EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 4, 2024, 3109-3128
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Computational Analysis of Reverse Degree-Based
Topological Indices in Hex-Derived Networks

Khalid A. Alsatami1, Haidar Ali2,∗, Bilal Ali3, Parvez Ali4

1 Department of Mathematics, College of Science, Qassim University, Buraydah,
Saudi Arabia
2 Department of Mathematics, Riphah International University, Faisalabad, Pakistan
3 Department of Mathematics, Government College University, Faisalabad, Pakistan
4 Department of Mechanical Engineering, College of Engineering, Qassim University,
Buraydah 51452, Saudi Arabia

Abstract. Topology is the mathematical study of the geometric and spatial properties that re-
main unchanged under continuous transformations of a graph’s shape and size. In chemical graph
theory, topological indices are used to quantify various chemical properties of molecules. These
indices are derived from the topological structure of a graph and are crucial in understanding
the valency of a chemical substance, which is determined by the number of surrounding atoms
in its molecular structure. Topological indices are connected to numerous physicochemical prop-
erties, such as vapor pressure, stability, and elastic energy. In molecular structures, topological
indices provide a numerical representation of the connections between molecules. In theoretical
chemistry, these indices are widely used to simulate the physicochemical characteristics of com-
plex compounds. QSAR/QSPR studies rely heavily on topological indices to predict physical
and chemical properties. This article explores the hex-derived network and its first two types,
calculating reversed degree-based topological indices for these networks.

2020 Mathematics Subject Classifications: 05C12, 05C90, 92E10

Key Words and Phrases: Chemical graph theory, Topological indices, Molecular structure,
Valency, Quantitative analysis, Hex-derived network, Theoretical chemistry, Chemical properties
prediction

1. Introduction

Graph theory is the study of graphs and a subfield of combinatorics. It is related with
applied mathematics and information technology. It is combination of mathematics, op-
erational research, information technology and electrical engineering. In graph theory, the
concept graph doesn’t donate the data infect it denotes the structures of molecules.

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5451

Email addresses: satamy@qu.edu.sa (K. A. Alsatami), haidar3830@gmail.com (H. Ali),
bilalali0462@gmail.com (B. Ali), p.ali@qu.edu.sa (P. Ali)

https://www.ejpam.com 3109 Copyright: © 2024 The Author(s). (CC BY-NC 4.0)



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3110

Cheminformatics is the combination of mathematics, chemistry and information science.
In this subject, we study the QSAR/QSPR relationship, characterisation, physical and
bioactivities of chemical compounds [15, 26].
Topological indices basically are the numerical values, polynomials or matrix to represent
a chemical graph. Topological indices are assumed to be the building blocks for the predic-
tion of physico-chemical properties of chemical compounds. They based on the topology
of chemical networks depend upon the distance, degree and eccentricity.
Graphs discussed in this article are undirected and finite. A graph is a structure made
up of vertices which are connected with edges. A graph is a pair of sets (U , E), where U
is the set of vertices and E is the set of edges, formed by pair of vertices. Order of G is
represented by |U| and size of G represented by |E|. The degree of a vertex v̌ is the number
of edges incident of that vertex and is denoted by dv̌. The reverse degree of a vertex v̌
is represented by Rv̌. It was introduced by Kulli [14]. If ∆ is the maximum degree of a
graph then reverse degree is defined as Rv̌ = 1− dv̌ +∆.
Topological indices are categorised into mainly two types distance based and degree based
topological indices [1, 3, 12, 16–20, 28]. Our work is based on degree based topological
indices. The theory of topological indices begin with the working of Wiener [30]. It is
defined as,

W (G) =
∑

(ǔ,v̌)∈E(G)

d(ǔ, v̌). (1)

Randić index is defined in [2, 6, 21], and its reverse Randić index is,

RRα(G) =
∑

(ǔ,v̌)∈E(G)

(Rǔ ×Rv̌)
α, α = −1, 1,

1

2
,−1

2
. (2)

ABC index is defined in [8], and its reverse ABC index is,

RABC(G) =
∑

(ǔ,v̌)∈E(G)

√
Rǔ +Rv̌ − 2

Rǔ ×Rv̌
(3)

GA index is defined in [29], and its reverse GA index is,

RGA(G) =
∑

(ǔ,v̌)∈E(G)

2
√
Rǔ ×Rv̌

Rǔ +Rv̌
. (4)

The first Zagreb index is defined in [10], and its reverse Zagreb index is,

RM1(G) =
∑

(ǔ,v̌)∈E(G)

(Rǔ +Rv̌). (5)

The hyper Zagrab index is defined in [24], and its reverse hyper Zagrab index is,

RHM(G) =
∑

(ǔ,v̌)∈E(G)

(Rǔ +Rv̌)
2. (6)



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3111

The forgotten index is defined in [9], and its reverse forgotten index is,

RF (G) =
∑

(ǔ,v̌)∈E(G)

((Rǔ)
2 + (Rv̌)

2). (7)

The first, second and third redefined Zagreb index is defined in [22, 27], and its reverse
redefined Zagreb index is,

RRZ1(G) =
∑

(ǔ,v̌)∈E(G)

