EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 16, No. 3, 2023, 1580-1591
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Schultz and Modified Schultz Polynomials of Edges
Induce Chain and Ring for Hexagonal Graphs

Asmaa S. Aziz1,∗, Haitham N. Mohammed2, Ahmed M. Ali 1

1Department of Mathematics, College of Computer Sciences and Mathematics,
Mosul University, Mosul, Iraq
2Nineveh Education Directorate, Mosul, Iraq

Abstract. Schultz polynomial is one of the must significant formulas that represent a relationship
between the degree’s of vertices in a simple connected graph G and the distances between these
vertices. In this work, Schultz and modified Schultz polynomials, as well as their topological indices
of chain and ring hexagonal graphs, have been successfully identified.

2020 Mathematics Subject Classifications: 05C09, 05C31

Key Words and Phrases: Graphical Indices, Graph polynomials

1. Introduction

Mathematical chemistry is a field of theoretical chemistry that examines and pre-
dicts molecule structure using mathematical approaches rather than quantum mechanics.
Chemical graph theory is a powerful method for determining molecule structures that has
made significant contributions to the advancement of chemical science. In the molecular
graph G, atoms and bonds are represented by vertices and edges, respectively. The order of
the graph G in graph theory is p = p(G) = |V (G)|, and the size of G is q = q(G) = |E(G)|,
while the degree of η ∈ V (G) is the number of vertices joining to η and denoted by δη.
Furthermore, the distance d(µ, η) = d(µ, η | G) between any two vertices µ and η is the
length of the shortest path connecting them in G. The greatest distance in G is the diam-
eter denoted by diam(G), [7]. Let d(G, ξ) express the number of random pair of vertices in
G with ξ distance. Let aξ(G) is the set of all these pairs such that |aξ(G)| = |aξ| = d(G, ξ)

and
∑diam(G)

ξ=1 d(G, ξ) = (
p
2
), where (

p
2
) represents the number of unordered pairs of

different vertices in G,[12].

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v16i3.4783

Email addresses: asmaas982@uomosul.edu.iq(A. S. Aziz),
haytham.nashwan@yahoo.com(H. N. Mohammed), ahmedgraph@uomosul.edu.iq(A. M. Ali)

https://www.ejpam.com 1580 © 2023 EJPAM All rights reserved.



1581

Topological indices were first used in Biology and Chemistry in 1947 when scientist
Harold Wiener [20] created the Wiener index to show connections between the physico-
chemical features of organic molecules in molecular graphs.

The Wiener index, abbreviated W(G), is the sum of all distances between all unordered
(µ, η) pairs in a connected graph G :

W (G) =
∑

{µ,η}⊆V (G)

d(µ, η);

where, d(µ, η) indicates the distance between µ and η.
Based on the Wiener index, Hosoya in 1988 [14] invented the new Hosoya polynomial

H(G;x) which is defined as:

H(G;x) =
∑

{µ,η}⊆V (G)

xd(µ,η);

Recently, many other polynomials have been obtainted such as detour polynomial [3],
m-polynomial [16],[17].

The Schultz index (Sc) is another based structure which was first introduced by Harry
Schultz [18] the molecular topological index and is characterized by:

Sc(G) =
∑

{µ,η}⊆V (G)

(δµ + δη)d(µ, η),

where δµ is the degree of the vertex µ and δη is the degree of η .
The Schultz index is based on this. In 1997, Klavžar and Gutman [15] proposed the

Mod. Sch. index, which is defined as:

Sc∗(G) =
∑

{µ,η}⊆V (G)

(δµδη)d(µ, η),

There are two significant polynomials for these structural descriptors in chemical graph
theory, “Schultz polynomial(ScP)” and ”Modified Schultz polynomial(MScP)” of G are
respectively defined as:

Sc(G;x) =
∑

µ,η⊆V (G)

(δµ + δη)x
d(µ,η);

Sc∗(G;x) =
∑

µ,η⊆V (G)

(δµδη)x
d(µ,η),

In 2009, Hassani et al. [13] computed the (ScP) and (MScP) of C100 fullerene isomers
using the GAP program, while Behmaram et al. [6] obtained (ScP) of some graph opera-
tions. Farahani [8] discovered hosoya, (ScP) and (MScP) and their topological indices for
benzene, followed by (ScP) and (MScP)of coronene polycyclic aromatic hydrocarbons in
a subsequent study [9]. Many researchers have worked over the last decade to determine
(ScP) and (MScP) and their indices for graphs consisting of chains and rings of special
graphs with chemical applications [10],[19],[5],[11], [4], [1], [2].



