EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5711
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

M-Polynomial and Degree-Based Topological Indices for
Iterative Graphs

Nihad Titan Sarhan1, Didar Abdulkhaleq Ali2,∗, Gohdar Hashem Mohiaddin3

1 Department of Mathematics, College of Education, Akre University for Applied Sciences,
Akre, Iraq
2 Department of Mathematics, College of Science, University of Zakho, Zakho, Iraq
3 Department of Mathematics, College of Education, University of Zakho, Zakho, Iraq

Abstract. Iterative graph have several applications in social network analysis, optimization prob-
lems, machine learning, and game theory. Such graphs are also commonly used in chemistry,
physics, and mathematics. In this article, we derive the M-polynomial for the fractal growth pat-
terns of benzene (FGBn, n ≥ 1), the Pythagoras tree (PTn, n ≥ 1), and the benzene dendrimer
(DBn, n ≥ 2). Moreover, we compute some degree-based topological indices based on the M-
polynomials, such as the first Zagreb index, the second Zagreb index, the modified second Zagreb
index, the general Randić index, the harmonic index, the inverse sum index, and the symmetric
division degree index. Finally, we presented our work graphically and compared the sketches of
M-polynomials and degree-based topological indices.

2020 Mathematics Subject Classifications: 05C05, 05C07, 05C10, 94C15

Key Words and Phrases: M-Polynomial, Topological Indices, Fractal, Pythagoras Tree, Benzene
Dendrimer

1. Introduction

Polynomials are tools in chemical graph theory used to collect information about molec-
ular graphs and thus display properties of the molecular graph without using quantum
mechanics. A molecular graph is generated by the conversion of a chemical molecule into
a graph in which vertices and edges are represented by atoms and bonds, respectively. Let
H(V, E) be a simple connected undirected graphs, where V(H) denotes the set of vertices
and E(H) denotes the set of edges. Number of vertices |V (H)| is the order of graph and
number of edges |E(H)| is the size of graph. connected graph is a graph in which there
exists a path between every pair of vertices. A vertex u’s degree, denoted as du, refers

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5711

Email addresses: nihad.titan@auas.edu.krd (N. T. Sarhan),
didar.ali@uoz.edu.krd (D. A. Ali), gohdar.mohiaddin@uoz.edu.krd (G. H. Mohiaddin)

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



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 2 of 15

to the number of edges that are adjacent to vertex u. Topological indices are numerical
parameters associated with a molecular graph that characterizes its topology.

The role of algebraic polynomials is central to the development of chemical graph
theory, such as the Hosoya polynomial [12], and the acyclic polynomial of a graph [10].
Deutsch and Klavžar [6], introduced the M-polynomial with the aim of achieving sim-
ilar goals as the Hosoya polynomial, specifically in deriving closed-form expressions for
numerous degree-based topological indices. Several researchers have calculated the M-
polynomial and associated topological indices for well-known graphs.Examples include
the M-polynomial and topological indices of nanostar dendrimers and polyhex nanotubes
[17, 18], linear chains of benzene, naphthalene, and anthracene [14], benzene rings em-
bedded in p-type surface networks, zigzag and rhombic benzenoid systems [1], generalized
Sierpinski networks [8], hourglass, triangular, as well as jagged-rectangle benzenoid sys-
tems detailed [13], Generalized Zegrab index, fourth version of atom-bond connectivity
and fifth version of geometric-arithmetic index for an infinite class of Titania nanotubes
TiO−2[m,n] [15], computation of Benzenoid planar octahedron networks by using topo-
logical indices [2], and topological indices of third types of hex-derived networks [3]. In
this paper, we derive the M-polynomial for the fractal growth patterns of benzene (FGBn,
n ≥ 1), the Pythagoras tree (PTn, n ≥ 1), and the benzene dendrimer (DBn, n ≥ 2).
Moreover, we compute some degree-based topological indices based on the M-polynomials.

