EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 4, 2023, 2187-2197
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Topological Characterization of Hexagonal Network and
Non-Kekulean Benzenoid Hydrocarbon

Hassan Kanj1, Hifza Iqbal2, Muhammad Haroon Aftab2,∗, Hasnain Raza2,
Kamel Jebreen3,4,5, Mohammed Issa Sowaity6

1 College of Engineering and Technology, American University of the Middle East, Egaila,
Kuwait
2 Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
3 Department of Mathematics, Palestine Technical University-Kadoorie, Hebron, Palestine
4 Department of Mathematics, An-Najah National University, Nablus, Palestine
5 Biostatistics and Clinical Research Department, University Hospital, Lariboisière, AP-HP,
Universite’ Paris, France
6 Department of Mathematics, Palestine Polytechnic University, Hebron, Palestine

Abstract. Graph theory has great deal of application in disciplines like biology, computer science,
informatics, engineering, mathematics, social science, linguistics and medicine. Topological indices
are created by labeling the chemical structure through numerical parameters. These notations
link chemical compounds physicochemical features such as boiling temperature, stability, strain
energy etc, to its molecular structure. Different topological features of chemical structure have
been explored. One of the molecular graphs topological indices are the connection indices defined
on the number of connection between vertices of distance two. In this work, we compute the first
Zagreb connection index, second Zagreb connection index, and modified first Zagreb connection
index for hexagonal network and non kekulean benzenoid hydrocarbon.

2020 Mathematics Subject Classifications: 0508, 05C92, 92E10

Key Words and Phrases: Topological Indices, Connection Index, Hexagonal Network, Non-
Kekulean Benzenoid Hydrocarbon

1. Introduction

Graph theory has been recognized as a separate branch of mathematics which has
given chemists several useful tips, like topological indices and graph energy. The structural
formula of a chemical compound is represented by a molecular graph. Topological indices
are most important part of graph theory we use these indices for the understanding and

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i4.4888

Email addresses: hassan.kanj@aum.edu.kw (H. Kanj), iqbalhifza3@gmail.com (H. Iqbal),
haroonuet@gmail.com (M. H. Aftab) , hasnainrazadgk1@gmail.com (H. Raza),
k.jebreen@yahoo.com (K. Jabreen), mohammad d2007@hotmail.com (M. I. Sowaity),

https://www.ejpam.com 2187 © 2023 EJPAM All rights reserved.



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2188

representation of chemical properties of different structures. In this branch of mathematics
we study the behavior of molecular structure, by using numerical.
Graph can indicate local relationship between interacting pieces of a system in statistical
physics. The hexagon unit cell is distinguished by the presence of a single line, known as
a six-fold symmetrical axis, along which cells can be turned by 60Â◦ or 120Â◦ without
changing their shape. Hexagon is a worldwide part of digital reality systems that combine
sensor, software, and self-driving technology. The Zhou et al. defined the basis of a novel
connection index, sum connectivity index and topological indices [30, 31]. Vukicevik, and
Furtula computed topological index based on the ratios of geometrical and arithmetical
means of end-vertex degrees of edges in [28]. Naji and Soner worked on first leap Zagreb
indices of some graph operations [26]. Based on vertex degrees, the idea of Randic (R),
atom-bond connectivity (ABC), and geometric-arithmetic (GA) topological indices was
formed in chemical graph theory. This theory is related to quantitative analysis, statistics,
topology, and complex mathematics [27, 29]. In addition to graph models, such dynamic
systems can be modeled using multi-agent systems, Markov chain and/or petri-nets, [4,
5, 11, 18]. Among other aspects of mathematics this theory has been widely applied
in subjects like computer science, economics, physics and chemistry. It is used to debate
molecules in chemistry and physics, in condensed matter physics, 3D structure of advanced
simulated crystal structures can be investigated quantitatively by gathering data on curve
properties related to the atom’s topology.
In chemistry, a graph is a natural representation of a molecule, with vertices representing
atoms and edges signifying bonds. For certain nanostructures their ABC and GA indices
for this critical class of networks are studied in [10, 23, 24]. One can also see the literature
related to structural invariants of graphs in [3, 6, 12, 13, 16, 17, 21, 25, 32]. Iqbal et al.
defined degree based indices and eccentricity based topological indices of some graphs in
[19, 20]. Ducoffe et al. computed first three maximal modified first Zagreb connection
indices, [7]. Farahani, et al. computed first Zagreb index, second Zagreb index, atomic
bond connectivity index, redefined first Zagreb connection index, second Zagreb connection
index and third Zagreb connection index in [8, 9].
Graph is a mathematical rendition of a system and define connection between vertices
and edges. A connected graph is a graph that has connection in sense of a topological
space there is a path from one point to any other point in the graph. Zagreb indices
are one of the oldest and most studied indices. Zagreb connection indices provide better
correlation coefficient then the classical Zagreb indices. Based on vertex and edge data,
the topological indices of connection number are define in [15]. The first Zagreb connection
index is stated as,

