which atoms are denoted with vertices and edges show the chemical bonding between them. Lately, a latest topic that has piqued the interest of researchers is cheminformatics, which is a composite of chemistry, information science, and mathematics, in which the QSAR/QSPR relationship, bioactivity, and classification of chemical compounds are investigated.
The topological descriptor is a real number associated with chemical compositions that maintains the correlation of chemical structures with a variety of physicochemical properties, chemical reactivity, or biological activity. Topological indices are classified into three types: distance-based topological indices, degree-based topological indices, and counting-related topological indices. Numerous researchers have recently discovered topological indices for the study of fundamental properties of molecular graphs or networks. These networks have very appealing topological properties, which have been considered in various characteristics such as [1–8].
Chen et al. [9] explained the construction of hexagonal mesh networks that consist of triangles, as shown in Figure 1. Furthermore, we gather the hexagonal mesh by putting triangles around the boundary of each hexagon. Imran et al. defined the new hex-derived networks, namely, first type (see Figure 2) and second type (see Figure 3); for detailed construction, see [10]. Simonraj and George [11] created the new network which is named as third type of hex-derived networks. Koam et al. [12] computed the vertex-edge-based topological indices of some hex-derived networks. There are some works related to hex-derived networks which can be seen in [13–15]. Related research papers that contain the theoretical as well as application aspects for new research directions can be found in [16–22].
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2. Preliminaries
Let be a simple connected graph with being the edge set and being the vertex set. The symbol denoted the concept of degree of a vertex , and it is defined by the number of attached edges with . The symbol denoted the number of all vertices adjacent to and is called as the open neighborhood of a vertex . On the contrary, the symbol is the union of and and called as the closed neighborhood of . The concept of -degree denoted by , and can be defined as follows: for any vertex from the vertex set of a graph, is the number of different edges that are attached to any vertex from the closed neighborhood of . In this research work, we elaborated different -degree-associated topological descriptors. In [23], vertex/edge-degree-based topological indices are defined in which they computed the degree of an edge as . In this article, we consider -degree which is the degree of a vertex and is calculated by adding the degrees of its all-neighboring vertices.
The researchers in [24–32] detailed different -degree topological invariants. This research work contains the computational exact results of given above descriptors.
3. Hex-Derived Network
Let be the notation for the hex-derived network of the first type, and it is shown in Figure 2. The original hex-derived network contains total number of vertices in which there are vertices of degree 3, 6 vertices of degree 5, vertices of degree 7, and vertices of degree 12. There are count of edges for the graph ; all these edges are partitioned into eight subsets according to their degrees and corresponding -degrees of end vertices elaborated in equations (9)–(16).
Let be the edge partition of according to its degrees and -degrees . It is defined as
And represents the number of edges in .
Given below are some -degree-based indices, for example, index, index, index, index, index, index, and index for .
Theorem 1. Let be the first type of hex-derived network; then,(i)(ii)(iii)
Proof. Let be the notation for the hex-derived network of the first type, and it is shown in Figure 2. The original hex-derived network contains total number of vertices in which there are vertices of degree 3, 6 vertices of degree 5, vertices of degree 7, and vertices of degree 12. There are count of edges for the graph ; all these edges are partitioned into eight subsets according to their degrees and relative -degrees of both end vertices elaborated in equations (9)–(16). Evaluating equation (2), we can determine the first -degree Zagreb index asEvaluating equations (9)–(16) and after simplifications, we will have the required results asEvaluating equation (3), we can compute the second -degree Zagreb index asEvaluating equations (9)–(16) and after simplifications, we will have the required result asEvaluating equation (4), we can determine the -degree harmonic index asEvaluating equations (9)–(16) and after simplifications, we get
Theorem 2. Let be the first type of hex-derived network; then,(i)(ii)
Proof. The -degree Randic index can be determined by evaluating the edge partitions in equation (5):The methodology of edge partitions can be determined from equations (9)–(16); after mathematical calculations, we get the following result:The -degree sum-connectivity index can be determined by using the values from equations (9)–(16) in equation (6); we get the following:After simplification, we obtain
Theorem 3. Let be the third type of hex-derived network; then,(i)(ii)
Proof. The numerical descriptor of the -degree atom-bond connectivity index can be calculated by evaluating the values of edge partitions in equation (7):Evaluating equations (9)–(16) and after mathematical calculations, we will haveThe -degree geometric-arithmetic index can be calculated by evaluating the methodology of edge partitions in (1):Evaluating equations (9)–(16) and after mathematical calculations, we will have
4. Hex-Derived Network
Let be the notation of the hex-derived network of the second type, and it is shown in Figure 3. This hex-derived network contained total number of vertices in which there are vertices of degree 5, vertices of degree 6, vertices of degree 7, and vertices of degree 12.
The total number of edges of this network is , and they can be partitioned into ten different subsets according to their degrees and associated with -degrees of both end vertices, which are shown in equations (17)–(26). Given below are some -degree-based indices, for example, index, index, index, index, index, index, and index for the network. Let be the edge partition of according to its degrees and -degrees . It is defined as
And represents the number of edges in .
Theorem 4. Let be the second type of the hex-derived network; then,(i)(ii)(iii)
Proof. Let be the second type of the hex-derived network which is shown in Figure 3. The hex-derived network has vertices. There are number of edges of which are partitioned into ten partitions based on their degrees and corresponding -degrees of end vertices given in equations (17)–(26).
The first -degree Zagreb index can be calculated asEvaluating equations (17)–(22) and after simplifications, we will have the final result as follows:The second -degree Zagreb index can be calculated asEvaluating equations (17)–(22) and after simplifications, we will have the final result as follows:The -degree harmonic index can be determined asEvaluating equations (17)–(22) and after some calculations, we have the final result as follows:
Theorem 5. Let be the second type of the hex-derived network; then,(i)(i)(ii)
Proof. The -degree Randic index is measured by evaluating equations (17)–(26) in the following:After simplification, we obtainThe -degree sum-connectivity index is measured by evaluating equations (17)–(26) in the given following formula:After simplification, we obtain
Theorem 6. Let be the second type of the hex-derived network; then,(i)(i)(ii)
Proof. The -degree atom-bond connectivity index is measured asBy putting the values from equations (17)–(26) and after simplification, we obtainThe -degree geometric-arithmetic index is measured asBy putting the values from equations (17)–(22) and after simplification, we obtain
5. Conclusion
In this paper, certain -degree-based topological indices, namely, first -degree Zagreb index, second -degree Zagreb index, -degree harmonic index, -degree Randic index, -degree sum-connectivity index, -degree atom-bond connectivity index, and -degree geometric-arithmetic index, for the first type of hex-derived network and second type of hex-derived network are studied. We calculated exact formulas of the aforementioned -degree-based topological indices for these derived networks. These results provide a basis to comprehend the deep network topology of these vital networks. The numerical comparison of the first type of hex-derived network and second type of hex-derived network is shown in Tables 1 and 2.
Data Availability
No data were associated with this article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research was supported by the University Program of Advanced Research (UPAR) and UAEU-AUA grants of United Arab Emirates University (UAEU) via Grant nos. G00003271 and G00003461.