EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5599
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Data Analysis and Irregularity Measurements in the
Identical Structures of Carbon Nanocones CNCt(m)

Kamel Jebreen1,2,3,∗, Hassan Kanj4, Inad Nawajah5, Nancy Chendeb6,
Rami M. Amro1

1 Faculty of Applied Sciences, Palestine Technical University - Kadoorie, Hebron, Palestine
2 Department of Mathematics, An-Najah National University, Nablus, Palestine
3 Unité de Recherche Clinique Saint-Louis Fernand-Widal Lariboisière, APHP, Paris, France
4 College of Engineering and Technology, American University of the Middle East,
Egaila 54200, Kuwait
5 Department of Mathematics, Hebron University, Hebron, Palestine
6 Department of Computer Science, DeVinci Engineering Shcool, La Défense, 92916 Paris,
France , France

Abstract. Irregularity indices are topological indices by nature. They are highly helpful for
determining the quantitative topography of nonregular graphs’ molecular structures. Both the
quantitative structure-property relationship (QSPR) and the quantitative structure-activity rela-
tionship (QSAR) depend heavily on the computation of abnormalities in a graph. It is made up of
several chemical and physical characteristics, including resistance, enthalpy, entropy, toxicity, melt-
ing and boiling points, and entropy. This paper examines the application of different irregularity
indices to identify irregularity measurements (IMs) in the network of carbon nanocone molecules
CNCt(m), for t = 4, 5, and t. We have used different irregularity indices such as Irdif (ξt) ,Al (ξt) ,
Irl (ξt) , Irlu (ξt) , Irlf (ξt) , Irf (ξt) , Irla (ξt), Ird 1 (ξt) , Ira (ξt) , Irga (ξt) , Irb (ξt)& Irr t (ξt). Compar-
ative graphic measures of irregularity in CNC4(m), CNC5(m) and CNCt(m) have also been ex-
amined and presented. We are interested in creating new formulas to gain a better understanding
of irregularity measures in carbon nanocones using the indices described above.

2020 Mathematics Subject Classifications: 05C09, 05C10, 05C12, 05C31, 05C07, 05C10,
05C90, 05C92

Key Words and Phrases: Carbon nanocones, graphical measurements, vertex degree, edges,
irregularity

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5599

Email addresses: Kamel.Jebreen@ptuk.edu.ps (K. Jebreen),
hassan.kanj@aum.edu.kw (H. Kanj), nancy.chendeb@devinci.fr (N. Chendeb),
inadn@hebron.edu (I. Nawaja), r.amro@ptuk.edu.ps (R. Amro)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 2 of 19

1. Introduction

The most significant and fascinating element in existence is carbon. Without carbon,
life on Earth would not be conceivable. It makes up the majority of the body’s fats, pro-
teins, carbohydrates, muscular tissue, and DNA. It is a non-metallic molecular component
that is produced cosmically when helium burns. Technology based on atoms and molecules
is known as nanotechnology. It comprises the design and decision-making involved in cre-
ating nanodevices. The use of carbon nanoparticles in nanotechnology [3, 9, 28] is crucial.

Carbon nanocones are structures that are conical in shape. They were invented in
1968 [12, 30]. They are useful for capping thin carbon on ultrafine gold needles that
are used in scanning probe microscopy [10, 15, 26]. Carbon nanostructures have various
real-life applications, such as biosensors, energy storage, and gas sensors [2, 27]. Carbon
nanocones are typically symbolized as CNCt(m). A topological descriptor (TD) is known
as the irregularity index of a graph ξt if (TD (ξt) ≥ 0 and TD (ξt) = 0 iff ξt is a regular
graph. Many researchers have described that the prediction and discrimination of irregu-
larity indices are low [13, 25, 31]. The main motivation of this study is how irregularity
indices give accurate results that are used in many fields like enthalpy and entropy. The
irregularity indices [1, 14, 29, 33] are depicted in Table 1.