1582

2. Results

2.1. The Edges Induce Chain For Hexagonal Graphs Ce(C6)γ

The edges induce chain for hexagonal graphs which is denoted by Ce(C6)γ is a graph
consisting of m hexagonal rings, m ≥ 3, γ = 4m − 1 every two successive rings have a
common edge induce, forming a chain as shown in Fig. 1.

Figure 1: Edges Induce Chain For Hexagonal Graphs Ce(C6)γ .

From Fig.1 we note, p(Ce(C6)γ) = 6m, q(Ce(C6)γ) = 7m − 1 and diam(Ce(C6)γ) =
4m− 1.

Table 1: Degree Matrix of Ce(C6)γ For 1 ≤ i, j ≤ γ − 2,and 1 ≤ r, s ≤ m− 1, r ̸= s and i, j ̸= r, s.

Theorem 1. For m ≥ 3, γ = 4m− 1 , then:

(i) Sc(Ce(C6)γ ;x) =
1
2(17γ−3)x+12(γ−1)x2+ 1

2(25γ−51)x3+
∑

ξ=4,8,...,γ−3(11γ−11ξ

+9)xξ + 1
2

∑
ξ=5,9,...,γ−2 (21γ − 21ξ + 22)xξ + 2

∑
ξ=6,10,...,γ−1 (5γ − 5ξ + 3)xξ

+1
2

∑
ξ=7,11,...,γ (21γ − 21ξ + 8)xξ.

(ii) Sc∗(Ce(C6)γ ;x) =
1
4(41γ−27)x+2(7γ−9)x2+ 1

4(57γ−127)x3+ 1
2

∑
ξ=4,8,...,γ−3(25γ

−25ξ+13)xξ+ 1
4

∑
ξ=5,9,...,γ−2 (49γ − 49ξ + 30)xξ+4

∑
ξ=6,10,...,γ−1 (3γ − 3ξ + 1)xξ+

1
4

∑
ξ=7,11,...,γ−4 (49γ − 49ξ + 12)xξ + 4xγ.

Proof. For vertex y, z ∈ V (Ce(C6)γ) there is d(y, z) = ξ, 1 ≤ ξ ≤ γ. And obviously:∑γ
i=1 |ai| = (18γ2 + 7γ + 193)/16.
The proof is consist of the following twelve cases:

(i) If d(y, z) = ξ = 1, then |a1| = (7γ + 3)/4, also is equal to q(Ce(C6)γ), we have four
subsets of it:

(a)
∣∣{(µ1(γ−1), ω0(γ)), (η1(γ−1), ω0(γ))

}∣∣ = 4.

(b) |{(µ4i+1, µ4i+2), (η4i+1, η4i+2) : 0 ≤ i ≤ (γ − 3)/4}| = (γ + 1)/2.



2.1 The Edges Induce Chain For Hexagonal Graphs Ce(C6)γ 1583

(c) |{(µ4i+2, ω4i+3), (η4i+2, ω4i+3) : 0 ≤ i ≤ (γ − 7)/4}| = (γ − 3)/2.

(d) |{(µ4i+1, ω4i), (η4i+1, ω4i), (ω4i−1, ω4i) : 1 ≤ i ≤ (γ − 3)/4}| = 3(γ − 3)/4.

(ii) If d(y, z) = ξ = 2, then |a2| = (5γ − 3)/2, we have six subsets of it:

(a) |{(µξ(ηξ), ω0)}| = 2.