ZC1(G) =
∑

a∈V (G)

(τG(a))
2. (1)

The second Zagreb connection index is stated as,

ZC2(G) =
∑

ăa∈E(G)

(τG(ă)× τG(a)). (2)



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2189

The modified first Zagreb connection index is stated as,

ZC∗1(G) =
∑

ăa∈E(G)

(τG(ă) + τG(a)). (3)

Where, τG represents the connection number(vertex count at distance two).

2. Hexagonal Network

The world of information technology has been witnessing the evolvement of computer
networks to serve many different real-life applications. The hexagonal system is one of the
major structural categories that may be thoroughly understood crystal pieces are defined
in this system using four axes, three separate axes positioned at 120Â◦ with one another,
and a fourth plane perpendicular to the axes of the other three. The lattice will define
the margins of an orderly stack of blocks, or unit cells, the atoms or molecular groups of
the solid are represented by locations and the points are linked by lines.

Figure 1: Hexagonal Network.

2.1. Related Theorems and Results

The total number of vertices and edge in hexagonal network are 3n(n-1)+1 and (9n-
15)+6 respectively, where n⩾ 3, n represent the number of molecules. Now we define
the vertices partition and edges partition by labelling the vertices with their respective
connection number in Table 1 and 2. Further we calculate the indices. We also computed
some numerical value of these indices in Table 3, to show their trend in the Figure 2 for
this structure.

Theorem 1. Consider a hexagonal network, then its first Zagreb connection index is,

ZC1(Q) = 2700n2 − 9150n+ 9126.

Proof. Let Q be a graph of a hexagonal network, now by using the information in
Table 1 and the Equation 1 we have,

ZC1(Q) =
∑

ā∈V (Q)

(τQ(ā))
2,



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2190

Table 1: Vertices Partition of Hexagonal Network Based on Connection Number.

V ertices Partition Count

11 6
14 6n− 12
23 6n− 12
30 3n2 − 15n+ 19

Table 2: Edges Partition of Hexagonal Network Based on Connection Number.

Edges Partition Count

(11, 14) 12
(11, 23) 6
(14, 14) 6n− 18
(14, 23) 12n− 24
(23, 23) 6n− 12
(23, 30) 12n− 30
(30, 30) 9n2 − 51n+ 72

= (6)(11)2 + (6n− 12)(14)2 + (6n− 12)(23)2 + (3n2 − 15n+ 19)(30)2,

ZC1(Q) = 2700n2 − 9150n+ 9126.

Theorem 2. Consider a hexagonal network, then its second Zagreb connection index is,

ZC2(Q) = 8100n2 − 29406n+ 29862.

Proof. Let Q graph be a hexagonal network, now by using the information in Table 2
and the Equation 2 we have,

ZC2(Q) =
∑

ǎā∈E(Q)

(τQ(ǎ)× τQ(ā)),

ZC2(Q) = (12)(11× 14) + (6)(11× 23) + (6n− 18)(14× 14)

+(12n− 24)(14× 23) + (6n− 12)(23× 23)

+(12n− 30)(23× 30) + (9n2 − 51 + 72)(30× 30),

= 8100n2 − 29406n+ 29862.