Definition 1. [6] Let H be a graph and mα,β be the count of edges e = uv ∈ E(H) such
that (du, dv) = (α, β), then the M-polynomial of H is defined as:

M(H;x, y) =
∑

δ≤α≤β≤∆

mα,βx
αyβ.

Here, δ denotes the minimum degree in H, while ∆ denotes the maximum degree in H.

Several indices are derived from polynomials. The first index is the Wiener index, also
known as the path number, introduced by Wiener in [22]. Gutman and Trinajstić in [11]
introduced two new indices labeled as the first Zagreb index, denoted as M1, and the
second Zagreb index, denoted as M2, defined as:

M1(H) =
∑

uv∈E(H)

(du + dv),

M2(H) =
∑

uv∈E(H)

(dudv).

Nikolic et al. in [19], studied a modified second Zagrab index MM2, formulated as:

MM2(H) =
∑

uv∈E(H)

1
(dudv)

.

Randic in [20], introduced a bound-additive topological index as:

R−(1/2)(H) =
∑

uv∈E(H)

1√
dudv

.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 3 of 15

The general Randic index was introduced by Bollobas et. al in [5],

Rγ(H) =
∑

uv∈E(H)

(dudv)
γ .

Another Randic index is called harmonic index defined by Fajtlowics in [7], and the inverse
sum index established by Vukicevic and Graovav in [21],

H(H) =
∑

uv∈E(H)

2
du+dv

,

I(H) =
∑

uv∈E(H)

dudv
du+dv

.

Gupta et al. [9] introduced symmetric division degree index SSD(H), formulated as:

SSD(H) =
∑

uv∈E(H)

(Min(du,dv)
Max(du,dv)

+ Max(du,dv)
Min(du,dv)

).

Computational of degree-based topological indices is immediately from the rules written
above or use M-polynomial with next notations to compute degree-based topological in-
dices.

Dx = x
∂M(H;x, y)

∂x
.

Dy = y
∂M(H;x, y)

∂y
.

Sx =

∫ x

0

M(H;x, y)

t
dt.

Sy =

∫ y

0

M(H;x, y)

t
dt.

JM(H;x, y) = M(H;x, x).

2. Main Results

This section comprises three subsections, each dedicated to distinct aspects of our
analysis. In the first subsection, We investigate the M-polynomial and degree-based topo-
logical indices associated with the fractal growth pattern of benzene. Subsection two
presents our findings on the M-polynomial of Pythagoras tree as well as the degree-based
topological indices of the Pythagoras tree. Finally, we explore the dendrimer of benzene
and its associated topological indices in the concluding subsection.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 4 of 15

2.1. Fractal Growth of Benzene

The term fractal was coined in 1975 by Benoit Mandelbrot to describe a new idea in
geometry. Geometric figures in regular shapes are characterized by mathematical equa-
tions that define their dimensions such as length, width, and height. Irregular shapes can
not be measured in this way. A Fractal is a type of mathematical shape that are infinitely
complex. In essence, a Fractal is a pattern that repeats forever, and every part of the
Fractal, regardless of how zoomed in, or zoomed out you are, it looks very similar to the
whole image. Fractals are currently employed in various applications to generate textured
landscapes and images resembling natural scenes, including lunar surfaces and mountain
ranges [4, 16]. Moreover, the order and size of all graphs is given by geometric series.

 
Figure 1: Fractal Growth of Benzene FGB1 , FGB2 , FGB3 and FGB4.

The following theorem computes the M-polynomial of fractal growth of benzene.

Theorem 1. Let FGBn be a fractal growth of benzene where n is the number of iterative
fractal growth.

(i) If n = 1, then M(FGB1;x, y) = 6x2y2.

(ii) If n ≥ 2, then M(FGBn;x, y) = [20(7n−2) + 4]x2y2 + [16(7n−2)− 4]x2y4

+6(7n−2)x4y4.