Irregularity Indices

Irrs (ξt) = 1
2

∑
ρµ∈E(ξt)

∣∣dρ − dµ
∣∣ Ira (ξt) =

∑
ρµ∈E(ξt)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

Irga (ξt) =
∑

ρµ∈E(ξt)
ln

dρ+dµ
2
√

dρ×dµ

Al (ξt) =
∑

ρµ∈E(ξt)

∣∣dρ − dµ
∣∣ Irdif (ξt) =

∑
ρµ∈E(ξt)

∣∣∣∣ dρdµ
− dµ

dρ

∣∣∣∣2 Irla (ξt) = 2
∑

ρµ∈E(ξt)
|dρ−dµ|
dρ+dµ

Irf (ξt) =
∑

ρµ∈E(ξt)

(
dρ − dµ

)2 Irlu (ξt) =
∑

ρµ∈E(ξt)
|dρ−dµ|

min(dρ,dµ)
Ird1 (ξt) =

∑
ρµ∈E(ξt)

ln
{
1 +

∣∣dρ − dµ
∣∣}

Irb (ξt) =
∑

ρµ∈E(ξt)

(
d

1
2
ρ − d

1
2
µ

)2

Irlf (ξt) =
∑

ρµ∈E(ξt)
|dρ−dµ|√

dρ×dµ
Irl (ξt) =

∑
ρµ∈E(ξt)

∣∣ln dρ − ln dµ
∣∣

Table 1: Irregularity indices for ξt = CNCt(m), for t = 4, 5 and t.

2. Material and Methods

Let ξt be an undirected, connected, simple, and finite graph of Carbon nanocones where
dρ and dµ are the valencies of the nodes ρ and µ, respectively [18, 23]. It is symbolized
as CNCt(m). We take the molecular structures of Carbon nanocones CNCt(m), for
t = 4, 5 and t, and then different irregularity indices such as Irdif (ξt) , Al (ξt) , Irl (ξt) ,
Irlu (ξt) , Irlf (ξt) , Irf (ξt) , Irla (ξt) , Ird1 (ξt) , Ira (ξt) , Irga (ξt) , Irb (ξt) and Irrs (ξt) are
computed. We have used the comparative study method of testing by choosing the different
molecular structures of Carbon nanocones. We have applied twelve irregularity indices to
four different Carbon nanocones that are CNCt(m), for t = 4, 5 and t. [4, 20].

3. Applications

The above-calculated irregularity descriptors for the chemical compound Carbon nanocones
network symbolically represented by ξt = CNCt(m), for t = 4, 5 and t, can be used to



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 3 of 19

examine many physiochemical topographies [17, 19, 22]. Some of these topographies are
given below:

(i) Standard enthalpy of vaporization (DHV AP )

(ii) Boiling point (BP )

(iii) Density

(iv) Solubility.

(v) Partition Coefficient.

(vi) Ionization.

(vii) Hydrogen Bonding.

(viii) Chelation.

(ix) Isosterism.

(x) Entropy

(xi) Surface activity.

(xii) Enthalpy of vaporization (HV AP )

(xiii) Melting point

(xiv) Ionization energy.

Irregularity descriptors describe specific consequences about the above-mentioned chemical
features [11, 16, 24]. Irregularity descriptors may support to explore also the chemical,
biological, and nano properties that are extensively familiarized in developing parts of
any country. Therefore, we calculate irregularity measurements for generalized version of
CNCt(m), t = 4, 5, . . . n & m = 1, 2, 3, . . . n.

4. Main Results and Discussion

We consider the molecular structures of Carbon nanocones CNCt(m), for t = 4, 5 and
t shown in Figures 1-3. Let us choose CNCt(m) for t = 4 i.e., CNC4(m). We construct
its molecular structure and apply different irregularity indices on it to get irregularity
measurements. [6, 7, 21].



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 4 of 19

Figure 1: Graphical representation of CNC4(m)

dµ |V (dµ)| (dρ, dµ) |V (dρ, dµ)|
2 4(m+ 1) (2, 2) 4
3 4m2 + 4m (2, 3) 8m
- - (3, 3) 2

(
3m2 +m

)
Table 2: Depiction of the partition of edges of CNC4(m).