(b) |{(µγ−ξ(ηγ−ξ), ωγ)}| = 2.

(c) |{(µ4i+1(η4i+1), ω4i+ξ+1) : 0 ≤ i ≤ (γ − ξ − 5)/4}| = (γ − ξ − 1)/2.

(d) |{(ω4i−1, µ4i+ξ−1(η4i+ξ−1)) : 1 ≤ i ≤ (γ − ξ − 1)/4}| = (γ − ξ − 1)/2.

(e) |{(µ4i+2(η4i+2), ω4i+ξ+2) : 0 ≤ i ≤ (γ − ξ − 5)/4}| = (γ − ξ − 1)/2.

(f) |{(ω4i, µ4i+ξ(η4i+ξ)) : 1 ≤ i ≤ (γ − ξ − 1)/4}| = (γ − ξ − 1)/2.

(iii) If d(y, z) = 3, then |a3| = (11γ − 21)/4, we have eight subsets of it:

(a)
∣∣{(ω0(γ−ξ), ωξ(γ))

}∣∣ = 2.

(b) |{(ω4i, ω4i+ξ) : 1 ≤ i ≤ (γ − ξ − 4)/4}| = (γ − ξ − 4)/4.

(c) |{(µ4i−2, µ4i−2+ξ(η4i−2+ξ)) : 1 ≤ i ≤ (γ − ξ)/4}| = (γ − ξ)/2.

(d) |{(η4i−2, µ4i−2+ξ(η4i−2+ξ)) : 1 ≤ i ≤ (γ − ξ)/4}| = (γ − ξ)/2.

(e) |{(µ4i+1, η4i+2) : 0 ≤ i ≤ (γ − ξ)/4}| = (γ + 1)/4.

(f) |{(µ4i+2, η4i+1) : 0 ≤ i ≤ (γ − ξ)/4}| = (γ + 1)/4.

(g) |{(µ4i−3(η4i−3), ω4i−3+ξ) : 1 ≤ i ≤ (γ − ξ)/4}| = (γ − ξ)/2.

(h)
∣∣{(ω4i−1, µ4i+ξ−1(η4i+ξ−1)) : 1 ≤ i ≤ (γ − ξ)/4

}∣∣ = (γ − ξ)/2.

(iv) If d(y, z) = ξ, ξ = 4, 8, . . . , γ − 7, then
∑

ξ=4,8,...,γ−7 |aξ| = 5(γ − 7)(γ + 5)/16, we
have seven subsets of it:

(a) |{(µ4i+1, µ4i+1+ξ(η4i+1+ξ)) : 0 ≤ i ≤ (γ − ξ − 3)/4}| = (γ + 1− ξ)/2.

(b) |{(µ4i+2, µ4i+2+ξ(η4i+2+ξ)) : 0 ≤ i ≤ (γ − ξ − 3)/4}| = (γ + 1− ξ)/2.

(c) |{(η4i+1, η4i+1+ξ(µ4i+1+ξ)) : 0 ≤ i ≤ (γ − ξ − 3)/4}| = (γ + 1− ξ)/2.

(d) |{(η4i+2, η4i+2+ξ(µ4i+2+ξ)) : 0 ≤ i ≤ (γ − ξ − 3)/4}| = (γ + 1− ξ)/2.

(e)
∣∣{(ω0(γ), ωξ(γ−ξ))

}∣∣ = 2.

(f) |{(ω4i−1, ω4i−1+ξ) : 1 ≤ i ≤ (γ − ξ − 3)/4}| = (γ − 3− ξ)/4.

(g) |{(ω4i, ω4i+ξ) : 1 ≤ i ≤ (γ − ξ − 3)/4}| = (γ − 3− ξ)/4.

(v) If d(y, z) = ξ, ξ = 5, 9, . . . , γ − 6, then
∑

ξ=5,9,...,γ−6 |aξ| = (γ − 7)(9γ + 37)/32, we
have seven subsets of it:

(a) |{(µ4i+1, µ4i+1+ξ(η4i+1+ξ)) : 0 ≤ i ≤ (γ − ξ − 2)/4}| = (γ + 2− ξ)/2.