Theorem 3. Consider a hexagonal network, then its modified first Zagreb connection
index is,

ZC∗
1 (Q) = 540n2 − 1536n+ 1290.



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2191

Proof. Let Q be a graph of hexagonal network, now by using the information in Table
2 and the Equation 3 we have,

ZC∗
1 (Q) =

∑
ǎā∈E(Q)

(τQ(ǎ) + τQ(ā)),

= (12)(11 + 14) + (6)(11 + 23) + (6n− 18)(14 + 14)

+(12n− 24)(14 + 23) + (6n− 12)(23 + 23)

+(12n− 30)(23 + 30) + (9n2 − 15n+ 72)(30 + 30),

ZC∗
1 (Q) = 540n2 − 1536n+ 1290.

Table 3: Numerical Behaviour of ZC1(Q), ZC2(Q), ZC∗
1 (Q), for Hexagonal Network .

n ZC1(Q) ZC2(Q) ZC∗
1 (Q)

4 15726 41838 3786
5 30876 85332 7110
6 51426 145026 11514
7 77376 220920 16998
8 108726 313014 23562
9 145476 421308 31206
10 187626 545802 39930

5 6 7 8 9 1 0
0

1 0 0 0 0 0

2 0 0 0 0 0

3 0 0 0 0 0

4 0 0 0 0 0

5 0 0 0 0 0

6 0 0 0 0 0

Co
mp

ari
so

n o
f T

op
olo

gic
al 

Ind
ice

s

n

 Z C 1 ( Q )
 Z C 2 ( Q )
 Z C * 1 ( Q )

Figure 2: ZC1(Q), ZC2(Q) , and ZC∗
1 (Q).

3. Non-Kekulean Benzenoid Hydrocarbon

These numerical parameters assist us to recognize physical features, chemical reactivity
and biological activities of chemical structures. The interesting fact is that they are struc-
ture invariant. The results can also play key role in the identification of the significance
of non-kekulean benzenoid hydrocarbon in the chemical and pharmaceutical industries.



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2192

Figure 3: Non-Kekulean Benzenoid Hydrocarbon.

3.1. Related Theorems and Results

The total number of vertices and edge in non-kekulean benzenoid hydrocorban are
12n+14 and 17n+14 respectively, where n⩾3, n represent the number of molecules. Now
we define the vertices partition and edges partition by labelling the vertices with their
respective connection number in Table 4 and 5. Further we calculate the indices. We also
computed some numerical value of these indices in Table 6, to show their trend in the
Figure 4 for this structure.

Table 4: Vertices Partition of Non-Kekulean Benzenoid Hydrocarbon Based on Connection Number.

V ertices Partition Count

2 4
3 8
4 4n+ 6
5 4
6 8n− 8

Table 5: Edges Partition of Non-Kekulean Benzenoid Hydrocarbon Based on Connection Number.

Edges Partition Count

(2, 3) 8
(3, 4) 8
(4, 4) 4n
(4, 5) 4
(4, 6) 2n+ 4
(5, 5) 2
(6, 5) 4
(6, 6) 11n− 16

Theorem 4. Consider a non-Kekulean benzenoid hydrocarbon network then its first Zagreb



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2193

connection index is,

ZC1(K) = 352n− 4.

Proof. Let K be a graph of non-kekulean benzenoid hydrocarbon, now by using the
information in Table 4 and the Equation 1 we have,

ZC1(K) =
∑

ā∈V (K)

(τK(ā))2,

= (4)(2)2 + (8)(3)2 + (4n+ 6)(4)2 + (4)(5)2 + (8n− 8)(6)2,

= 352n− 4.