With the help of Table 1 and Table 2, we have the following theorems:
Theorem 1 Let ξt be the molecular graph of Carbon nanocones CNCt(m) with t = 4,

then the irregularity indices are given by:



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 5 of 19

Irdif (ξt=4) = 50/9m

Al (ξt=4) = 8m

Irl (ξt=4) = 3.2437m

Irlu (ξt=4) = 4m

Irlf (ξt=4) = 3.2659m

Irf (ξt=4) = 8m

Irla (ξt=4) = 3.2m

Ird 1 (ξt=4) = 5.5451m

Ira (ξt=4) = 0.1346m

Irga (ξt=4) = 0.1632m

Irb (ξt=4) = 0.8081m

Ir rs (ξt=4) = 4m

Proof. The cardinality of ξt=4 with respect to the vertices is |V (ξt=4)| = 4(m + 1)2

and the cardinality of ξt=4 with respect to the edges is |E (ξt=4)| = 2
(
3m2 + 5m+ 2

)
. For

this graph we identify (2, 2), (2, 3) and (3, 3) types of edges.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 6 of 19

Irdif (ξt=4) =
∑

ρµ∈E(ξt=4)

∣∣∣∣∣ dρdµ −
dµ

dρ

∣∣∣∣∣ =
 ∑

ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣∣∣∣ dρdµ −
dµ

dρ

∣∣∣∣∣ = 6.67m.

Al (ξt=4) =
∑

ρµ∈E(ξt=4)

∣∣dρ − dµ
∣∣ =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣dρ − dµ
∣∣ = 8m.

Irl (ξt=4) =
∑

ρµ∈E(ξt)

∣∣ln dρ − ln dµ
∣∣ =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣ln dρ − ln dµ
∣∣ = 3.243m.

Irlu (ξt=4) =
∑

ρµ∈E(ξt)

∣∣dρ − dµ
∣∣

min
(
dρ, dµ

) =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

.

.

.;

∣∣dρ − dµ
∣∣

min
(
dρ, dµ

) = 4 m.

Irlf (ξt=4) =
∑

ρµ∈E(ξt)

∣∣dρ − dµ
∣∣√

dρ × dµ
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣dρ − dµ
∣∣√

dρ × dµ
= 3.2659 m.

Irf (ξt=4) =
∑

ρµ∈E(ξt)

(
dρ − dµ

)2
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(dρ − dµ
)2

= 8m.

Irla (ξt=4) =2
∑

ρµ∈E(ξt)

∣∣dρ − dµ
∣∣

dρ + dµ
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣dρ − dµ
∣∣

dρ + dµ
= 3.2m.

Ird 1 (ξt=4) =
∑

ρµ∈E(ξt)

ln
{
1 +

∣∣dρ − dµ
∣∣} =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ln
{
1 +

∣∣dρ − dµ
∣∣} = 5.5451 m.

Ira (ξt=4) =
∑

ρµ∈E(ξt)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(d
− 1

2
ρ − d

− 1
2

µ

)2

= 0.1346m.

Irga (ξt=4) =
∑

ρµ∈E(ξt)

ln
dρ + dµ

2
√

dρ × dµ
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ln
dρ + dµ

2
√

dρ × dµ
= 0.1632 m.

Irb (ξt=4) =
∑

ρµ∈E(ξt)

(
d

1
2
ρ − d

1
2
µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(d
1
2
ρ − d

1
2
µ

)2

= 0.8081 m.

Irrs (ξt=4) =
1

2

∑
ρµ∈E(ξt)

∣∣dρ − dµ
∣∣ =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ∣∣dρ − dµ
∣∣ = 4m.

All irregularity measurements for the Carbon nanocones network CNC4(m) by using
the test values of parameter t, are shown in Table 3.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 7 of 19

Irregularity Indices m = 1 m = 2 m = 3 m = 4

Irdif (ξt) 5.58 11.12 16.68 22.24
Al (ξt) 8 16 24 32
Irl (ξt) 3.24 6.49 9.73 12.97
Irlu (ξt) 4 8 12 16
Irlf (ξt) 3.27 6.53 9.80 13.06
Irf (ξt) 8 16 24 32
Irla (ξt) 3.20 6.40 9.60 12.8
Ird 1 (ξt) 5.55 11.10 16.65 22.2
Ira (ξt) 0.13 0.27 0.40 0.54
Irga (ξt) 0.16 0.33 0.49 0.65
Irb (ξt) 0.81 1.62 2.42 3.23
Irr (ξt) 4 8 12 16

Table 3: Depiction of irregularity indices with some test values for ξ4 = CNC4( m).

Graph I: Graphical Measurements of irregularity in CNC4(m).
Now, let us choose CNCt(m) for t = 5 i.e., CNC5(m). We construct its molecular

structure and apply different irregularity indices to obtain irregularity measurements.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 8 of 19

Figure 2: Molecular depiction of CNC5( m)

dµ |V (dµ)| (dρ, dµ) |V (dρ, dµ)|
2 5(m+ 1) (2, 2) 5
3 5m2 + 5m (2, 3) 10m
- - (3, 3) 2.5

(
3m2 +m

)
Table 4: Depiction of the partition of edges of CNC5( m).