(b) |{(η4i+1, η4i+1+ξ(µ4i+1+ξ)) : 0 ≤ i ≤ (γ − ξ − 2)/4}| = (γ + 2− ξ)/2.

(c) |{(µξ(ηξ), ω0)}| = 2.



2.2 The Edges Induce Ring For Hexagonal Graphs Re(C6)γ 1584

(d) |{(µγ−ξ(ηγ−ξ), ωγ)}| = 2.

(e) |{(µ4i+2(η4i+2), ω4i+2+ξ) : 0 ≤ i ≤ (γ − ξ − 6)/4}| = (γ − 2− ξ)/2.

(f) |{(ω4i, µ4i+ξ(η4i+ξ)) : 0 ≤ i ≤ (γ − ξ − 2)/4}| = (γ − 2− ξ)/2.

(g) |{(ω4i−1, ω4i−1+ξ) : 1 ≤ i ≤ (γ − ξ − 2)/4}| = (γ − 2− ξ)/4.

(vi) If d(y, z) = ξ, ξ = 6, 10, ..., γ−5, then
∑

ξ=7,11,...,γ−5 |aξ| = (γ − 7)(γ + 1)/4, we have
six subsets of it: Similar to (ii(a− f)), put ξ = 6, 10, ..., γ − 5.

(vii) If d(y, z) = ξ, ξ = 7, 11, ..., γ − 8, then
∑

ξ=7,11,...,γ−8 |aξ| = (γ − 11)(9γ + 17)/32, we
have six subsets of it: Similar to (iii(a− d), (g) and (h)), put ξ = 7, 11, ..., γ − 9.

(viii) If d(y, z) = γ−4, |aγ−4| = 10 then we have six subsets of it: Similar to (iii(a−d), (f)
and (h)), put ξ = γ − 4.

(ix) If d(y, z) = γ− 3, |aγ−3| = 10, there are five subsets of it: Similar to (iv(a− e)), put
ξ = γ − 3.

(x) If d(y, z) = γ − 2, |aγ−2| = 8, the four subsets of it: Similar to (iv(a − d)), put
ξ = γ − 2.

(xi) If d(y, z) = γ − 1, |aγ−1| = 4 then two subsets of it: Similar to (ii(a − b)), put
ξ = γ − 1.

(xii) If d(µ, η) = γ then |aγ | = 1, we have: |{(ω0, ωγ)}| = 1.

Corollary 1. For m ≥ 3, γ = 4m− 1, then:

(i) Sc(Ce(C6)γ) = (7γ3 + 15γ2 + 29γ + 21)/4.

(ii) Sc∗(Ce(C6)γ) = (49γ3 + 63γ2 + 185γ + 147)/24.

2.2. The Edges Induce Ring For Hexagonal Graphs Re(C6)γ

This graph is said to be a hexagonal bracelet graphs Re(C6)γ which is a connected
graph consisting of m ≥ 3, hexagonal rings such that two hexagons are joined by exactly
one added edge as shown in Fig. 2.

Figure 2: Edges Induce Ring for Hexagonal graphs Re(C6)γ .



2.2 The Edges Induce Ring For Hexagonal Graphs Re(C6)γ 1585

Table 2: Degree Matrix of Re(C6)γ For 1 ≤ i, j ≤ 2γ − 1, and 1 ≤ r, s ≤ m− 1, r ̸= s, and i, j ̸= r, s.

Theorem 2. For m ≥ 3, γ = m, then:

(i) Sc(Re(C6);x) = 34γx+ 48γx2 + 50γx3 +

[ ∑2γ−4
ξ=4,8 44γx

ξ, γ is an even∑2γ−2
ξ=4,8 44γx

ξ, γ is an odd

+
∑2γ−1

ξ=5,7 42γx
ξ

[ ∑2γ−2
ξ=4,8 40γx

ξ, γ is an even∑2γ−2
ξ=4,8 40γx

ξ, γ is an odd
+

[
22γx2γ , γ is an even,
20γx2γ , γ is an odd.

