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

Authors: Ali N. A. Koam; Ali Ahmad; Ashfaq Ahmed Qummer
Journal: Journal of Mathematics
 (2021)

---

## 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 dimensiont 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.


---

## Body

## 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 dimensionp, 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.Letp and q represent the number of rows and number of triangles in each row of third-type pth chain hex-derived networks Gp,q3, 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 dθ be the degree of a vertex θ∈V in G and ℜθ be its reverse degree that was introduced by Kulli [13] and defined as ℜθ=1+Δ−dθ, where Δ denoted the maximum degree of the given graph. Let Eℜθ,ℜϑ represent the edge partition of the given graph based on reverse degree of end vertices of an edge θϑ∈E and Eℜθ,ℜϑ represent its cardinality.Figure 1
Third-type chain hex-derived networkGp,q3.We define general reverse degree topological invariantTG as follows:(1)TG=∑θϑ∈EGλℜθ,ℜϑ,(2)Ifλℜθ,ℜϑ=ℜθ×ℜϑα,thenTGrepresents the general reverse Randic´indexℜℝαGforα=12,−12,1,−1,Ifα=1,then it is known as the second reverse Zagreb indexℜM2G,(3)Ifλℜθ,ℜϑ=ℜθ+ℜϑ−2ℜθ×ℜϑ,thenTGrepresents the reverse atom−bond connectivity indexℜABℂG,(4)Ifλℜθ,ℜϑ=2ℜθ×ℜϑℜθ+ℜϑ,thenTGrepresents the reverse geometric−arithmetic indexℜGAG,(5)Ifλℜθ,ℜϑ=ℜθ+ℜϑ,thenTGrepresents the first reverse Zagreb indexℜM1G,(6)Ifλℜθ,ℜϑ=ℜθ+ℜϑ2,thenTGrepresents the reverse hyper Zagreb indexℜℍMG,(7)Ifλℜθ,ℜϑ=ℜθ2+ℜϑ2,thenTGrepresents the reverse forgotten indexℜFG,(8)Ifλℜθ,ℜϑ=ℜθ+ℜϑαℜθ×ℜϑβ,thenTGrepresentsthe first reverse redefined indexℜℝℤ1Gforα=1,β=−1,the second reverse redefined indexℜℝℤ2Gforα=−1,β=1,the third reverse redefined indexℜℝℤ3Gforα=1,β=1.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 (K3), Chen et al. [26] assembled a hexagonal mesh. In terms of chemistry, these K3 graphs are also called oxide graphs. Figure 1 is obtained by joining these K3 graphs. Two-dimensional mesh graph HX(2) (see Figure 2(a)) is obtained by joining six K3 graphs and three-dimensional mesh graph HX(3) (see Figure 2(b)) is obtained by putting K3 graphs around all sides of HX(2) [27]. Furthermore, repeating the same process by putting the tK3 graph around each hexagon, we obtained the tth hexagonal mesh. We have to note that the one-dimensional hexagonal mesh graph does not exist.Figure 2
Hexagonal meshes: (a) HX (2) and (b) HX (3).
(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 HHDN3t. Huo el at. [28] explained the graphical construction algorithm for mth chain hex-derived network of third type. In this paper, we denote it by Gp,q3, and different priorities of p and q 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.Figure 3
Chain hex-derived networksGp,q3 for different priorities of p and q.
(a)(b)(c)(d)(e)