With the help of Table 1 and Table 4, we have the following theorems.
Theorem 2. Let ξt be the molecular graph of Carbon nanocones CNCt(m) with t = 5,

then the irregularity indices are given by:

Irdif (ξt=5) =6.94 m

Al (ξt=5) =10 m

Irl (ξt=5) =4.0546 m

Irlu (ξt=5) =5 m

Irlf (ξt=5) =4.0824 m

Irf (ξt=5) =10 m

Irla (ξt=5) =4 m

Ird 1 (ξt=5) =6.9314 m

Ira (ξt=5) =0.1683 m

Irga (ξt=5) =0.2041 m

Irb (ξt=5) =1.0102 m

Irs (ξt=5) =5 m.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 9 of 19

Proof. The cardinality of ξt=5 with respect to the vertices is |V (ξt=5)| = 5(m+ 1)2 and
the cardinality of ξt=5 with respect to the edges is |E (ξt=5)| = 5

2

(
3m2 + 5m+ 2

)
. We

have found (2, 2), (2, 3) and (3, 3) types of edges.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 10 of 19

Irdif (ξt=5) =
∑

ρµ∈E(ξt=4)

∣∣∣∣dρdµ − dµ
dρ

∣∣∣∣ =
 ∑

ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

.̄..;

∣∣∣∣dρdµ − dµ
dρ

∣∣∣∣ = 8.33 m.

Al (ξt=5) =
∑

ρµ∈E(ξt=4)

|dρ − dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ| = 10 m.

Irl (ξt=5) =
∑

ρµ∈E(ξt)

|ln dρ − ln dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

ī

 |ln dρ − ln dµ| = 4.0546m.

Irlu (ξt=5) =
∑

ρµ∈E(ξt)

|dρ − dµ|
min (dρ, dµ)

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ|
min (dρ, dµ)

= 5 m.

Irlf (ξt=5) =
∑

ρµ∈E(ξt)

|dρ − dµ|√
dρ × dµ

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ|√
dρ × dµ

= 4.0824 m.

Irf (ξt=5) =
∑

ρµ∈E(ξt)

(dρ − dµ)
2 =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 (dρ − dµ)
2 = 10 m.

Irla (ξt=5) = 2
∑

ρµ∈E(ξt)

|dρ − dµ|
dρ + dµ

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

;
|dρ − dµ|
dρ + dµ

= 4m.

Ird 1 (ξt=5) =
∑

ρµ∈E(ξt)

ln {1 + |dρ − dµ|} =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

; ln {1 + |dρ − dµ|}

= 6.9314 m.

Ira (ξt=5) =
∑

ρµ∈E(ξt)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

= 0.1683m.

Irga (ξt=5) =
∑

ρµ∈E(ξt)

ln
dρ + dµ

2
√

dρ × dµ
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

... ln
dρ + dµ

2
√
dρ × dµ

= 0.2041 m.

Irb (ξt=5) =
∑

ρµ∈E(ξt)

(
d

1
2
ρ − d

1
2
µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(
d

1
2
ρ − d

1
2
µ

)2

= 1.0102m.

Irrs (ξt=5) =
1

2

∑
ρµ∈E(ξt)

|dρ − dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

; dρ − dµ |= 5m.

All irregularity measurements for the Carbon nanocones network CNC5( m) by using
test values of parameter t, are shown in Table 5.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 11 of 19

Irregularity Indices m = 1 m = 2 m = 3 m = 4

Irdif (ξt) 6.94 13.88 20.80 27.76
Al (ξt) 10 20 30 40
Irl (ξt) 4.05 8.10 12.15 16.2
Irlu (ξt) 5 10 15 20
Irlf (ξt) 4.08 8.16 12.24 16.32
Irf (ξt) 10 20 30 40
Irla (ξt) 4 8 12 16
Ird1 (ξt) 6.93 13.86 20.79 27.72
Ira (ξt) 0.17 0.34 0.51 0.68
Irga (ξt) 0.20 0.40 0.60 0.80
Irb (ξt) 1.01 2.02 3.03 4.04
Irr rt (ξt) 5 10 15 20

Table 5: Depiction of irregularity indices with some test values for ξ5 = CNC5( m).