(ii) Sc∗(Re(C6);x) = 41γx+ 56γx2 + 57γx3 +

[ ∑2γ−4
ξ=4,8 50γx

ξ, γ is an even∑2γ−2
ξ=4,8 50γx

ξ, γ is an odd

+
∑2γ−1

ξ=5,7 49γx
ξ +

[ ∑2γ−2
ξ=4,8 48γx

ξ, γ is an even∑2γ−2
ξ=4,8 48γx

ξ, γ is an odd
+

[
25γx2γ , γ is an even,
24γx2γ , γ is an odd.

Proof. For any two vertices y, z ∈ V (Re(C6)) there is d(y, z) = ξ, 1 ≤ ξ ≤ 2γ.

And clearly
∑γ

ξ=1 d(Re(C6)γ , ξ) =

[
3γ(6γ − 1), γ is an even,
(1/2)γ(36γ + 11), γ is an odd.

Proof will be divided into eight cases:

(i) If d(y, z) = 1, then |a1| = 7γ, there are three subsets of it:

(a) |{(ωi, µi), (ωi, ηi) : 1 ≤ i ≤ 2γ}| = 4γ.

(b) |{(µi, µi+1), (ηi, ηi+1) : i = 1, 3, 5, . . . , 2γ − 1}| = 2γ.

(c) |{(ωi, ωi+1) : i = 2, 4, 6, . . . , 2γ, (ω2γ+1 ≡ ω1)}| = γ.

(ii) If d(y, z) = 2, then |a2|= 10γ, there are three subsets of it:

(a) |{(µi, ηi) : 1 ≤ i ≤ 2γ}| = 2γ.

(b) |{(ωi, µi+1), (ωi, ηi+1) : i = 1, 3, 5, . . . , 2γ − 1}| = 2γ.

(c) |{(ωi, µi−1), (ωi, ηi−1), (µi, ωi+1), (ηi, ωi+1), (ωi, µi+1), (ωi, ηi+1) :
i = 2, 4, 6, . . . , 2γ, (ω2γ+1 ≡ ω1), (µ2γ+1 ≡ u1), (η2γ+1 ≡ η1)}| = 6γ.

(iii) If d(y, z) = 3, then |a3| = 11γ, there are three subsets:

(a) |{ (ωi, ωi+1) : i = 1, 3, 5, . . . , 2γ − 1 }| = γ.



2.2 The Edges Induce Ring For Hexagonal Graphs Re(C6)γ 1586

(b) |{(µi, ωi+2), (ηi, ωi+2), (µi, ηi+1), (ηi, µi+1) :
i = 1, 3, 5, . . . , 2γ − 1, (ω2γ+1 ≡ ω1)}| = 4γ.

(c) |{ (µi, µi+1), (µi, ηi+1), (ηi, µi+1), (ηi, ηi+1), (ωi, µi+2), (ωi, ηi+2) :
i = 2, 4, 6, . . . , 2γ, (µ2γ+1 ≡ µ1), (η2γ+1 ≡ η1) , (µ2γ+2 ≡ µ2), (η2γ+2 ≡ η2)}| =
6γ.

(iv) If d(y, z) = ξ, ξ = 4, 8, . . . , 2γ − 4, then we have: A: If γ is an even number, then
we have two subsets of it

(a) |{ (ωi, ωi+(ξ/2)), (µi, µi+(ξ/2)), (µi, ηi+(ξ/2)), (ηi, µi+(ξ/2)), (ηi, ηi+(ξ/2)) :
i = 1, 3, 5, . . . , 2γ − 1 }| = 5γ.
If i+ ξ

2 > 2γ, then (ωi+(ξ/2) ≡ ωi+(ξ/2)−2γ), (µi+(ξ/2) ≡ µi+(ξ/2)−2γ), (ηi+(ξ/2) ≡
ηi+(ξ/2)−2γ).