Rǔ +Rv̌

Rǔ ×Rv̌
. (8)

RRZ2(G) =
∑

(ǔ,v̌)∈E(G)

Rǔ ×Rv̌

Rǔ +Rv̌
. (9)

RRZ3(G) =
∑

(ǔ,v̌)∈E(G)

(Rǔ +Rv̌)(Rǔ ×Rv̌). (10)

Hex derived network is derived from hexagonal mesh shown in Figure 1, by adding a layer
of triangles around its boundary, after that connecting the faces of HX(n), with the ver-
tices we get HDN1(n) shown in Figure 2 . By connecting the vertices of HDN1(n) with
each other we get HDN2(n) shown in Figure 3. For the detail construction of Hex derived
network we refer the reader to concern [7, 13, 25].

Figure 1: Hexagonal Meshes

2. Main Results

Hex-derived network has a lot of applications in material sciences. Shao Z et al. [23],
computed the metric dimensions of hex derived network and Imran et al. [13], computed



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3112

Figure 2: Hex-derived network HDN1(4)

Figure 3: Hex-derived network HDN2(n)

the topological indices of it. Here we discuss the first two types of HDN and find the
reversed degree based indices of it. The symbols used in this articles is from the book
[4, 5, 11].

2.1. Results on Hex-derived network of Type 1

In this section, we will compute reverse randić, ABC, GA, Zagreb, redined Zagreb, hyper
Zagreb and forgotten index for hex-derive network of type 1. The reversed edge partition
of HDN1(n) is written in Table1.



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3113

(Rǔ,Rv̌) Number of Edges

(1, 1) 9n2 − 33n+ 30

(6, 1) 12n− 24

(6, 6) 6n− 18

(8, 1) 6

(8, 6) 12

(10, 1) 18n2 − 54n+ 42

(10, 6) 18n− 36

(10, 8) 12

Table 1: Edge Partition of first type of hex-derived network

Theorem 2.1.1. Let G1 be the first type of hex-derived network then

RRα(HDN1(n)) =


189n2 + 795n− 918, if α = 1,
54
5 n

2 − 539
15 n+ 121

4 , if α = −1,

65.920998n2 + 1.058284n− 75.386629, if α = 1
2 ,

14.6921n2 − 41.85353n+ 31.031039, if α = −1
2 .

Proof. Let G1
∼= HDN1(n), then by using equation 2 and Table 1, we have

RRα(G1) = (1)α|E1,1(G1)|+ (6)α|E6,1(G1)|+ (36)α|E6,6(G1)|+ (8)α|E8,1(G1)|
+(48)α|E8,6(G1)|+ (10)α|E10,1(G1)|+ (60)α|E10,6(G1)|
+(80)α|E10,8(G1)|,

RRα(G1) = (1)α(9n2 − 33n+ 30) + (6)α(12n− 24) + (36)α(6n− 18)

+(8)α(6) + (48)α(12) + (10)α(18n2 − 54n+ 42)

+(60)α(18n− 36) + (80)α(12),

for α = 1,

RR1(G1) = (1)(9n2 − 33n+ 30) + (6)(12n− 24) + (36)(6n− 18)

+(8)(6) + (48)(12) + (10)(18n2 − 54n+ 42)

+(60)(18n− 36) + (80)(12),

⇒ RR1(G1) = 189n2 + 795n− 918.

for α = −1,

RR−1(G1) = (1)−1(9n2 − 33n+ 30) + (6)−1(12n− 24) + (36)−1(6n− 18)

+(8)−1(6) + (48)−1(12) + (10)−1(18n2 − 54n+ 42)

+(60)−1(18n− 36) + (80)−1(12),



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3114

⇒ RR−1(G1) =
54

5
n2 − 539

15
n+

121

4
.

for α = 1
2 ,

RR 1
2
(G1) = (1)

1
2 (9n2 − 33n+ 30) + (6)

1
2 (12n− 24) + (36)

1
2 (6n− 18)

+(8)
1
2 (6) + (48)

1
2 (12) + (10)

1
2 (18n2 − 54n+ 42)

+(60)
1
2 (18n− 36) + (80)

1
2 (12),

⇒ RR 1
2
(G1) = 65.920998n2 + 1.058284n− 75.386629.

for α = −1
2 ,

RR− 1
2
(G1) = (1)−

1
2 (9n2 − 33n+ 30) + (6)−

1
2 (12n− 24) + (36)−

1
2 (6n− 18)

+(8)−
1
2 (6) + (48)−

1
2 (12) + (10)−

1
2 (18n2 − 54n+ 42)

+(60)−
1
2 (18n− 36) + (80)−

1
2 (12),

⇒ RR− 1
2
(G1) = 14.6921n2 − 41.85353n+ 31.031039.

Theorem 2.1.2. Let G1 be the first type of hex-derived network then

RABC(G1) = 17.076299n2 − 28.417343n+ 8.03836.

RGA(G1) = 19.349272n2 − 32.221141n+ 12.068803.

