On the Study of Reverse Degree-Based Topological Properties for the Third Type of p th Chain Hex-Derived Network

Vertices and edges are made from a network, with the degree of a vertex referring to the number of connected edges.*e 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 t 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.


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. e 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 p, 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. e 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]. e 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][6][7][8][9][10][11][12], these networks have extremely compelling topological qualities that have been examined in distinct characteristics.
Let p and q represent the number of rows and number of triangles in each row of third-type p th chain hex-derived networks G 3 p,q , respectively, shown in Figure 1. Let G be a simple connected network, with a set of vertex and edges denoted by V and E, respectively. |V| represents the order of G and |E| represents the size of G. Let d θ be the degree of a vertex θ ∈ V in G and R θ be its reverse degree that was introduced by Kulli [13] and defined as where Δ denoted the maximum degree of the given graph. Let E R θ ,R ϑ represent the edge partition of the given graph based on reverse degree of end vertices of an edge θϑ ∈ E and |E R θ ,R ϑ | represent its cardinality.

Structure of Third-Type Hex-Derived Networks
With the help of complete graphs of order 3 (K 3 ), Chen et al. [26] assembled a hexagonal mesh. In terms of chemistry, these K 3 graphs are also called oxide graphs. Figure 1 is obtained by joining these K 3 graphs. Two-dimensional mesh graph HX(2) (see Figure 2(a)) is obtained by joining six K 3 graphs and three-dimensional mesh graph HX(3) (see Figure 2(b)) is obtained by putting K 3 graphs around all sides of HX(2) [27]. Furthermore, repeating the same process by putting the tK 3 graph around each hexagon, we obtained the t th hexagonal mesh. We have to note that the one-dimensional hexagonal mesh graph does not exist. e novel network, labeled the third category of hexderived networks, was developed in [1]. In [2,3], they defined the graphically construction algorithm for the third type of hexagonal hex-derived network HHDN3(t). Huo el at. [28] explained the graphical construction algorithm for m th chain hex-derived network of third type. In this paper, we denote it by G 3 p,q , and different priorities of p and q the chain hex-derived networks are shown in Figure 3. In [29][30][31][32][33], you may find related research that utilizes this idea and that may benefit from the new research's visions.

Main Results
In this section, we study the third-type p th chain hex-derived networks G 3 p,q in the following three cases. (i) Case 1: for p � q, (p, q) ≥ 1.
(ii) Case 2: for p < q, p is odd and q is a natural number.
For p > q, p is odd and q is a natural number. For p < q, p and q both are even. For p > q, p and q both are even. (iii) Case 3: for p < q, p is even and q is odd. For p > q, p is even and q is odd.

Results for Case 1.
We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 1 for G 3 p,q .

Lemma 1.
Let G 3 p,q be a third-type p th chain hex-derived networks. en, Proof. e graph G 3 p,q contains 12pq edges and maximum degree in G 3 p,q graph is 8. ere are two types of reverse degree vertices in G 3 p,q that are 1 and 5. Let us partition the edges of G 3 p,q according to its reverse degrees according to Case 1 as Note that After simplification, we obtain Journal of Mathematics Put α � 1, and we have Put α � (1/2), and we have Put α � (−1/2), and we have Put α � −1, and we have □ Theorem 2. Let G 3 p,q be a third-type p th chain hex-derived networks. en, the reverse atom-bond connectivity index is e reverse geometric-arithmetic index is e first reverse Zagreb index is e reverse hyper-Zagreb index is e reverse forgotten index is Proof. For RABC(G 3 p,q ) which is the reverse atom-bond connectivity index of G 3 p,q , from equation (3), we have  (3).

Results for Case 3.
We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 3 for G 3 p,q .
(64) □ Theorem 7. e general reverse Randić index of G 3 p,q is equal to Proof. For RR α (G 3 p,q ) which is the general reverse Randić index of G 3 p,q , from equation (2), we have Put α � 1, and we have Put α � (1/2), and we have Put α � (−1/2), and we have Put α � −1, and we have □ Theorem 8. Let G 3 p,q be a third-type p th chain hex-derived networks. en, the reverse atom-bond connectivity index is e reverse geometric-arithmetic index is e first reverse Zagreb index is e reverse hyper-Zagreb index is e reverse forgotten index is Proof. For RABC(G 3 p,q ) which is the reverse atom-bond connectivity index of G 3 p,q , from equation (3), we have us, by Lemma 3 and after simplification, For RGA(G 3 p,q ) which is the reverse geometric-arithmetic index of G 3 p,q , from equation (4), we have /3), and λ(5, 5) � 1. us, by Lemma 3 and after simplification, 8

Numerical and Graphical Representation
In this section, we determine the numerical values of RABC, RGA, RM 1 , RHM, and RF 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
e authors declare that they have no conflicts of interest.