(b) |{ (ωi, ωi+(ξ/2)), (µi, ηi+(ξ/2)), (µi, µi+(ξ/2)), (ηi, µi+(ξ/2)), (ηi, ηi+(ξ/2)) :
i = 2, 4, 6, . . . , 2γ }| = 5γ.
If i+(ξ/2) > 2γ, then (ωi+(ξ/2) ≡ ωi+(ξ/2)−2γ), (µi+(ξ/2) ≡ µi+(ξ/2)−2γ), (ηi+(ξ/2) ≡
ηi+(ξ/2)−2γ).

Hence
∑2γ−4

ξ=4,8 |aξ| = 10γ

B: If γ is an odd number, we get
∑2γ−2

ξ=4,8 |aξ| = 10γ, note we add the distance
of ξ = 2γ − 2.

(v) If d(y, z) = ξ, ξ = 5, 9, 13, . . . , 2γ − 3, then we have: A: If γ is an even number,
then we have four subsets of it

(a) |{ (ωi, µi+(ξ−1)/2), (ωi, ηi+(ξ−1)/2) : i = 1, 3, 5, . . . , 2γ − 1 }| = 2γ.
If i+(ξ−1/2) > 2γ, then (µi+(ξ−1/2) ≡ µi+(ξ−1)/2−2γ), (ηi+(ξ−1)/2 ≡ ηi+(ξ−1)/2−2γ).

(b) |{ (µi, µi+(ξ−1)/2+1), (µi, ηi+(ξ−1)/2+1), (ηi, µi+(ξ−1)/2+1), (ηi, ηi+(ξ−1)/2+1) :
i = 1, 3, 5, . . . , 2γ − 1 }| = 4γ.
If i+(ξ− 1)/2+ 1 > 2γ, then (µi+(ξ−1)/2+1 ≡ µi+(ξ−1)/2+1−2γ), (ηi+(ξ−1)/2+1 ≡
ηi+(ξ−1)/2+1−2γ).

(c) |{ (µi, ωi+(ξ−1)/2), (ηi, ωi+(ξ−1)/2) :
i = 2, 4, 6, . . . , 2γ }| = 2γ.
If i+ (ξ − 1)/2 > 2γ, then (ωi+(ξ−1)/2 ≡ ωi+(ξ−1)/2−2γ).

(d) |{ (ωi, ωi+(ξ−1)/2+1) : i = 2, 4, 6, . . . , 2γ }| = γ. If i + (ξ − 1)/2 + 1 > 2γ, then
(ωi+(ξ−1)/2+1 ≡ ωi+(ξ−1)/2+1−2γ).

Hence
∑2γ−3

ξ=5,9 |aξ| = 9γ, B: If γ is an odd number, we get
∑2γ−1

ξ=5,9 |aξ| = 9γ, note
we add the distance of ξ = 2γ − 1. If d(y, z) = ξ, ξ = 6, 10, 14, . . . , 2γ − 2, then∑2γ−2

ξ=6,10 |aξ| = 8γ, we have two subsets of it:

(a) |{ (ωi, µi+(ξ/2)), (ωi, ηi+(ξ/2)), (µi, ωi+(ξ/2)), (ηi, ωi+(ξ/2)) :
i = 1, 3, 5, . . . , 2γ − 1 }| = 4γ.



2.2 The Edges Induce Ring For Hexagonal Graphs Re(C6)γ 1587

If i+(ξ/2) > 2γ, then (µi+(ξ/2) ≡ µi+(ξ/2)−2γ), (ηi+(ξ/2) ≡ ηi+(ξ/2)−2γ), (ωi+(ξ/2) ≡
ωi+(ξ/2)−2γ).

(b) |{ (µi, ωi+(ξ/2)), (ηi, ωi+(ξ/2)), (ωi, µi+(ξ/2)), (ωi, ηi+(ξ/2)) :
i = 2, 4, 6, . . . , 2γ }| = 4γ.
If i+(ξ/2) > 2γ, then (ωi+(ξ/2) ≡ ωi+(ξ/2)−2γ), (µi+(ξ/2) ≡ µi+(ξ/2)−2γ), (ηi+(ξ/2) ≡
ηi+(ξ/2)−2γ).