Proof. Assume that FGBn is the fractal growth of benzene as shown in Figure 4,
|V(FGBn)| = |V(FGBn−1)|+6|V(FGBn−1)− 1| and |E(FGBn)| = 6(7n−1), where n ≥ 2.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 5 of 15

(i) For n = 1, the edge set E(FGB1) has only one partition

|E(2, 2)| = |e = uv ∈ E(FGB1) : du = 2anddv = 2| = 6.
Hence, the M-polynomial for FGB1 is

M(FGB1;x, y) =
∑

δ≤α≤β≤∆

mα,βx
αyβ

=
∑
2≤2

m2,2x
2y2

= 6x2y2.

(ii) For n ≥ 2, the edge set E(FGBn) can be divided into the following three parts
|E(2, 2)| = |e = uv ∈ E(FGBn) : du = 2 and dv = 2| = 20(7n−2) + 4, |E(2, 4)| =
|e = uv ∈ E(FGBn) : du = 2 and dv = 4| = 16(7n−2) − 4, and |E(4, 4)| = |e = uv ∈
E(FGBn) : du = 4 and dv = 4| = 6(7n−2).

Hence, the M-polynomial of FGBn is

M(FGBn;x, y) =
∑

δ≤α≤β≤∆

mα,βx
αyβ

=
∑
2≤2

m2,2x
2y2 +

∑
2≤4

m2,4x
2y4 +

∑
4≤4

m4,4x
4y4

= [20(7n−2) + 4]x2y2 + [16(7n−2)− 4]x2y4 + 6(7n−2)x4y4.

Degree-based topological indices of fractal growth of benzene FGBn,n ≥ 2 are given in
the next proposition.

Proposition 1. Let FGBn be a fractal growth of benzene where n ≥ 2, then

(i) M1(FGBn) = 32(7n−1)− 8.

(ii) M2(FGBn) = 304(7n−2)− 16.

(iii) MM2(FGBn) =
59
8 (7

n−2) + 1
2 .

(iv) Rγ(FGBn) = [5(22γ+2) + 23γ+4 + 3(24γ+1)](7n−2) + 22γ+2 − 23γ+2.

(v) SSD(FGBn) = 92(7n−2)− 2.

(vi) H(FGBn) =
101
6 (7n−2) + 2

3 .

(vii) I(FGBn) =
160
3 (7n−2)− 4

3 .

Proof. Since, M(FGBn;x, y) = [20(7n−2)+4]x2y2+[16(7n−2)− 4]x2y4+6(7n−2)x4y4,
using above operators we get:



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 6 of 15

Dx(FGBn) = 2[20(7n−2) + 4]x2y2 + 2[16(7n−2)− 4]x2y4 + 24(7n−2)x4y4,

Dy(FGBn) = 2[20(7n−2) + 4]x2y2 + 4[16(7n−2)− 4]x2y4 + 24(7n−2)x4y4,

DxDy(FGBn) = 4[20(7n−2) + 4]x2y2 + 8[16(7n−2)− 4]x2y4 + 96(7n−2)x4y4,

Sx(FGBn) = [10(7n−2) + 2]x2y2 + [8(7n−2)− 2]x2y4 + 3
2(7

n−2)x4y4,

Sy(FGBn) = [10(7n−2) + 2]x2y2 + [4(7n−2)− 1]x2y4 + 3
2(7

n−2)x4y4,

SxSy(FGBn) = [5(7n−2) + 1]x2y2 + 1
2 [4(7

n−2)− 1]x2y4 + 3
8(7

n−2)x4y4,

SxDy(FGBn) = [20(7n−2) + 4]x2y2 + 2[16(7n−2)− 4]x2y4 + 6(7n−2)x4y4,

SyDx(FGBn) = [20(7n−2) + 4]x2y2 + 1
2 [16(7

n−2)− 4]x2y4 + 6(7n−2)x4y4,

SxJ(FGBn) = [5(7n−2) + 1]x4 + 2
3 [4(7

n−2)− 1]x6 + 3
4(7

n−2)x8,

SxJ(DxDy(FGBn)) = [20(7n−2) + 4]x4 + 4
3 [16(7

n−2)− 4]x6 + 12(7n−2)x8,

Dγ
xD

γ
y (FGBn) = 22γ+2[5(7n−2) + 1]x2y2 + 23γ+2[4(7n−2)− 1]x2y4 + 3(24γ+1)(7n−2)x4y4.