## 3. Main Results

In this section, we study the third-typepth chain hex-derived networks Gp,q3 in the following three cases.(i)
Case 1: forp=q,p,q≥1.(ii)
Case 2: forp<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: forp<q, p is even and q is odd. For p>q, p is even and q 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 forGp,q3.Lemma 1.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(9)T1Gp,q3=3pqλ1,1+2λ1,5+λ5,5+3p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Proof.
The graphGp,q3 contains 12pq edges and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 1 as(10)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−3p−3q+2, E1,5=6pq−4, and E5,5=3pq+3p+3q+2. Hence,(11)T1Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−3p−3q+2λ1,1+6pq−4λ1,5+3pq+3p+3q+2λ5,5.
After simplification, we obtain(12)T1Gp,q3=3pqλ1,1+2λ1,5+λ5,5+3p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Theorem 1.
The general reverse Randić index ofGp,q3 is equal to(13)ℜℝαGp,q3=108pq+72p+q+32,forα=1,18+65pq+12p+q+12−45,forα=12,185+655pq−125p+q+125−455,forα=−12,10825pq−7225p+q+3225,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 1,(14)ℜℝαGp,q3=3pq1+25α+25α+3p+q25α−1+21−25α+25α.
Putα=1, and we have(15)ℜℝ1Gp,q3=108pq+72p+q+32.
Putα=1/2, and we have(16)ℜℝ1/2Gp,q3=18+65pq+12p+q+12−45.
Putα=−1/2, and we have(17)ℜℝ−1/2Gp,q3=185+655pq−125p+q+125−455.
Putα=−1, and we have(18)ℜℝ−1Gp,q3=10825pq−7225p+q+3225.Theorem 2.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(19)ℜABℂGp,q3=1255+625pq−855+23p+3q+225.The reverse geometric-arithmetic index is(20)ℜGAGp,q3=6+25pq+4−453.The first reverse Zagreb index is(21)ℜM1Gp,q3=72pq+24p+24q.The reverse hyper-Zagreb index is(22)ℜℍMGp,q3=528pq+288p+288q+64.The reverse forgotten index is(23)ℜFGp,q3=312pq+144p+144q.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 1 and after simplification,(24)ℜABℂGp,q3=1255+625pq−855+23p+3q+225.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 1 and after simplification,(25)ℜGAGp,q3=6+25pq+4−453.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 1 and after simplification,(26)ℜM1Gp,q3=72pq+24p+24q.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 1 and after simplification,(27)ℜℍMGp,q3=528pq+288p+288q+64.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26, and λ5,5=50. Thus, by Lemma 1 and after simplification,(28)ℜFGp,q3=312pq+144p+144q.Theorem 3.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(29)ℜℝℤ1Gp,q3=72pq5−24p5−24q5.The second reverse redefined index is(30)ℜℝℤ2Gp,q3=14pq+6p+6q+83.The third reverse redefined index is(31)ℜℝℤ3Gp,q3=936pq+744p+744q+384.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 1 and after simplification,(32)ℜℝℤ1Gp,q3=72pq5−24p5−24q5.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 1 and after simplification,(33)ℜℝℤ2Gp,q3=14pq+6p+6q+83.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 1 and after simplification,(34)ℜℝℤ3Gp,q3=936pq+744p+744q+384.