Graph2: Graphical Measurements of irregularity in CNC5( m).
Now, let us choose CNCt(m) for t = t i.e., CNCt(m). We construct its molecular

structure and apply different irregularity indices to obtain irregularity measurements.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 12 of 19

Figure 3: Graphical formation display of generalized form of CNCt( m)

dµ |V (dµ)| (dρ, dµ) |V (dρ, dµ)|
2 t(m+ 1) (2, 2) t
3 t

(
m2 +m

)
(2, 3) 2tm

- - (3, 3) 0.5t
(
3m2 +m

)
Table 6: Depiction of the partition of edges of CNCt(m).

With the help of Table 1 and Table 6, we have the following theorems.

Theorem 3. Let ξt be the molecular graph of Carbon nanocones CNCt(m) with t = t,
then the irregularity indices are given by:



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 13 of 19

Irdif (ξt) = 1.3888tm

Al (ξt) = 2tm

Irl (ξt) = 0.8109tm

Irlu (ξt) = tm

Irlf (ξt) = 0.8164tm

Irf (ξt) = 2tm

Irla (ξt) = 0.4tm

Ird 1 (ξt) = 1.3862tm

Ira (ξt) = 0.0336tm

Irga (ξt) = 0.0408tm

Irb (ξt) = 0.2020tm

Irr (ξt) = tm

Proof. The cardinality of ξt with respect to the vertices is |V (ξt)| = t(m + 1)2 and
the cardinality of ξt with respect to the edges is |E (ξt)| = t

2

(
3m2 + 5m+ 2

)
. We have

found (2, 2), (2, 3) and (3, 3) types of edges.



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 14 of 19

Irdif (ξt) =
∑

ρµ∈E(ξt=4)

∣∣∣∣dρdµ − dµ
dρ

∣∣∣∣ =
 ∑

ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

.̄..

∣∣∣∣dρdµ − dµ
dρ

∣∣∣∣ = 1.6666tm.

Al (ξt) =
∑

ρµ∈E(ξt=4)

|dρ − dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ| = 2tm.

Irl (ξt) =
∑

ρµ∈E(ξt)

|ln dρ − ln dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

.̄..

 |ln dρ − ln dµ| = 0.8109tm.

Irlu (ξt) =
∑

ρµ∈E(ξt)

|dρ − dµ|
min (dρ, dµ)

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

;

 |dρ − dµ|
min (dρ, dµ)

= tm.

Irlf (ξt) =
∑

ρµ∈E(ξt)

|dρ − dµ|√
dρ × dµ

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ|√
dρ × dµ

= 0.8164tm.

Irf (ξt) =
∑

ρµ∈E(ξt)

(dρ − dµ)
2
=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 (dρ − dµ)
2
= 2tm.

Irla (ξt) =2
∑

ρµ∈E(ξt)

|dρ − dµ|
dρ + dµ

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 |dρ − dµ|
dρ + dµ

= 0.8tm.

Ird 1 (ξt) =
∑

ρµ∈E(ξt)

ln {1 + |dρ − dµ|} =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

; ln {1 + |dρ − dµ|} = 1.3862tm.

Ira (ξt) =
∑

ρµ∈E(ξt)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(
d
− 1

2
ρ − d

− 1
2

µ

)2

= 0.0336tm.

Irga (ξt) =
∑

ρµ∈E(ξt)

ln
dρ + dµ

2
√
dρ × dµ

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

 ln
dρ + dµ

2
√

dρ × dµ
= 0.0408tm.

Irb (ξt) =
∑

ρµ∈E(ξt)

(
d

1
2
ρ − d

1
2
µ

)2

=

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

(
d

1
2
ρ − d

1
2
µ

)2

= 0.2020tm.

Irrs (ξt) =
1

2

∑
ρµ∈E(ξt)

|dρ − dµ| =

 ∑
ρµ∈E22(ξt=4)

+
∑

ρµ∈E23(ξt=4)

+
∑

ρµ∈E33(ξt=4)

|dρ − dµ| = tm

All irregularity measurements for the Carbon nanocones network CNCt(m) by using test
values of parameter m, are shown in Table 7 with their graphical representation in graph

3 .