Proof. Let G1
∼= HDN1(n), then by using equation 3 and Table 1, we have

RABC(G1) =

√
1 + 1− 2

1× 1
|E1,1(G1)|+

√
6 + 1− 2

6× 1
|E6,1(G1)|

+

√
6 + 6− 2

6× 6
|E6,6(G1)|+

√
8 + 1− 2

8× 1
|E8,1(G1)|

+

√
8 + 6− 2

8× 6
|E8,6(G1)|+

√
10 + 1− 2

10× 1
|E10,1(G1)|

+

√
10 + 6− 2

10× 6
|E10,6(G1)|+

√
10 + 8− 2

10× 8
|E10,8(G1)|,

RABC(G1) =

√
1 + 1− 2

1× 1
(9n2 − 33n+ 30) +

√
6 + 1− 2

6× 1
(12n− 24)

+

√
6 + 6− 2

6× 6
(6n− 18) +

√
8 + 1− 2

8× 1
(6)

+

√
8 + 6− 2

8× 6
(12) +

√
10 + 1− 2

10× 1
(18n2 − 54n+ 42)



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3115

+

√
10 + 6− 2

10× 6
(18n− 36) +

√
10 + 8− 2

10× 8
(12),

⇒ RABC(G1) = 17.076299n2 − 28.417343n+ 8.03836.

Now let G1
∼= HDN1(n), then by using equation 4 and Table 1, we have

RGA(G1) =
2
√
1× 1

1 + 1
|E1,1(G1)|+

2
√
6× 1

6 + 1
|E6,1(G1)|

+
2
√
6× 6

6 + 6
|E6,6(G1)|+

2
√
8× 1

8 + 1
|E8,1(G1)|

+
2
√
8× 6

8 + 6
|E8,6(G1)|+

2
√
10× 1

10 + 1
|E10,1(G1)|

+
2
√
10× 6

10 + 6
|E10,6(G1)|+

2
√
10× 8

10 + 8
|E10,8(G1)|,

RGA(G1) =
2
√
1× 1

1 + 1
(9n2 − 33n+ 30) +

2
√
6× 1

6 + 1
(12n− 24)

+
2
√
6× 6

6 + 6
(6n− 18) +

2
√
8× 1

8 + 1
(6)

+
2
√
8× 6

8 + 6
(12) +

2
√
10× 1

10 + 1
(18n2 − 54n+ 42)

+
2
√
10× 6

10 + 6
(18n− 36) +

2
√
10× 8

10 + 8
(12),

⇒ RGA(G1) = 19.349272n2 − 32.221141n+ 12.068803.

Theorem 2.1.3. Let G1 be the first type of hex-derived network then

RM1(G1) = 216n2 − 216n

RHM(G1) = 2214n2 − 606n− 1056

Proof. Let G1
∼= HDN1(n), then by using equation 5 and Table 1, we have

RM1(G1) = (1 + 1)|E1,1(G1)|+ (6 + 1)|E6,1(G1)|+ (6 + 6)|E6,6(G1)|
+(8 + 1)|E8,1(G1)|+ (8 + 6)|E8,6(G1)|+ (10 + 1)|E10,1(G1)|
+(10 + 6)|E10,6(G1)|+ (10 + 8)|E10,8(G1)|,

RM1(G1) = (1 + 1)(9n2 − 33n+ 30) + (6 + 1)(12n− 24) + (6 + 6)(6n− 18)

+(8 + 1)(6) + (8 + 6)(12) + (10 + 1)(18n2 − 54n+ 42)

+(10 + 6)(18n− 36) + (10 + 8)(12),



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3116

⇒ RM1(G1) = 216n2 − 216n.

Now let G1
∼= HDN1(n), then by using equation 6 and Table 1, we have

RHM(G1) = (1 + 1)2|E1,1(G1)|+ (6 + 1)2|E6,1(G1)|+ (6 + 6)2|E6,6(G1)|
+(8 + 1)2|E8,1(G1)|+ (8 + 6)2|E8,6(G1)|+ (10 + 1)2|E10,1(G1)|
+(10 + 6)2|E10,6(G1)|+ (10 + 8)2|E10,8(G1)|,

RHM(G1) = (1 + 1)2(9n2 − 33n+ 30) + (6 + 1)2(12n− 24) + (6 + 6)2(6n− 18)

+(8 + 1)2(6) + (8 + 6)2(12) + (10 + 1)2(18n2 − 54n+ 42)

+(10 + 6)2(18n− 36) + (10 + 8)2(12),

⇒ RHM(G1) = 2214n2 − 606n− 1056.

Theorem 2.1.4. Let G1 be the first type of hex-derived network then

RF (G1) = 1836n2 − 2196n+ 780.

Proof. Let G1
∼= HDN1(n), then by using equation 7 and Table 1, we have

RF (G1) = ((1)2 + (1)2)|E1,1(G1)|+ ((6)2 + (1)2)|E6,1(G1)|
+((6)2 + (6)2)|E6,6(G1)|+ ((8)2 + (1)2)|E8,1(G1)|
+((8)2 + (6)2)|E8,6(G1)|+ ((10)2 + (1)2)|E10,1(G1)|
+((10)2 + (6)2)|E10,6(G1)|+ ((10)2 + (8)2)|E10,8(G1)|,

RF (G1) = ((1)2 + (1)2)(9n2 − 33n+ 30) + ((6)2 + (1)2)(12n− 24)

+((6)2 + (6)2)(6n− 18) + ((8)2 + (1)2)(6) + ((8)2 + (6)2)(12)

+((10)2 + (1)2)(18n2 − 54n+ 42) + ((10)2 + (6)2)(18n− 36)

+((10)2 + (8)2)(12),

⇒ RF (G1) = 1836n2 − 2196n+ 780.