### 3.2. Results for Case 2

We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 2 forGp,q3.Lemma 2.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(35)T2Gp,q3=3pqλ1,1+2λ1,5+λ5,5+22p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Proof.
The graphGp,q3 contains 12pq edges and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 2 as(36)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−4p−2q+2, E1,5=6pq−4, and E5,5=3pq+4p+2q+2. Hence,(37)T2Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−4p−2q+2λ1,1+6pq−4λ1,5+3pq+4p+2q+2λ5,5.
After simplification, we obtain(38)T2Gp,q3=3pqλ1,1+2λ1,5+λ5,5+22p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Theorem 4.
The general reverse Randić index ofGp,q3 is equal to(39)ℜℝαGp,q3=108pq+96p+48q+32,forα=1,18+65pq+16p+8q+12−45,forα=12,185+655pq−16p5−85q+125−455,forα=−12,10825pq−9625p−4825q+3225,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 2,(40)ℜℝαGp,q3=3pq1+25α+25α+22p+q25α−1+21−25α+25α.
Putα=1, and we have(41)ℜℝ1Gp,q3=108pq+96p+48q+32.
Putα=1/2, and we have(42)ℜℝ1/2Gp,q3=18+65pq+16p+8q+12−45.
Putα=−1/2, and we have(43)ℜℝ−1/2Gp,q3=185+655pq−16p5−85q+125−455.
Putα=−1, and we have(44)ℜℝ−1Gp,q3=10825pq−9625p−4825q+3225.Theorem 5.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(45)ℜABℂGp,q3=1255+625pq−855+24p+2q+225.The reverse geometric-arithmetic index is(46)ℜGAGp,q3=6+25pq+4−453.The first reverse Zagreb index is(47)ℜM1Gp,q3=72pq+32p+16q.The reverse hyper-Zagreb index is(48)ℜℍMGp,q3=528pq+384p+192q+64.The reverse forgotten index is(49)ℜFGp,q3=312pq+192p+96q.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 2 and after simplification,(50)ℜABℂGp,q3=1255+625pq−855+24p+2q+225.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 2 and after simplification,(51)ℜGAGp,q3=6+25pq+4−453.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 2 and after simplification,(52)ℜM1Gp,q3=72pq+32p+16q.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 2 and after simplification,(53)ℜℍMGp,q3=528pq+384p+192q+64.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26, and λ5,5=50. Thus, by Lemma 2 and after simplification,(54)ℜFGp,q3=312pq+192p+96q.Theorem 6.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(55)ℜℝℤ1Gp,q3=72pq5−32p5−16q5.The second reverse redefined index is(56)ℜℝℤ2Gp,q3=14pq+8p+4q+83.The third reverse redefined index is(57)ℜℝℤ3Gp,q3=936pq+992p+496q+384.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 2 and after simplification,(58)ℜℝℤ1Gp,q3=72pq5−32p5−16q5.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 2 and after simplification,(59)ℜℝℤ2Gp,q3=14pq+8p+4q+83.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 2 and after simplification,(60)ℜℝℤ3Gp,q3=936pq+992p+496q+384.


### 3.3. Results for Case 3

We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 3 forGp,q3.Lemma 3.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(61)T3Gp,q3=3pqλ1,1+2λ1,5+λ5,5+2λ5,5−λ1,12p+q+2λ1,1−4λ1,5+3λ5,5.Proof.
The graphGp,q3 contains 12pq edges, and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 3 as(62)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−4p−2q+2, E1,5=6pq−8, and E5,5=3pq+4p+2q+6. Hence,(63)T1Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−4p−2q+2λ1,1+6pq−8λ1,5+3pq+4p+2q+6λ5,5.
After simplification, we obtain(64)T3Gp,q3=3pqλ1,1+2λ1,5+λ5,5+2λ5,5−λ1,12p+q+2λ1,1−4λ1,5+3λ5,5.Theorem 7.
The general reverse Randić index ofGp,q3 is equal to(65)ℜℝαGp,q3=108pq+96p+48q+112,forα=1,18+65pq+16p+8q+32−85,forα=12,185+655pq−16p5−85q+165−855,forα=−12,108pq25−96p25−48q25+1625,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 3,(66)ℜℝαGp,q3=3pq1+25α+25α+225α−12p+q+21−45α+325α.
Putα=1, and we have(67)ℜℝ1Gp,q3=108pq+96p+48q+112.
Putα=1/2, and we have(68)ℜℝ1/2Gp,q3=18+65pq+16p+8q+32−85.
Putα=−1/2, and we have(69)ℜℝ−1/2Gp,q3=185+655pq−16p5−85q+165−855.
Putα=−1, and we have(70)ℜℝ−1Gp,q3=108pq25−96p25−48q25+1625.Theorem 8.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(71)ℜABℂGp,q3=1255+625pq−1655+24p+2q+625.The reverse geometric-arithmetic index is(72)ℜGAGp,q3=6+25pq+8−853.The first reverse Zagreb index is(73)ℜM1Gp,q3=72pq+32p+16q+16.The reverse hyper-Zagreb index is(74)ℜℍMGp,q3=528pq+384p+192q+320.The reverse forgotten index is(75)ℜFGp,q3=312pq+192p+96q+96.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 3 and after simplification,(76)ℜABℂGp,q3=1255+625pq−1655+24p+2q+625.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 3 and after simplification,(77)ℜGAGp,q3=6+25pq+8−853.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 3 and after simplification,(78)ℜM1Gp,q3=72pq+32p+16q+16.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 3 and after simplification,(79)ℜℍMGp,q3=528pq+384p+192q+320.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26 and λ5,5=50. Thus, by Lemma 3 and after simplification,(80)ℜFGp,q3=312pq+192p+96q+96.Theorem 9.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(81)ℜℝℤ1Gp,q3=72pq5−32p5−16q5−165.The second reverse redefined index is(82)ℜℝℤ2Gp,q3=14pq+8p+4q+283.The third reverse redefined index is(83)ℜℝℤ3Gp,q3=936pq+992p+496q+1264.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 3 and after simplification,(84)ℜℝℤ1Gp,q3=72pq5−32p5−16q5−165.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 3 and after simplification,(85)ℜℝℤ2Gp,q3=14pq+8p+4q+283.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 3 and after simplification,(86)ℜℝℤ3Gp,q3=936pq+992p+496q+1264.


