Abstract
Vertices and edges are made from a network, with the degree of a vertex referring to the number of connected edges. The chance of every vertex possessing a given degree is represented by a network’s degree appropriation, which reveals important global network characteristics. Many fields, including sociology, public health, business, medicine, engineering, computer science, and basic sciences, use network theory. Logistical networks, gene regulatory networks, metabolic networks, social networks, and driven networks are some of the most significant networks. In physical, theoretical, and environmental chemistry, a topological index is a numerical value assigned to a molecular structure/network that is used for correlation analysis. Hexagonal networks of dimension are used to build hex-derived networks, which have a wide range of applications in computer science, medicine, and engineering. For the third type of hex-derived networks, topological indices of reverse degree based are discussed in this study.
1. Introduction
A topological descriptor is a numerical value that represents the complete structure of a graph. In the study of topological descriptors, graph theory has shown to be a fruitful field of study. The primary elements of topological indices link the many chemical and physical characteristics of fundamental chemical substances. Vertex-edge-based topological indices are employed in the research of QSAR/QSPR for the prediction of bio-activity of different chemical compounds. With the dimension , hexagonal networks create hex-derived networks, which have a wide range of implementations in engineering, computer science, and also medicine. In [1], researchers created a new form of graph known as a “third type of hex-derived networks” [2, 3] and continued this work by calculating degree-based topological descriptors for these networks, in which they computed exact values of some vertex-edge named topological indices for this network.
Researchers have used graph theory to develop a range of helpful tools, including graph labeling, topological indices, and finding numbers. The subject of graph theory has several applications and implementations in various fields of study, including chemistry, medicine, and engineering. A polynomial, a series of integers, a numeric value, or a matrix can all be used to identify a graph. A chemical compound can be represented as a graph (or a diagram) or usually denoted as a molecular graph, nodes played a role of atoms, and the bonding between atoms is usually labeled as edges in the molecular graph theory. Recently, a new topic called cheminformatics was established, which is a mix of chemistry, information science, and mathematics, in which the QSPR/QSAR connection, bio-activity, and characterization of chemical compounds are investigated and reported in [4].
The topological descriptor is a numerical number associated with chemical compositions that maintain the relationship between chemical structures and a variety of physico-chemical characteristics, biological activity, and chemical reactivity. To describe the topology of a chemical network, it translated into a number, which is further used to create topological indices. Distance-based topological indices, degree-based topological indices, and counting-related topological indices are some of the most common forms of topological indices for graphs. Many academics have recently discovered topological indices for studying basic features of molecular graphs or networks. In [5–12], these networks have extremely compelling topological qualities that have been examined in distinct characteristics.
Let and represent the number of rows and number of triangles in each row of third-type chain hex-derived networks , respectively, shown in Figure 1. Let be a simple connected network, with a set of vertex and edges denoted by and , respectively. represents the order of and represents the size of . Let be the degree of a vertex in and be its reverse degree that was introduced by Kulli [13] and defined as , where denoted the maximum degree of the given graph. Let represent the edge partition of the given graph based on reverse degree of end vertices of an edge and represent its cardinality.
We define general reverse degree topological invariant as follows:
For latest results on topological descriptors for different chemical and computer networks and for general graphs, we refer to see [14–25]. In this current research work, we determine the exact values of all the above reverse indices.
2. Structure of Third-Type Hex-Derived Networks
With the help of complete graphs of order 3 (), Chen et al. [26] assembled a hexagonal mesh. In terms of chemistry, these graphs are also called oxide graphs. Figure 1 is obtained by joining these graphs. Two-dimensional mesh graph HX(2) (see Figure 2(a)) is obtained by joining six graphs and three-dimensional mesh graph HX(3) (see Figure 2(b)) is obtained by putting graphs around all sides of HX(2) [27]. Furthermore, repeating the same process by putting the graph around each hexagon, we obtained the hexagonal mesh. We have to note that the one-dimensional hexagonal mesh graph does not exist.
(a)
(b)
The novel network, labeled the third category of hex-derived networks, was developed in [1]. In [2, 3], they defined the graphically construction algorithm for the third type of hexagonal hex-derived network . Huo el at. [28] explained the graphical construction algorithm for chain hex-derived network of third type. In this paper, we denote it by , and different priorities of and the chain hex-derived networks are shown in Figure 3. In [29–33], you may find related research that utilizes this idea and that may benefit from the new research’s visions.
(a)
(b)
(c)
(d)
(e)
3. Main Results
In this section, we study the third-type chain hex-derived networks in the following three cases.(i)Case 1: for .(ii)Case 2: for , is odd and is a natural number. For , is odd and is a natural number. For , and both are even. For , and both are even.(iii)Case 3: for , is even and is odd. For , is even and is odd.