Theorem 2.1.5. Let G1 be the first type of hex-derived network then

RRZ1(G1) =
189

5
n2 − 523

5
n+

1511

20

RRZ2(G1) =
459

22
n2 +

2325

77
n− 13070

231
RRZ3(G1) = 1998n2 + 14370n− 12888



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3117

Proof. Let G1
∼= HDN1(n), then by using equation 8 and Table 1, we have

RRZ1(G1) =

(
1 + 1

1× 1

)
|E1,1(G1)|+

(
6 + 1

6× 1

)
|E6,1(G1)|+

(
6 + 6

6× 6

)
|E6,6(G1)|

+

(
8 + 1

8× 1

)
|E8,1(G1)|+

(
8 + 6

8× 6

)
|E8,6(G1)|+

(
10 + 1

10× 1

)
|E10,1(G1)|

+

(
10 + 6

10× 6

)
|E10,6(G1)|+

(
10 + 8

10× 8

)
|E10,8(G1)|,

RRZ1(G1) =

(
1 + 1

1× 1

)
(9n2 − 33n+ 30) +

(
6 + 1

6× 1

)
(12n− 24) +

(
6 + 6

6× 6

)
(6n− 18)

+

(
8 + 1

8× 1

)
(6) +

(
8 + 6

8× 6

)
(12) +

(
10 + 1

10× 1

)
(18n2 − 54n+ 42)

+

(
10 + 6

10× 6

)
(18n− 36) +

(
10 + 8

10× 8

)
(12),

⇒ RRZ1(G1) =
189

5
n2 − 523

5
n+

1511

20
.

Now let G1
∼= HDN1(n), then by using equation 9 and Table 1, we have

RRZ2(G1) =

(
1× 1

1 + 1

)
|E1,1(G1)|+

(
6× 1

6 + 1

)
|E6,1(G1)|+

(
6× 6

6 + 6

)
|E6,6(G1)|

+

(
8× 1

8 + 1

)
|E8,1(G1)|+

(
8× 6

8 + 6

)
|E8,6(G1)|+

(
10× 1

10 + 1

)
|E10,1(G1)|

+

(
10× 6

10 + 6

)
|E10,6(G1)|+

(
10× 8

10 + 8

)
|E10,8(G1)|,

RRZ2(G1) =

(
1× 1

1 + 1

)
(9n2 − 33n+ 30) +

(
6× 1

6 + 1

)
(12n− 24) +

(
6× 6

6 + 6

)
(6n− 18)

+

(
8× 1

8 + 1

)
(6) +

(
8× 6

8 + 6

)
(12) +

(
10× 1

10 + 1

)
(18n2 − 54n+ 42)

+

(
10× 6

10 + 6

)
(18n− 36) +

(
10× 8

10 + 8

)
(12),

⇒ RRZ2(G1) =
459

22
n2 +

2325

77
n− 13070

231
.

Again let G1
∼= HDN1(n), then by using equation 10 and Table 1, we have

RRZ3(G1) = (1 + 1)(1× 1)|E1,1(G1)|+ (6 + 1)(6× 1)|E6,1(G1)|
+(6 + 6)(6× 6)|E6,6(G1)|+ (8 + 1)(8× 1)|E8,1(G1)|
+(8 + 6)(8× 6)|E8,6(G1)|+ (10 + 1)(10× 1)|E10,1(G1)|



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3118

+(10 + 6)(10× 6)|E10,6(G1)|+ (10 + 8)(10× 8)|E10,8(G1)|,

RRZ3(G1) = (1 + 1)(1× 1)(9n2 − 33n+ 30) + (6 + 1)(6× 1)(12n− 24)

+(6 + 6)(6× 6)(6n− 18) + (8 + 1)(8× 1)(6)

+(8 + 6)(8× 6)(12) + (10 + 1)(10× 1)(18n2 − 54n+ 42)

+(10 + 6)(10× 6)(18n− 36) + (10 + 8)(10× 8)(12),

⇒ RRZ3(G1) = 1998n2 + 14370n− 12888.

2.2. Results on Hex-derived network of Type 2

In this section, we will compute reverse randić, ABC, GA, Zagreb, redined Zagreb,
hyper Zagreb and forgotten index for hex-derive network of type 2. The reversed edge
partition of HDN2(n) is written in Table 2.