(vi) If d(y, z) = ξ, ξ = 7, 11, 15, . . . , 2γ − 1, then
∑2γ−1

ξ=7,11 |aξ| = 9γ,

we have three subsets of it:

(a) |{ (ωi, ωi+(ξ−1)/2), (µi, ωi+(ξ+1)/2), (ηi, ωi+(ξ+1)/2) :
i = 1, 3, 5, . . . , 2γ − 1 }| = 3γ.
If i+(ξ−1)/2, i+(ξ+1)/2 > 2γ, then (ωi+(ξ−1)/2 ≡ ωi+(ξ−1)/2−2γ), (ωi+(ξ+1)/2 ≡
ωi+(ξ+1)/2−2γ).

(b) |{ (µi, µi+(ξ−1)/2), (µi, ηi+(ξ−1)/2), (ηi, µi+(ξ−1)/2), (ηi, ηi+(ξ−1)/2) :
i = 2, 4, 6, . . . , 2γ }| = 4γ.
If i+(ξ−1)/2 > 2γ, then (µi+(ξ−1)/2 ≡ µi+(ξ−1)/2−2γ), (ηi+(ξ−1)/2 ≡ ηi+(ξ−1)/2−2γ).

(c) |{ (ωi, µi+(ξ+1)/2), (ωi, ηi+(ξ+1)/2) : i = 2, 4, 6, . . . , 2γ }| = 2γ.
If i+(ξ+1)/2 > 2γ, then (µi+(ξ+1)/2 ≡ µi+(ξ+1)/2−2γ), (ηi+(ξ+1)/2 ≡ ηi+(ξ+1)/2−2γ).

(vii) If d(y, z) = 2γ, then we have:

A: If γ is an even number, then we have two subsets of it

(a) |{ (ωi, ωi+γ), (µi, µi+γ), (µi, ηi+γ), (µi, ηi+γ), (ηi, ηi+γ) :
i = 1, 3, 5, . . . , γ − 1 }| = 5γ/2.

(b) |{ (µi, µi+γ), (µi, ηi+γ), (ηi, µi+γ), (ηi, ηi+γ), (ωi, ωi+γ) :
i = 2, 4, 6, . . . , γ }| = 5γ/2.

Hence |a2γ | = 5γ.

B: If n is an odd number, then we have: |{ (ωi, µi+γ), (ωi, ηi+γ), (µi, ωi+γ), (ηi, ωi+γ) :
i = 1, 3, 5, . . . , 2γ − 1 }| = 4γ.
If i+ γ > 2γ, then (µi+γ ≡ µi−γ), (ηi+γ ≡ ηi−γ), (ωi+γ ≡ ωi−γ). Hence |a2γ | = 4γ.

Corollary 2. For m ≥ 3, γ = m, we have:

Sc(Re(C6)γ) =

[
4γ(21γ2 + 8), γ is an even,
6γ(14γ2 + 5), γ is an odd.

]
.

Sc∗(Re(C6)γ) =

[
2γ(49γ2 + 16), γ is an even,
γ(98γ2 + 31), γ is an odd.

]
.



1588

3. Examples:

To clarify the previous results, some examples were taken, which were verified pro-
grammatically using the Mathematica program.

Table 3: The Edges Induce Chain for Hexagonal Graphs Ce(C6)γ , γ = 15, 19.



1589

Table 4: The Edges Induce Chain for Hexagonal Graphs Ce(C6)γ , γ = 15, 19.



1590

4. Conclusion

In this paper, we were able to obtain general formulas for the Schultz and modified
Schultz polynomials with their indices for both types of hexagonal rings joining to each
other by an edge or bridge, and we compared the results using Mathematica program for
many examples, and the results were identical

Acknowledgements

This paper was supported by Mosul University-College of Computer Sciences and
Mathematics and the general directorate of Education in Nineveh governorate/ Iraq.

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