Thus,

(i) M1(FGBn) = (Dx +Dy)(M(FGBn;x, y))|x=y=1 = 32(7n−1)− 8.

(ii) M2(FGBn) = (DxDy)(M(FGBn;x, y))|x=y=1 = 304(7n−2)− 16.

(iii) MM2(FGBn) = (SxSy)(M(FGBn;x, y))|x=y=1 = 59
8 (7

n−2) + 1
2 .

(iv) Rγ(FGBn) = (Dγ
xD

γ
y )(M(FGBn;x, y))|x=y=1 = [5(22γ+2) + 23γ+4

+ 3(24γ+1)](7n−2) + 22γ+2 − 23γ+2.

(v) SSD(FGBn) = (SyDx + SxDy)(M(FGBn;x, y))|x=y=1 = 92(7n−2)− 2.

(vi) H(FGBn) = (2SxJ)(M(FGBn; x, y))|x=1 = 101
6 (7n−2) + 2

3 .

(vii) I(FGBn) = (SxJ)(DxDy(M(FGBn;x, y)))|x=1 = 160
3 (7n−2)− 4

3 .

2.2. Pythagoras tree

Pythagoras tree (PTn), n ≥ 1 is a fractal that begins with a square.It involves creating
a right isosceles triangle with its hypotenuse along the top edge of the square. Squares
are then constructed along the other two sides of this triangle. This process is repeated
recursively for each new square created. See Figure 2.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 7 of 15

 

 

 PT0 PT1 PT2 PT8 

Figure 2: Pythagoras tree PT0 , PT1, PT2 and PT8.

In the rest of this subsection, we determine the M-polynomial of the Pythagoras tree and
with the help of the M-polynomial, we find some degree-based topological indices.The next
theorem presents the calculation of the M-polynomial for the Pythagoras tree.

Theorem 2. Let PTn, n ≥ 1 be the Pythagoras tree. Then, M(PTn;x, y) = (1+2n)x2y2+
(2 + 2n+1)x2y4 + (5 · 2n − 7)x4y4.

Proof. Let PTn, n ≥ 1 be the Pythagoras tree as shown in Figure 2. Then,

|V(PTn)| = 5 · 2n − 1 and |E(PTn)| = (2n+3 − 4).
The edge set E(PTn) can be divided into the following three parts:

|E(2, 2)| = |e = uv ∈ E(PTn) : du = 2 and dv = 2| = 1 + 2n,

|E(2, 4)| = |e = uv ∈ E(PTn) : du = 2 and dv = 4| = 2 + 2n+1, and

|E(4, 4)| = |e = uv ∈ E(PTn) : du = 4 and dv = 4| = 5 · 2n − 7.
Thus, the M-polynomial of PTn is given as:

M(PTn;x, y) =
∑

δ≤α≤β≤∆

mα,βx
αyβ

=
∑
2≤2

m2,2x
2y2 +

∑
2≤4

m2,4x
2y4 +

∑
4≤4

m4,4x
4y4

= |E(2, 2)|x2y2 + |E(2, 4)|x2y4 + |E(4, 4)|x4y4

= (1 + 2n)x2y2 + (2 + 2n+1)x2y4 + (5 · 2n − 7)x4y4.

Now, We provide several topological indices based on degrees using the M-polynomial of
the Pythagoras tree.

Proposition 2. Let PTn, n ≥ 1 be the Pythagoras tree. Then,



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 8 of 15

(i) M1(PTn) = 56 · 2n − 40.