(Rǔ,Rv̌) Number of Edges

(1, 1) 9n2 − 33n+ 30

(6, 1) 12n− 24

(6, 6) 6n− 18

(7, 1) 18n2 − 60n+ 48

(7, 6) 6n− 12

(7, 7) 9n2 − 33n+ 30

(8, 1) 6n

(8, 6) 12n− 12

(8, 7) 12n− 24

(8, 8) 18

Table 2: Edge Partition second type of hex-derived network

Theorem 2.2.1. Let G2 be the second type of hex-derived network, then

RRα(HDN2(n)) =


576n2 − 234n− 228, if α = 1,
576
49 n

2 − 5692
147 n+ 50625

1568 , if α = −1,

119.623524n2 − 128.557979n+ 3.701427, if α = 1
2 ,

17.089075n2 − 48.110416n+ 35.089224, if α = −1
2 .

Proof. Let G2
∼= HDN2(n), then by using equation 2 and Table 2, we have

RRα(G2) = (1)α|E1,1(G2)|+ (6)α|E6,1(G2)|+ (36)α|E6,6(G2)|+ (7)α|E7,1(G2)|
+(42)α|E7,6(G2)|+ (49)α|E7,7(G2)|+ (8)α|E8,1(G2)|
+(48)α|E8,6(G2)|+ (56)α|E8,7(G2)|+ (64)α|E8,8(G2)|,



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3119

RRα(G2) = (1)α(9n2 − 33n+ 30) + (6)α(12n− 24) + (36)α(6n− 18)

+(7)α(18n2 − 60n+ 48) + (42)α(6n− 12) + (49)α(9n2 − 33n+ 30)

+(8)α(6n) + (48)α(12n− 12) + (56)α(12n− 24) + (64)α(18),

for α = 1,

RR1(G2) = (1)(9n2 − 33n+ 30) + (6)(12n− 24) + (36)(6n− 18)

+(7)(18n2 − 60n+ 48) + (42)(6n− 12) + (49)(9n2 − 33n+ 30)

+(8)(6n) + (48)(12n− 12) + (56)(12n− 24) + (64)(18),

⇒ RR1(G2) = 576n2 − 234n− 228.

for α = −1,

RR−1(G2) = (1)−1(9n2 − 33n+ 30) + (6)−1(12n− 24) + (36)−1(6n− 18)

+(7)−1(18n2 − 60n+ 48) + (42)−1(6n− 12) + (49)−1(9n2 − 33n+ 30)

+(8)−1(6n) + (48)−1(12n− 12) + (56)−1(12n− 24) + (64)−1(18),

⇒ RR−1(G2) =
576

49
n2 − 5692

147
n+

50625

1568
.

for α = 1
2 ,

RR 1
2
(G2) = (1)

1
2 (9n2 − 33n+ 30) + (6)

1
2 (12n− 24) + (36)

1
2 (6n− 18)

+(7)
1
2 (18n2 − 60n+ 48) + (42)

1
2 (6n− 12) + (49)

1
2 (9n2 − 33n+ 30)

+(8)
1
2 (6n) + (48)

1
2 (12n− 12) + (56)

1
2 (12n− 24) + (64)

1
2 (18),

⇒ RR 1
2
(G2) = 119.623524n2 − 128.557979n+ 3.701427.

for α = −1
2 ,

RR− 1
2
(G2) = (1)−

1
2 (9n2 − 33n+ 30) + (6)−

1
2 (12n− 24) + (36)−

1
2 (6n− 18)

+(7)−
1
2 (18n2 − 60n+ 48) + (42)−

1
2 (6n− 12) + (49)−

1
2 (9n2 − 33n+ 30)

+(8)−
1
2 (6n) + (48)−

1
2 (12n− 12) + (56)−

1
2 (12n− 24) + (64)−

1
2 (18),

⇒ RR− 1
2
(G2) = 17.089075n2 − 48.110416n+ 35.089224.

Theorem 2.2.2. Let G2 be the second type of hex-derived network then

RABC(G2) = 21.118607n2 − 37.298413n+ 12.603823.

RGA(G2) = 29.905881n2 − 57.684337n+ 27.164543.



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3120

Proof. Let G2
∼= HDN2(n), then by using equation 3 and Table 2, we have

RABC(G2) =

√
1 + 1− 2

1× 1
|E1,1(G2)|+

√
6 + 1− 2

6× 1
|E6,1(G2)|

+

√
6 + 6− 2

6× 6
|E6,6(G2)|+

√
7 + 1− 2

7× 1
|E7,1(G2)|

+

√
7 + 6− 2

7× 6
|E7,6(G2)|+

√
7 + 7− 2

7× 7
|E7,7(G2)|

+

√
8 + 1− 2

8× 1
|E8,1(G2)|+

√
8 + 6− 2

8× 6
|E8,6(G2)|

+

√
8 + 7− 2

8× 7
|E8,7(G2)|+

√
8 + 8− 2

8× 8
|E8,8(G2)|,

RABC(G2) =

√
1 + 1− 2

1× 1
(9n2 − 33n+ 30) +

√
6 + 1− 2

6× 1
(12n− 24)

+

√
6 + 6− 2

6× 6
(6n− 18) +

√
7 + 1− 2

7× 1
(18n2 − 60n+ 48)

+

√
7 + 6− 2

7× 6
(6n− 12) +

√
7 + 7− 2

7× 7
(9n2 − 33n+ 30)

+

√
8 + 1− 2

8× 1
(6n) +

√
8 + 6− 2

8× 6
(12n− 12)

+

√
8 + 7− 2

8× 7
(12n− 24) +

√
8 + 8− 2

8× 8
(18),

⇒ RABC(G2) = 21.118607n2 − 37.298413n+ 12.603823.