## 3.1. Results for Case 1

We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 1 forGp,q3.Lemma 1.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(9)T1Gp,q3=3pqλ1,1+2λ1,5+λ5,5+3p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Proof.
The graphGp,q3 contains 12pq edges and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 1 as(10)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−3p−3q+2, E1,5=6pq−4, and E5,5=3pq+3p+3q+2. Hence,(11)T1Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−3p−3q+2λ1,1+6pq−4λ1,5+3pq+3p+3q+2λ5,5.
After simplification, we obtain(12)T1Gp,q3=3pqλ1,1+2λ1,5+λ5,5+3p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Theorem 1.
The general reverse Randić index ofGp,q3 is equal to(13)ℜℝαGp,q3=108pq+72p+q+32,forα=1,18+65pq+12p+q+12−45,forα=12,185+655pq−125p+q+125−455,forα=−12,10825pq−7225p+q+3225,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 1,(14)ℜℝαGp,q3=3pq1+25α+25α+3p+q25α−1+21−25α+25α.
Putα=1, and we have(15)ℜℝ1Gp,q3=108pq+72p+q+32.
Putα=1/2, and we have(16)ℜℝ1/2Gp,q3=18+65pq+12p+q+12−45.
Putα=−1/2, and we have(17)ℜℝ−1/2Gp,q3=185+655pq−125p+q+125−455.
Putα=−1, and we have(18)ℜℝ−1Gp,q3=10825pq−7225p+q+3225.Theorem 2.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(19)ℜABℂGp,q3=1255+625pq−855+23p+3q+225.The reverse geometric-arithmetic index is(20)ℜGAGp,q3=6+25pq+4−453.The first reverse Zagreb index is(21)ℜM1Gp,q3=72pq+24p+24q.The reverse hyper-Zagreb index is(22)ℜℍMGp,q3=528pq+288p+288q+64.The reverse forgotten index is(23)ℜFGp,q3=312pq+144p+144q.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 1 and after simplification,(24)ℜABℂGp,q3=1255+625pq−855+23p+3q+225.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 1 and after simplification,(25)ℜGAGp,q3=6+25pq+4−453.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 1 and after simplification,(26)ℜM1Gp,q3=72pq+24p+24q.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 1 and after simplification,(27)ℜℍMGp,q3=528pq+288p+288q+64.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26, and λ5,5=50. Thus, by Lemma 1 and after simplification,(28)ℜFGp,q3=312pq+144p+144q.Theorem 3.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(29)ℜℝℤ1Gp,q3=72pq5−24p5−24q5.The second reverse redefined index is(30)ℜℝℤ2Gp,q3=14pq+6p+6q+83.The third reverse redefined index is(31)ℜℝℤ3Gp,q3=936pq+744p+744q+384.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 1 and after simplification,(32)ℜℝℤ1Gp,q3=72pq5−24p5−24q5.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 1 and after simplification,(33)ℜℝℤ2Gp,q3=14pq+6p+6q+83.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 1 and after simplification,(34)ℜℝℤ3Gp,q3=936pq+744p+744q+384.