(ii) M2(PTn) = 100 · 2n − 92.

(iii) MM2(PTn) =
13
16 · 2n + 1

16 .

(iv) Rγ(PTn) = (1 + 2γ+1 − 7 · 22γ)22γ + (1 + 2γ+1 + 5 · 22γ)22γ+n.

(v) SSD(PTn) = 17 · 2n − 7.

(vi) H(PTn) =
29
12(2

n)− 7
12 .

(vii) I(PTn) =
41
3 2

n − 31
3 .

Proof. Since, M(PTn;x, y) = (1 + 2n)x2y2 + (2 + 2n+1)x2y4 + (5 · 2n − 7)x4y4. Then,
we have

Dx(PTn) = 2(1 + 2n)x2y2 + 2(2 + 2n+1)x2y4 + 4(5 · 2n − 7)x4y4,

Dy(PTn) = 2(1 + 2n)x2y2 + 4(2 + 2n+1)x2y4 + 4(5 · 2n − 7)x4y4,

DxDy(PTn) = 4(1 + 2n)x2y2 + 8(2 + 2n+1)x2y4 + 16(5 · 2n − 7)x4y4,

D3
xD

3
y(PTn) = 64(1 + 2n)x2y2 + 512(2 + 2n+1)x2y4 + 4096(5 · 2n − 7)x4y4,

Sx(PTn) =
1
2(1 + 2n)x2y2 + 1

2(2 + 2n+1)x2y4 + 1
4(5 · 2

n − 7)x4y4,

Sy(PTn) =
1
2(1 + 2n)x2y2 + 1

4(2 + 2n+1)x2y4 + 1
4(5 · 2

n − 7)x4y4,

SxSy(PTn) =
1
4(1 + 2n)x2y2 + 1

8(2 + 2n+1)x2y4 + 1
16(5 · 2

n − 7)x4y4,

SxDy(PTn) = (1 + 2n)x2y2 + 2(2 + 2n+1)x2y4 + (5 · 2n − 7)x4y4,

SyDx(PTn) = (1 + 2n)x2y2 + 1
2(2 + 2n+1)x2y4 + (5 · 2n − 7)x4y4,

SxJ(PTn) =
i
4(1 + 2n)x4 + 1

6(2 + 2n+1)x6 + 1
8(5 · 2

n − 7)x8,

SxJ(DxDy(PTn)) = (1 + 2n)x4 + 4
3(2 + 2n+1)x6 + 2(5 · 2n − 7)x8,

Dγ
xD

γ
y (PTn) = 22γ(1 + 2n)x2y2 + 23γ(2 + 2n+1)x2y4 + 24γ(5 · 2n − 7)x4y4,

Thus,

(i) M1(PTn) = (Dx +Dy)(M(FGBn;x, y))|x=y=1 = 56 · 2n − 40.

(ii) M2(PTn) = (DxDy)(M(FGBn;x, y))|x=y=1 = 100 · 2n − 92.

(iii) MM2(PTn) = (SxSy)(M(FGBn;x, y))|x=y=1 = 13
16 · 2n + 1

16 .

(iv) Rγ(PTn) = (Dγ
xD

γ
y )(M(FGBn;x, y))|x=y=1 = (1 + 2γ+1 − 7 · 22γ)22γ + (1 + 2γ+1 +

5 · 22γ)22γ+n.

(v) SSD(PTn) = (SyDx + SxDy)(M(FGBn;x, y))|x=y=1 = 17 · 2n − 7.

(vi) H(PTn) = (2SxJ)(M(FGBn; x, y))|x=1 = 29
12(2

n)− 7
12 .

(vii) I(PTn) = SxJ(DxDy(M(FGBn;x, y)))|x=1 = 41
3 2

n − 31
3 .



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 9 of 15

2.3. Dendrimer of Benzene

A dendrimer is also one kind of a graph that is never-ending in its iterative growth,
it possesses molecular architecture has three domains central, branches, and terminal. As
shown in next Figure 3.