Now, let G2
∼= HDN2(n), then by using equation 3 and Table 2, we have

RGA(G2) =
2
√
1× 1

1 + 1
|E1,1(G2)|+

2
√
6× 1

6 + 1
|E6,1(G2)|

+
2
√
6× 6

6 + 6
|E6,6(G2)|+

2
√
7× 1

7 + 1
|E7,1(G2)|

+
2
√
7× 6

7 + 6
|E7,6(G2)|+

2
√
7× 7

7 + 7
|E7,7(G2)|

+
2
√
8× 1

8 + 1
|E8,1(G2)|+

2
√
8× 6

8 + 6
|E8,6(G2)|

+
2
√
8× 7

8 + 7
|E8,7(G2)|+

2
√
8× 8

8 + 8
|E8,8(G2)|,

RGA(G2) =
2
√
1× 1

1 + 1
(9n2 − 33n+ 30) +

2
√
6× 1

6 + 1
(12n− 24)



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3121

+
2
√
6× 6

6 + 6
(6n− 18) +

2
√
7× 1

7 + 1
(18n2 − 60n+ 48)

+
2
√
7× 6

7 + 6
(6n− 12) +

2
√
7× 7

7 + 7
(9n2 − 33n+ 30)

+
2
√
8× 1

8 + 1
(6n) +

2
√
8× 6

8 + 6
(12n− 12)

+
2
√
8× 7

8 + 7
(12n− 24) +

2
√
8× 8

8 + 8
(18),

⇒ RGA(G2) = 29.905881n2 − 57.684337n+ 27.164543.

Theorem 2.2.3. Let G2 be the second type of hex-derived network then

RM1(G2) = 288n2 − 372n+ 84

RHM(G2) = 2952n2 − 2436n+ 132

Proof. Let G2
∼= HDN2(n), then by using equation 5 and Table 2, we have

RM1(G2) = (1 + 1)|E1,1(G2)|+ (6 + 1)|E6,1(G2)|+ (6 + 6)|E6,6(G2)|
+(7 + 1)|E7,1(G2)|+ (7 + 6)|E7,6(G2)|+ (7 + 7)|E7,7(G2)|
+(8 + 1)|E8,1(G2)|+ (8 + 6)|E8,6(G2)|+ (8 + 7)|E8,7(G2)|
+(8 + 8)|E8,8(G2)|,

RM1(G2) = (1 + 1)(9n2 − 33n+ 30) + (6 + 1)(12n− 24) + (6 + 6)(6n− 18)

+(7 + 1)(18n2 − 60n+ 48) + (7 + 6)(6n− 12)

+(7 + 7)(9n2 − 33n+ 30) + (8 + 1)(6n) + (8 + 6)(12n− 12)

+(8 + 7)(12n− 24) + (8 + 8)(18),

⇒ RM1(G2) = 288n2 − 372n+ 84.

Now let G2
∼= HDN2(n), then by using equation 6 and Table 2, we have

RHM(G2) = (1 + 1)2|E1,1(G2)|+ (6 + 1)2|E6,1(G2)|+ (6 + 6)2|E6,6(G2)|
+(7 + 1)2|E7,1(G2)|+ (7 + 6)2|E7,6(G2)|+ (7 + 7)2|E7,7(G2)|
+(8 + 1)2|E8,1(G2)|+ (8 + 6)2|E8,6(G2)|+ (8 + 7)2|E8,7(G2)|
+(8 + 8)2|E8,8(G2)|,

RHM(G2) = (1 + 1)2(9n2 − 33n+ 30) + (6 + 1)2(12n− 24) + (6 + 6)2(6n− 18)

+(7 + 1)2(18n2 − 60n+ 48) + (7 + 6)2(6n− 12)

+(7 + 7)2(9n2 − 33n+ 30) + (8 + 1)2(6n) + (8 + 6)2(12n− 12)

+(8 + 7)2(12n− 24) + (8 + 8)2(18),

⇒ RHM(G2) = 2952n2 − 2436n+ 132.



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3122

Theorem 2.2.4. Let G2 be the second type of hex-derived network then

RF (G2) = 1800n2 − 1968n+ 588.

Proof. Let G2
∼= HDN2(n), then by using equation 7 and Table 2, we have

RF (G2) = ((1)2 + (1)2)|E1,1(G2)|+ ((6)2 + (1)2)|E6,1(G2)|
+((6)2 + (6)2)|E6,6(G2)|+ ((7)2 + (1)2)|E7,1(G2)|
+((7)2 + (6)2)|E7,6(G2)|+ ((7)2 + (7)2)|E7,7(G2)|
+((8)2 + (1)2)|E8,1(G2)|+ ((8)2 + (6)2)|E8,6(G2)|
+((8)2 + (7)2)|E8,7(G2)|+ ((8)2 + (8)2)|E8,8(G2)|,

RF (G2) = ((1)2 + (1)2)(9n2 − 33n+ 30) + ((6)2 + (1)2)(12n− 24)

+((6)2 + (6)2)(6n− 18) + ((7)2 + (1)2)(18n2 − 60n+ 48)

+((7)2 + (6)2)(6n− 12) + ((7)2 + (7)2)(9n2 − 33n+ 30)

+((8)2 + (1)2)(6n) + ((8)2 + (6)2)(12n− 12)

+((8)2 + (7)2)(12n− 24) + ((8)2 + (8)2)(18),

⇒ RF (G2) = 1800n2 − 1968n+ 588.