3.1. Results for Case 1
We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 1 for .
Lemma 1. Let be a third-type chain hex-derived networks. Then,
Proof. The graph contains edges and maximum degree in graph is 8. There are two types of reverse degree vertices in that are 1 and 5. Let us partition the edges of according to its reverse degrees according to Case 1 asNote that and , , and . Hence,After simplification, we obtain
Theorem 1. The general reverse Randić index of is equal to
Proof. For which is the general reverse Randić index of , from equation (2), we have ; therefore, , , and . Thus, by Lemma 1,Put , and we havePut , and we havePut , and we havePut , and we have
Theorem 2. Let be a third-type chain hex-derived networks. Then, the reverse atom-bond connectivity index is
The reverse geometric-arithmetic index is
The first reverse Zagreb index is
The reverse hyper-Zagreb index is
The reverse forgotten index is
Proof. For which is the reverse atom-bond connectivity index of , from equation (3), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the reverse geometric-arithmetic index of , from equation (4), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the first reverse Zagreb index of , from equation (5), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the first reverse hyper-Zagreb index of , from equation (6), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the reverse forgotten index of , from equation (7), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,
Theorem 3. Let be a third-type chain hex-derived networks. Then, the first reverse redefined index is
The second reverse redefined index is
The third reverse redefined index is
Proof. For which is the first reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the second reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,For which is the third reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 1 and after simplification,
3.2. Results for Case 2
We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 2 for .
Lemma 2. Let be a third-type chain hex-derived networks. Then,
Proof. The graph contains edges and maximum degree in graph is 8. There are two types of reverse degree vertices in that are 1 and 5. Let us partition the edges of according to its reverse degrees according to Case 2 asNote that and , , and . Hence,After simplification, we obtain
Theorem 4. The general reverse Randić index of is equal to
Proof. For which is the general reverse Randić index of , from equation (2), we have ; therefore, , , and . Thus, by Lemma 2,Put , and we havePut , and we havePut , and we havePut , and we have
Theorem 5. Let be a third-type chain hex-derived networks. Then, the reverse atom-bond connectivity index is
The reverse geometric-arithmetic index is
The first reverse Zagreb index is
The reverse hyper-Zagreb index is
The reverse forgotten index is
Proof. For which is the reverse atom-bond connectivity index of , from equation (3), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the reverse geometric-arithmetic index of , from equation (4), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the first reverse Zagreb index of , from equation (5), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the first reverse hyper-Zagreb index of , from equation (6), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the reverse forgotten index of , from equation (7), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,
Theorem 6. Let be a third-type chain hex-derived networks. Then, the first reverse redefined index is
The second reverse redefined index is
The third reverse redefined index is
Proof. For which is the first reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the second reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,For which is the third reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 2 and after simplification,
3.3. Results for Case 3
We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 3 for .
Lemma 3. Let be a third-type chain hex-derived networks. Then,
Proof. The graph contains edges, and maximum degree in graph is 8. There are two types of reverse degree vertices in that are 1 and 5. Let us partition the edges of according to its reverse degrees according to Case 3 asNote that and , , and . Hence,After simplification, we obtain
Theorem 7. The general reverse Randić index of is equal to
Proof. For which is the general reverse Randić index of , from equation (2), we have ; therefore, , , and . Thus, by Lemma 3,Put , and we havePut , and we havePut , and we havePut , and we have
Theorem 8. Let be a third-type chain hex-derived networks. Then, the reverse atom-bond connectivity index is
The reverse geometric-arithmetic index is
The first reverse Zagreb index is
The reverse hyper-Zagreb index is
The reverse forgotten index is
Proof. For which is the reverse atom-bond connectivity index of , from equation (3), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the reverse geometric-arithmetic index of , from equation (4), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the first reverse Zagreb index of , from equation (5), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the first reverse hyper-Zagreb index of , from equation (6), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the reverse forgotten index of , from equation (7), we have ; therefore, , and . Thus, by Lemma 3 and after simplification,
Theorem 9. Let be a third-type chain hex-derived networks. Then, the first reverse redefined index is
The second reverse redefined index is
The third reverse redefined index is
Proof. For which is the first reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the second reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,For which is the third reverse redefined index of , from equation (8), we have ; therefore, , , and . Thus, by Lemma 3 and after simplification,
4. Numerical and Graphical Representation
In this section, we determine the numerical values of , and in Tables 1–3, for Case 1, Case 2, and Case 3, respectively. We represent these results graphically in Figures 4–6.
Data Availability
No data were used to support the findings of the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.