 

Figure 3: Dendrimer of Benzene DB1 , DB2 and DB3.

M-polynomial of dendrimer graph is given in theorem 3.

Theorem 3. Let DBn be a dendrimer of benzene where n is the number of iterative growth
of the dendrimer.

(i) If n = 1, then M(DB1;x, y) = 6x2y2.

(ii) If n ≥ 2, then M(DBn;x, y) = 24(5n−2)x2y2 + 12(5n−2)x2y4

+(9 · 5n−2 − 3)x4y4.

Proof. Let DBn, n ≥ 1 is the dendrimer of benzene as shown in Figure 3, then
|V(DBn)| = 3

2(5
n − 1) and |E(DBn)| = 6 + 9 · 5n−1 − 3.

(i) For n = 1, the prove is same as in theorem 1.

(ii) For n ≥ 2, the edge set E(DBn) has three partitions:

|E(2, 2)| = |e = uv ∈ E(FGBn) : du = 2 and dv = 2| = 24(5n−2). |E(2, 4)| = |e = uv ∈
E(FGBn) : du = 2 and dv = 4| = 12(5n−2). |E(4, 4)| = |e = uv ∈ E(FGBn) : du = 4
and dv = 4| = 6 + 9 · 5n−2 − 3.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 10 of 15

Hence, the M-polynomial of DBn is

M(DBn;x, y) =
∑

δ≤α≤β≤∆

mα,βx
αyβ

=
∑
2≤2

m2,2x
2y2 +

∑
2≤4

m2,4x
2y4 +

∑
4≤4

m4,4x
4y4

= 24(5n−2)x2y2 + 12(5n−2)x2y4 + [6 + 9(5n−2 − 1)]x4y4.

The following proposition provides the degree-based topological indices for the benzene
dendrimer DBn, n ≥ 2.

Proposition 3. Let DBn be a dendrimer of benzene where n ≥ 2, then

(i) M1(DBn) = 48(5n−1)− 24.

(ii) M2(DBn) = 336(5n−2)− 48.

(iii) MM2(DBn) =
129
16 (5

n−2)− 3
16 .

(iv) Rγ(DBn) = 3 · 22γ · 5n−2[8 + 2γ+2 + 3 · 22γ ]− 3 · 24γ.

(v) SSD(DBn) = 96(5n−2)− 6.

(vi) H(DBn) =
73
4 (5

n−2)− 3
4 .

(vii) I(DBn) = 58(5n−2)− 6.

Proof. Since, M(DBn;x, y) = 24(5n−2)x2y2+12(5n−2)x2y4+(9 ·5n−2−3)x4y4., using
above operators we get:

Dx(DBn) = 48(5n−2)x2y2 + 24(5n−2)x2y4 + 4(9 · 5n−2 − 3)x4y4.,

Dy(DBn) = 48(5n−2)x2y2 + 48(5n−2)x2y4 + 4(9 · 5n−2 − 3)x4y4.,

DxDy(DBn) = 96(5n−2)x2y2 + 96(5n−2)x2y4 + 16(9 · 5n−2 − 3)x4y4.,

Sx(DBn) = 12(5n−2)x2y2 + 6(5n−2)x2y4 + 1
4(9 · 5

n−2 − 3)x4y4.,

Sy(DBn) = 12(5n−2)x2y2 + 3(5n−2)x2y4 + 1
4(9 · 5

n−2 − 3)x4y4.,

SxSy(DBn) = 6(5n−2)x2y2 + 3
2(5

n−2)x2y4 + 1
16(9 · 5

n−2 − 3)x4y4.,

SxDy(DBn) = 24(5n−2)x2y2 + 24(5n−2)x2y4 + (9 · 5n−2 − 3)x4y4.,

SyDx(DBn) = 24(5n−2)x2y2 + 6(5n−2)x2y4 + (9 · 5n−2 − 3)x4y4.,

SxJ(DBn) = 6(5n−2)x2y2 + 2(5n−2)x2y4 + 1
8(9 · 5

n−2 − 3)x4y4.,

SxJ(DxDy(DBn)) = 24(5n−2)x2y2 + 16(5n−2)x2y4 + 2(9 · 5n−2 − 3)x4y4.,

Dγ
xD

γ
y (DBn) = 3 · 22γ+3 · 5n−2x2y2 + 3 · 23γ+2 · 5n−2x2y4 + 24γ(9 · 5n−2 − 3)x4y4..