Theorem 2.2.5. Let G2 be the second type of hex-derived network then

RRZ1(G2) =
288

7
n2 − 3155

28
n+

562

7

RRZ2(G2) =
207

4
n2 − 124361

2730
n− 4588

455
RRZ3(G2) = 7200n2 − 1116n− 1800

Proof. Let G2
∼= HDN2(n), then by using equation 8 and Table 2, we have

RRZ1(G2) =

(
1 + 1

1× 1

)
|E1,1(G2)|+

(
6 + 1

6× 1

)
|E6,1(G2)|+

(
6 + 6

6× 6

)
|E6,6(G2)|

+

(
7 + 1

7× 1

)
|E7,1(G2)|+

(
7 + 6

7× 6

)
|E7,6(G2)|+

(
7 + 7

7× 7

)
|E7,7(G2)|

+

(
8 + 1

8× 1

)
|E8,1(G2)|+

(
8 + 6

8× 6

)
|E8,6(G2)|+

(
8 + 7

8× 7

)
|E8,7(G2)|

+

(
8 + 8

8× 8

)
|E8,8(G2)|,

RRZ1(G2) =

(
1 + 1

1× 1

)
(9n2 − 33n+ 30) +

(
6 + 1

6× 1

)
(12n− 24)



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3123

+

(
6 + 6

6× 6

)
(6n− 18) +

(
7 + 1

7× 1

)
(18n2 − 60n+ 48)

+

(
7 + 6

7× 6

)
(6n− 12) +

(
7 + 7

7× 7

)
(9n2 − 33n+ 30)

+

(
8 + 1

8× 1

)
(6n) +

(
8 + 6

8× 6

)
(12n− 12)

+

(
8 + 7

8× 7

)
(12n− 24) +

(
8 + 8

8× 8

)
(18),

⇒ RRZ1(G2) =
288

7
n2 − 3155

28
n+

562

7
.

Now let G2
∼= HDN2(n), then by using equation 9 and Table 2, we have

RRZ2(G2) =

(
1× 1

1 + 1

)
|E1,1(G2)|+

(
6× 1

6 + 1

)
|E6,1(G2)|+

(
6× 6

6 + 6

)
|E6,6(G2)|

+

(
7× 1

7 + 1

)
|E7,1(G2)|+

(
7× 6

7 + 6

)
|E7,6(G2)|+

(
7× 7

7 + 7

)
|E7,7(G2)|

+

(
8× 1

8 + 1

)
|E8,1(G2)|+

(
8× 6

8 + 6

)
|E8,6(G2)|+

(
8× 7

8 + 7

)
|E8,7(G2)|

+

(
8× 8

8 + 8

)
|E8,8(G2)|,

RRZ2(G2) =

(
1× 1

1 + 1

)
(9n2 − 33n+ 30) +

(
6× 1

6 + 1

)
(12n− 24)

+

(
6× 6

6 + 6

)
(6n− 18) +

(
7× 1

7 + 1

)
(18n2 − 60n+ 48)

+

(
7× 6

7 + 6

)
(6n− 12) +

(
7× 7

7 + 7

)
(9n2 − 33n+ 30)

+

(
8× 1

8 + 1

)
(6n) +

(
8× 6

8 + 6

)
(12n− 12)

+

(
8× 7

8 + 7

)
(12n− 24) +

(
8× 8

8 + 8

)
(18),

⇒ RRZ2(G2) =
207

4
n2 − 124361

2730
n− 4588

455
.

Again let G2
∼= HDN2(n), then by using equation 10 and Table 2, we have

RRZ3(G2) = (1 + 1)(1× 1)|E1,1(G2)|+ (6 + 1)(6× 1)|E6,1(G2)|
+(6 + 6)(6× 6)|E6,6(G2)|+ (7 + 1)(7× 1)|E7,1(G2)|
+(7 + 6)(7× 6)|E7,6(G2)|+ (7 + 7)(7× 7)|E7,7(G2)|
+(8 + 1)(8× 1)|E8,1(G2)|+ (8 + 6)(8× 6)|E8,6(G2)|



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3124

+(8 + 7)(8× 7)|E8,7(G2)|+ (8 + 8)(8× 8)|E8,8(G2)|,

RRZ3(G2) = (1 + 1)(1× 1)(9n2 − 33n+ 30) + (6 + 1)(6× 1)(12n− 24)

+(6 + 6)(6× 6)(6n− 18) + (7 + 1)(7× 1)(18n2 − 60n+ 48)