## 3.2. Results for Case 2

We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 2 forGp,q3.Lemma 2.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(35)T2Gp,q3=3pqλ1,1+2λ1,5+λ5,5+22p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Proof.
The graphGp,q3 contains 12pq edges and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 2 as(36)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−4p−2q+2, E1,5=6pq−4, and E5,5=3pq+4p+2q+2. Hence,(37)T2Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−4p−2q+2λ1,1+6pq−4λ1,5+3pq+4p+2q+2λ5,5.
After simplification, we obtain(38)T2Gp,q3=3pqλ1,1+2λ1,5+λ5,5+22p+qλ5,5−λ1,1+2λ1,1−2λ1,5+λ5,5.Theorem 4.
The general reverse Randić index ofGp,q3 is equal to(39)ℜℝαGp,q3=108pq+96p+48q+32,forα=1,18+65pq+16p+8q+12−45,forα=12,185+655pq−16p5−85q+125−455,forα=−12,10825pq−9625p−4825q+3225,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 2,(40)ℜℝαGp,q3=3pq1+25α+25α+22p+q25α−1+21−25α+25α.
Putα=1, and we have(41)ℜℝ1Gp,q3=108pq+96p+48q+32.
Putα=1/2, and we have(42)ℜℝ1/2Gp,q3=18+65pq+16p+8q+12−45.
Putα=−1/2, and we have(43)ℜℝ−1/2Gp,q3=185+655pq−16p5−85q+125−455.
Putα=−1, and we have(44)ℜℝ−1Gp,q3=10825pq−9625p−4825q+3225.Theorem 5.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(45)ℜABℂGp,q3=1255+625pq−855+24p+2q+225.The reverse geometric-arithmetic index is(46)ℜGAGp,q3=6+25pq+4−453.The first reverse Zagreb index is(47)ℜM1Gp,q3=72pq+32p+16q.The reverse hyper-Zagreb index is(48)ℜℍMGp,q3=528pq+384p+192q+64.The reverse forgotten index is(49)ℜFGp,q3=312pq+192p+96q.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 2 and after simplification,(50)ℜABℂGp,q3=1255+625pq−855+24p+2q+225.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 2 and after simplification,(51)ℜGAGp,q3=6+25pq+4−453.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 2 and after simplification,(52)ℜM1Gp,q3=72pq+32p+16q.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 2 and after simplification,(53)ℜℍMGp,q3=528pq+384p+192q+64.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26, and λ5,5=50. Thus, by Lemma 2 and after simplification,(54)ℜFGp,q3=312pq+192p+96q.Theorem 6.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(55)ℜℝℤ1Gp,q3=72pq5−32p5−16q5.The second reverse redefined index is(56)ℜℝℤ2Gp,q3=14pq+8p+4q+83.The third reverse redefined index is(57)ℜℝℤ3Gp,q3=936pq+992p+496q+384.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 2 and after simplification,(58)ℜℝℤ1Gp,q3=72pq5−32p5−16q5.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 2 and after simplification,(59)ℜℝℤ2Gp,q3=14pq+8p+4q+83.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 2 and after simplification,(60)ℜℝℤ3Gp,q3=936pq+992p+496q+384.