Thus,



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 11 of 15

(i) M1(FGBn) = (Dx +Dy)(M(FGBn;x, y))|x=y=1 = 48(5n−1)− 24.

(ii) M2(FGBn) = (DxDy)(M(FGBn;x, y))|x=y=1 = 336(5n−2)− 48.

(iii) MM2(FGBn) = (SxSy)(M(FGBn;x, y))|x=y=1 = 129
16 (5

n−2)− 3
16 .

(iv) Rγ(FGBn) = (Dγ
xD

γ
y )(M(FGBn;x, y))|x=y=1 = 3·22γ ·5n−2[8+2γ+2+3·22γ ]−3·24γ .

(v) SSD(FGBn) = (SyDx + SxDy)(M(FGBn;x, y))|x=y=1 = 96(5n−2)− 6.

(vi) H(FGBn) = (2SxJ)(M(FGBn; x, y))|x=1 = 73
4 (5

n−2)− 3
4 .

(vii) I(FGBn) = (SxJ)(DxDy(M(FGBn;x, y)))|x=1 = 58(5n−2)− 6.

Remark 1. The order and size of all graphs is given by geometric series.

2.4. Plotting Representation

Our figures 2D and 3D of M-polynomials and topological indices are obtained by
Wolfram Mathematica.

 

 

Figure 4: A 3D representation of the M-polynomial for benzene’s fractal growth with n=2,3 and 4.

-2 -1 1 2 3

50

100

150

200

Figure 5: A 2D representation of degree-based topological indices derived from the M-polynomial of benzene’s
fractal growth.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 12 of 15

 

 

Figure 6: A 3D representation of the M-polynomial for a Pythagoras tree with n=2,3 and 4.

-2 2 4 6 8 10

2000

4000

6000

8000

10000

12000

14000

Figure 7: A 2D representation of degree-based topological indices derived from the M-polynomial of a Pythagoras
tree.

 

 

Figure 8: A 3D representation of the M-polynomial for a benzene dendrimer with n=2,3 and 4.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 13 of 15

-2 -1 1 2 3

-50

50

100

150

200

Figure 9: A 2D representation of degree-based topological indices derived from the M-polynomial of a benzene
dendrimer.

  
 

 

Figure 10: A 3D representation of the general Randic index for above three graphs with n=3,6 and γ=1,2

3. Conclusions

In this article, we have derived the general forms of the M-polynomials for the fractal
growth patterns of benzene (FGBn, n ≥ 1), the Pythagoras tree (PTn, n ≥ 1), and the
benzene dendrimer (DBn, n ≥ 2). Additionally, we computed degree-based topological
indices associated with these polynomials. The graphical representations in Figures (4, 6,
8) illustrate the M-polynomial graphs for the fractal growth patterns of benzene FGBn,
the Pythagoras tree PTn, and the benzene dendrimer DBn for n = 2, 3, 4. It is evident
that as n increases, the graph in Figure 4 becomes larger than that in Figure 8, and both
surpass the graph in Figure 8. Similarly, the topological indices shown in Figure 5 are
greater than those in Figure 7, which in turn are larger than those in Figure 9. These
findings highlight the trends in graph growth and the corresponding indices as the fractal
structure evolves.



N. T. Sarhan, D. A. Ali, G. H. Mohiaddin / Eur. J. Pure Appl. Math, 18 (1) (2025), 5711 14 of 15

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