+(7 + 6)(7× 6)(6n− 12) + (7 + 7)(7× 7)(9n2 − 33n+ 30)

+(8 + 1)(8× 1)(6n) + (8 + 6)(8× 6)(12n− 12)

+(8 + 7)(8× 7)(12n− 24) + (8 + 8)(8× 8)(18),

⇒ RRZ3(G2) = 7200n2 − 1116n− 1800.

n RR1 RR−1 RR 1
2

RR− 1
2

RABC RGA

1 66 — −8.41 3.87 −3.30 −0.80

2 1428 1.58 190.41 6.09 19.51 25.02

3 3168 19.65 521.07 37.69 76.47 89.55

4 5286 59.32 983.58 98.69 167.59 192.77

5 7782 120.58 1577.93 189.06 292.86 334.69

6 10656 203.45 2304.12 308.83 452.28 515.32

7 13908 307.92 3162.15 457.97 645.85 734.64

8 17538 433.98 4152.02 636.49 873.58 992.65

9 21546 581.65 5273.74 844.41 1135.46 1289.37

10 25932 750.92 6527.29 1081.71 1431.49 1624.78

Table 3: Comparison Table for HDN1(n)

n RM1 RHM RF RRZ1 RRZ2 RRZ3

1 0 552 420 8.75 −5.52 3480

2 432 6588 3732 17.55 87.26 23844

3 1296 17052 10716 101.95 221.77 48204

4 2592 31944 21372 261.95 398.02 76560

5 4320 51264 35700 497.55 615.98 108912

6 6480 75012 53700 808.75 875.68 145260

7 9072 303188 75372 1195.55 1177.10 185604

8 12096 135792 100716 1657.95 1520.25 229944

9 15552 172824 129732 2195.95 1905.13 278280

10 19440 214284 162420 2809.55 2331.73 330612

Table 4: Comparison Table for HDN1(n)

3. Discussion

While this study demonstrates the potential of reversed degree-based topological in-
dices in understanding molecular structures and predicting chemical properties, limitations



K. A. Alsatami et al. / Eur. J. Pure Appl. Math, 17 (4) (2024), 3109-3128 3125

n RR1 RR−1 RR 1
2

RR− 1
2

RABC RGA

1 114 — −5.23 4.07 −3.57 −0.61

2 1608 1.86 225.08 7.22 22.48 31.42

3 4254 21.91 694.64 44.56 90.77 123.26

4 8052 65.48 1403.45 116.07 201.31 274.92

5 13002 132.56 2351.50 221.76 354.07 486.39

6 19104 223.14 3538.80 361.63 549.08 757.67

7 26358 337.24 4965.35 535.68 786.33 1088.76

8 34764 474.84 6631.14 743.96 1065.81 1479.66

9 44322 635.96 8536.18 986.31 1387.53 1930.38

10 55032 820.58 10680.47 1262.89 1751.48 2440.91

Table 5: Comparison Table for HDN2(n)

n RM1 RHM RF RRZ1 RRZ2 RRZ3

1 0 648 420 8.75 −3.88 4284

2 492 7068 3852 19.5 105.81 24768

3 1560 19392 10884 112.54 319.01 59652

4 3204 37620 21516 287.86 635.70 108936

5 5424 61752 35748 545.46 1055.89 172620

6 8220 91788 53580 885.36 1579.59 250704

7 11592 127728 75012 1307.54 2206.79 343188

8 15540 169572 100044 1812 2937.48 450072

9 20064 217320 128676 2398.75 3771.68 571356

10 25164 270972 160908 3067.78 4709.38 707040

Table 6: Comparison Table for HDN2(n)

include a focus on hex-derived networks, computational method limitations for large or
complex structures, and the need for experimental validation. Future directions include
applying topological indices to pharmacokinetics and pharmacodynamics, developing new
indices incorporating electronic or steric properties, using machine learning to improve
predictive power, and designing novel materials with specific properties, such as high-
temperature superconductors or nanomaterials, to further unlock the potential of topo-
logical indices in chemical graph theory and drive innovation in molecular design and
property prediction.

4. Conclusion

This study successfully computed reversed degree-based topological indices for hex-
derived networks of type 1 and 2, contributing to the advancement of quantitative structure-
property relationships (QSPRs) and quantitative structure-activity relationships (QSARs).
The findings of this research have significant implications for understanding the physical,
biomedical, and molecular properties of chemicals, as well as their biological activities.



REFERENCES 3126

Given the diverse applications of hex-derived networks in fields such as networking, phar-
macy, electronics, and data analysis, this study’s results can be applied to optimize molec-
ular structures and predict desired properties, ultimately driving innovation and discovery
in various scientific domains. We obtain some closed formulas for reversed Randić, ABC,
GA, first Zagreb, hyper Zagreb, forgotten, first, second, third redefined Zagreb index of
Hex-derived network. The numerical behavior of these indices are shown in Table 3, 4, 5,
6, for HDN1(n) and HDN2(n). It is clear that the values of these indices are directly
proportional to the value of n, as n increases the values of these indices also increases,
which is beneficial for the researchers. These results are helpful in the field of computer
science and chemistry.

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