## 3.3. Results for Case 3

We provide a formula that would be used to calculate any reverse degree topological descriptors of Case 3 forGp,q3.Lemma 3.
LetGp,q3 be a third-type pth chain hex-derived networks. Then,(61)T3Gp,q3=3pqλ1,1+2λ1,5+λ5,5+2λ5,5−λ1,12p+q+2λ1,1−4λ1,5+3λ5,5.Proof.
The graphGp,q3 contains 12pq edges, and maximum degree in Gp,q3 graph is 8. There are two types of reverse degree vertices in Gp,q3 that are 1 and 5. Let us partition the edges of Gp,q3 according to its reverse degrees according to Case 3 as(62)E1,1=θϑ∈EGp,q3:ℜθ=1,ℜϑ=1,E1,5=θϑ∈EGp,q3:ℜθ=1,ℜϑ=5,E5,5=θϑ∈EGp,q3:ℜθ=5,ℜϑ=5.
Note thatEGp,q3=E1,1∪E1,5∪E5,5 and E1,1=3pq−4p−2q+2, E1,5=6pq−8, and E5,5=3pq+4p+2q+6. Hence,(63)T1Gp,q3=∑θϑ∈EGp,q3λℜθ,ℜϑ=∑θϑ∈E1,1λ1,1+∑θϑ∈E1,5λ1,5+∑θϑ∈E5,5λ5,5=3pq−4p−2q+2λ1,1+6pq−8λ1,5+3pq+4p+2q+6λ5,5.
After simplification, we obtain(64)T3Gp,q3=3pqλ1,1+2λ1,5+λ5,5+2λ5,5−λ1,12p+q+2λ1,1−4λ1,5+3λ5,5.Theorem 7.
The general reverse Randić index ofGp,q3 is equal to(65)ℜℝαGp,q3=108pq+96p+48q+112,forα=1,18+65pq+16p+8q+32−85,forα=12,185+655pq−16p5−85q+165−855,forα=−12,108pq25−96p25−48q25+1625,forα=−1.Proof.
ForℜℝαGp,q3 which is the general reverse Randić index of Gp,q3, from equation (2), we have λℜθ,ℜϑ=ℜθ×ℜϑα; therefore, λ1,1=1, λ1,5=5α, and λ5,5=25α. Thus, by Lemma 3,(66)ℜℝαGp,q3=3pq1+25α+25α+225α−12p+q+21−45α+325α.
Putα=1, and we have(67)ℜℝ1Gp,q3=108pq+96p+48q+112.
Putα=1/2, and we have(68)ℜℝ1/2Gp,q3=18+65pq+16p+8q+32−85.
Putα=−1/2, and we have(69)ℜℝ−1/2Gp,q3=185+655pq−16p5−85q+165−855.
Putα=−1, and we have(70)ℜℝ−1Gp,q3=108pq25−96p25−48q25+1625.Theorem 8.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the reverse atom-bond connectivity index is(71)ℜABℂGp,q3=1255+625pq−1655+24p+2q+625.The reverse geometric-arithmetic index is(72)ℜGAGp,q3=6+25pq+8−853.The first reverse Zagreb index is(73)ℜM1Gp,q3=72pq+32p+16q+16.The reverse hyper-Zagreb index is(74)ℜℍMGp,q3=528pq+384p+192q+320.The reverse forgotten index is(75)ℜFGp,q3=312pq+192p+96q+96.Proof.
ForℜABℂGp,q3 which is the reverse atom-bond connectivity index of Gp,q3, from equation (3), we have λℜθ,ℜϑ=ℜθ+ℜϑ−2/ℜθ×ℜϑ; therefore, λ1,1=0, λ1,5=4/5, and λ5,5=8/5. Thus, by Lemma 3 and after simplification,(76)ℜABℂGp,q3=1255+625pq−1655+24p+2q+625.
ForℜGAGp,q3 which is the reverse geometric-arithmetic index of Gp,q3, from equation (4), we have λℜθ,ℜϑ=2ℜθ×ℜϑ/ℜθ+ℜϑ; therefore, λ1,1=1, λ1,5=5/3, and λ5,5=1. Thus, by Lemma 3 and after simplification,(77)ℜGAGp,q3=6+25pq+8−853.
ForℜM1Gp,q3 which is the first reverse Zagreb index of Gp,q3, from equation (5), we have λℜθ,ℜϑ=ℜθ+ℜϑ; therefore, λ1,1=2, λ1,5=6, and λ5,5=10. Thus, by Lemma 3 and after simplification,(78)ℜM1Gp,q3=72pq+32p+16q+16.
ForℜℍMGp,q3 which is the first reverse hyper-Zagreb index of Gp,q3, from equation (6), we have λℜθ,ℜϑ=ℜθ+ℜϑ2; therefore, λ1,1=4, λ1,5=36, and λ5,5=100. Thus, by Lemma 3 and after simplification,(79)ℜℍMGp,q3=528pq+384p+192q+320.
ForℜFGp,q3 which is the reverse forgotten index of Gp,q3, from equation (7), we have λℜθ,ℜϑ=ℜθ2+ℜϑ2; therefore, λ1,1=2, λ1,5=26 and λ5,5=50. Thus, by Lemma 3 and after simplification,(80)ℜFGp,q3=312pq+192p+96q+96.Theorem 9.
LetGp,q3 be a third-type pth chain hex-derived networks. Then, the first reverse redefined index is(81)ℜℝℤ1Gp,q3=72pq5−32p5−16q5−165.The second reverse redefined index is(82)ℜℝℤ2Gp,q3=14pq+8p+4q+283.The third reverse redefined index is(83)ℜℝℤ3Gp,q3=936pq+992p+496q+1264.Proof.
Forℜℝℤ1Gp,q3 which is the first reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ1ℜθ×ℜϑ−1; therefore, λ1,1=2, λ1,5=6/5, and λ5,5=2/5. Thus, by Lemma 3 and after simplification,(84)ℜℝℤ1Gp,q3=72pq5−32p5−16q5−165.
Forℜℝℤ2Gp,q3 which is the second reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑ−1ℜθ×ℜϑ1; therefore, λ1,1=1/2, λ1,5=5/6, and λ5,5=5/2. Thus, by Lemma 3 and after simplification,(85)ℜℝℤ2Gp,q3=14pq+8p+4q+283.
Forℜℝℤ3Gp,q3 which is the third reverse redefined index of Gp,q3, from equation (8), we have λℜθ,ℜϑ=ℜθ+ℜϑℜθ×ℜϑ; therefore, λ1,1=2, λ1,5=30, and λ5,5=250. Thus, by Lemma 3 and after simplification,(86)ℜℝℤ3Gp,q3=936pq+992p+496q+1264.