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 15 of 19

Irregularity Indices m = 1 m = 2 m = 3 m = 4

Irdif (ξt) 1.39 t 2.78 t 5.01 t 5.56 t
Al (ξt) 2 t 4 t 6 t 8 t
Irl (ξt) 0.81 t 1.62 t 2.43 t 3.24 t
Irlu (ξt) t 2 t 3 t 4 t
Irlf (ξt) 0.82 t 1.64 t 2.46 t 3.28 t
Irf (ξt) 2 t 4 t 6 t 8 t
Irla (ξt) 0.4 t 0.8 t 1.2 t 1.6 t
Ird1 (ξt) 1.39 t 2.78 t 4.17 t 5.56 t
Ira (ξt) 0.03 t 0.06 t 0.09 t 0.12 t
Irga (ξt) 0.04 t 0.08 t 0.12 t 0.16 t
Irb (ξt) 0.20 t 0.40 t 0.60 t 0.80 t
Irr t (ξt) t 2 t 3 t 4 t

Table 7: Depiction of irregularity indices with some test values for ξt = CNCt(m).

Graph 3: Graphical Measurements of irregularity in CNCt=1(m).

Irf (ξt) Ira (ξt)

CNC4( m) 32 0.54
CNC5( m) 40 0.68

Table 8: Shows the irregularity indices Irf (ξt) and Ira (ξt) for ξt = CNCt=4,5( m).

Graph4: Graphical Analysis of irregularity in CNCt=4,5( m).



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 16 of 19

IMs CNCt(m), t = 4, 5, . . . n&m = 1, 2, 3, . . . n

Irdif (ξt) 1.39 tm
Al (ξt) 2 tm
Irl (ξt) 0.81 tm
Irlu (ξt) tm
Irlf (ξt) 0.82 tm
Irf (ξt) 2 tm
Irla (ξt) 0.40tm
Ird 1 (ξt) 1.39 tm
Ira (ξt) 0.03 tm
Irga (ξt) 0.04ttm
Irb (ξt) 0.20 tm
Irr rs) tm

Table 9: Shows the irregularity measurements with generalized form of Carbon nancones.

5. Conclusion

We analyse the theoretical and graphical data we acquired for the Carbon Nanocones
network CNCt(m), for t = 4, 5 and t using the twelve irregularity indices described
above. Based on these results, we determine which molecular network is more irregu-
lar than the others according to a certain irregularity index. It has been noted that, for
CNCt(m), for t = 4, 5 and t, all of the generated graphs have a straight-line form, and
all formulas derived from irregularity indices are linear in m and CNCt( m) are depen-
dent on a parameter m . It has shown that increasing the parameter value causes the
irregularity indices’ value to grow as well. Graphs 1-2 show the graphic behaviour of IM
for CNCt(m), for t = 4, 5.
The calculated and shown irregularity indices for for ξt = CNCt=4,5,( m) are presented
in Table 8. The irregularity measurements for the generalized version of of CNCt(m), t =
4, 5, . . . n& m = 1, 2, 3, . . . n, are displayed in Table 9. The graphical behaviour of CNC4(m),
CNC5(m) for the irregularity index Ir f (ξt) is shown by the black line.On the other
hand, graph 4 shows the graphical behavior of CNC4( m), CNC5( m) for the irregu-
larity index Ira (ξt), as represented by the pink line.
Moreover, a variety of physiochemical parameters, including those listed in Section 5
above, can be found and compared using irregularity indices [5, 8, 32]. Using irregularity
indices, correlation, and the regression model P = a + bX, where X can be any topolog-
ical index - we determine the experimental values of the fourteen physiochemical param-
eters listed above and then compare them to our estimated values. Additionally, we can
contrast their major and non-significant errors. If we compute the fourteen physiochem-
ical parameters listed above more accurately and numerically using irregularity indices,
we do not require the experimental results.

It has also been observed that the difference of irregularity measurements for Irf (ξt)



K. Jebreen et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5599 17 of 19

among CNC4( m) and CNC5( m) are unified. Similarly, the difference of irregularity
measurements for Ira (ξt) among CNC4( m) and CNC5( m) are same.

Acknowledgements

The authors extend their heartfelt gratitude to Palestine Technical University-Kadoorie,
Palestine, American University of the Middle East, Egaila 54200, Kuwait, and ICTP-
Arab Fund Associates Programme (2024-2026) for their invaluable support in facilitat-
ing this research.

Data Availability: All data used are included inside the manuscript.

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