Theorem 5. Consider a non-Kekulean benzenoid hydrocarbon network, then its second
Zagreb connection index is,

ZC2(K) = 508n− 86.

Proof. Let K be a non-Kekulean benzenoid hydrocarbon network, now by using the
information in Table 5 and the Equation 2 we have,

ZC2(K) =
∑

ǎā∈E(K)

(τK(ǎ)× τK(ā)),

= (8)(2× 3) + (8)(3× 4) + (4n)(4× 4) + (4)(4× 5)

+(2n+ 4)(4× 6) + (2)(5× 5) + (4)(6× 5)

+(11n− 16)(6× 6),

= 508n− 86.

Theorem 6. Consider a non-Kekulean benzenoid hydrocarbon network, then its modified
first Zagreb connection index is,

ZC∗
1 (K) = 184n+ 44.

Proof. Let K be a graph of non-Kekulean benzenoid hydrocarbon network, now by
using the information in Table 5 and the Equation 3 we have,

ZC∗
1 (K) =

∑
ǎā∈E(K)

(τK(ǎ) + τK(ā)),

= (8)(2 + 3) + (8)(3 + 4) + (4n)(4 + 4) + (4)(4 + 5)

+(2n+ 4)(4 + 6) + (2)(5 + 5) + (4)(6 + 5)

+(11n− 16)(4 + 6),

= 184n+ 44.



M. H. Aftab et al. / Eur. J. Pure Appl. Math, 16 (4) (2023), 2187-2197 2194

Table 6: Numerical Behaviour of ZC1(Q), ZC2(Q), ZC∗
1 (Q), for Non-Kekulean Benzenoid Hydrocarbon.

n ZC1(Q) ZC2(Q) ZC∗
1 (Q)

3 1052 1438 596
4 1404 1946 780
5 1756 2454 964
6 2108 2962 1148
7 2460 3470 1332
8 2812 3978 1516
9 3164 4486 1700
10 3516 4994 1884

3 4 5 6 7 8 9 1 0
0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

Co
mp

ari
so

n o
f T

op
olo

gic
al 

Ind
ice

s

n

 Z C 1 ( K )
 Z C 2 ( K )
 Z C ^ * 1 ( K )

Figure 4: ZC1(Q), ZC2(Q) , and ZC∗
1 (Q).

4. Applications and Statement of Novelty

Topological indices are massively employed as structural descriptors for Quantitative
Structure Activity Relationships, Quantitative Structure Property Relationships, Quanti-
tative Structure Toxicity Relationships. These models are widely used in pharmaceutical
and agricultural industry for chemical screening. Once a correlation between physicochem-
ical properties and molecular structure is developed it is achievable to forecast the certain
properties of chemical networks for any molecular size, [4, 14]. Over the decades numerous
structural topological descriptors have been defined, keeping in view the importance and
applications of these structures we have defined Zagreb connection indices for hexagonal
network and non-kekulean benzenoid hydrocarbon.

5. Conclusions

In mathematics, graph theory [1, 2, 22] is defined as the study of graphs which are
mathematical structures employed to model pairwise relation between objects. In this
paper, we calculated the connection number based indices for hexagonal network and non
kekulean benzenoid hydrocarbon. We defined the first Zagreb connection index, the second
Zagreb connection index and modified first Zagreb connection index based on vertices and
edges. Further we have computed some numerical values of these indices with different n,



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and provided their graphical representation. The graphs show an increasing trend of the
indices. In future one can study different types of topological indices of these networks. It
will be fascinating to debate the distance-based important parameters of these structure
because topological indices are widely used in chemistry, it may be interesting to discover
new chemical characteristics and explore their mathematical features.

Acknowledgments

All authors are thankful to those who supported this work and expedite the process
of this work. We are also thankful to the reviewers for their valuable comments.

The authors are grateful to the American University of the Middle East, Egaila 54200,
Kuwait, and Palestine Technical University – Kadoorie, Palestine, and the University of
Lahore, Lahore, Pakistan, for their support in completing this research.

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