## 4. Numerical and Graphical Representation

In this section, we determine the numerical values ofℜABℂ,ℜGA,ℜM1,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.Table 1
Numerical comparison ofℜABℂ,ℜGA,ℜM1,ℜℍM, and ℜF for Case 1.
p,qℜABℂℜGAℜM1ℜℍMℜF1,18.011411.49112011686002,232.59742.907384332818243,371.30995.269792654436724,4124.15168.5713441081661445,5191.12262.8220401614492406,6272.21378.02288022528129607,7367.43514.16386429968173048,8476.78671.25499238464222729,9600.26849.276264480162786410,10737.871048.276805862434080Table 2
Numerical comparison ofℜABℂ,ℜGA,ℜM1,ℜℍM, and ℜF for Case 2.
p,qℜABℂℜGAℜM1ℜℍMℜF5,6227.56315.19241618976108965,7264.01367.55279221808125525,8300.47419.91316824640142084,4124.15168.5713441081661446,6272.21378.02288022528129608,8476.78671.25499238464222729,8535.55755.02560043072249609,7470.85660.774936381282205610,8594.32838.806208476802764812,10883.661257.791846995240704Table 3
Numerical comparison ofℜABℂ,ℜGA,ℜM1,ℜℍM, and ℜF for Case 3.
p,qℜABℂℜGAℜM1ℜℍMℜF4,5152.22211.4816641337675846,7314.41441.87334426144150248,9533.10756.035600431362496010,11808.311154.08432643523739212,131140.01635.71184089792523204,393.448127.701056876848966,5227.38316.20244819424110888,7417.83588.474416343041977610,9664.78944.536960534083096012,11968.241384.4100807673644640Figure 4
Graphical representation of Table1.Figure 5
Graphical representation of Table2.Figure 6
Graphical representation of Table3.



---

*Source: 4540276-2021-11-03.xml*