Degree Based Topological Indices On
Titania Nanotubes
October 5, 2021
K. Vijayalakshmi1 and N. Parvathi2
1Dept.of Mathematics, SRM Institute of Science and Technology,
Kattankulathur-603 203,Tamil Nadu, INDIA
suraj1719cyr@gmail.com
2Dept.of Mathematics, SRM Institute of Science and Technology,
Kattankulathur-603 203,Tamil Nadu, INDIA
parvathn@srmist.edu.in
Abstract
The topological index is a numerical quantity that can be characterized as a whole structure of a graph and it correlates physiochemical properties of the corresponding chemical compound. Moreover, these topological indices are related to graph theory mainly with
the help of vertex degrees and it has enormous application in several
fields. In this paper, topological indices are calculated for Titania
nanotubes and Zagreb index, Randic Index, Sum Connectivity Index,
Harmonic Index are estimated. Titania nanotubes (T iNT) combine
nanotubular structure with the chemical and electronic properties of
titania.
Key Words and Phrases: Molecular structure descriptor, Topological Indices, Randic Index, Sum-Connectivity Index, Titania nanotube.
1 Introduction
Graph theory is an important branch of mathematics and it is one of the
fastest expanding and blooming fields in the current era. At the same time
1
chemical graph theory is a part of mathematical chemistry that applies graph
theory and goes hand in hand with the mathematical model of chemical phenomena.[2].
Many of the current panoramas of chemical theory are basically investigated
from the graph theoretical approach in recent years. Though they belong to
a different category, when they combine, give us a colorful result in our daytoday-life. Chemical graphs were first introduced in the eighteenth century
[1]. Topological descriptors came into existence from the hydrogen suppressed
molecular graph in which atoms are represented by vertices and bonds are
represented by edges. The connection between the atoms and bonds is represented mathematically by a graph invariant and the distance between them is
calculated and it is named as topological indices. Graph theory when amalgamated with Chemistry produces fruitful results and its applications are not
restricted to a single field, its vast implementation makes the combination
more powerful in the real life.
A graph G consists of two sets namely vertex set V (G) and edge set E(G).
If two vertices are connected by an edge, we say it is adjacent. A graph G is
simple if it has multiple edges or no loops. The degree of a vertex v is the
number of vertices adjacent with vertex v, it is usually denoted by deg(v).The
distance between any two vertices u and v is denoted by d(u,v).The diameter
of a graph is the maximum distance between any two vertices of G.
Topological indices are nothing but real numbers which forms a bridge between graphs and chemical compounds and it is mainly useful in Quantity
Structure-Activity Relationship (QSAR) and Quantity Structure-Property
Relationship(QSPR).It is very interesting and surprising to know the fact
that without using lab we can predict approximately some properties of
chemical structures using the topological indices, take for instance boiling
point, viscosity, radius of gyration can be calculated approximately using
the indices.[1].In the same way as topological indices, polynomials such as
Hosoya polynomial and M-polynomial plays a vital role in the calculation
of degree- based topological indices.[1].(4,5,6,12] Wiener was the first person
to define the topological indices while examining the boiling point of some
kinds of paraffin.
Titania nanotube (T iNT) integrates the nanotubular structure with the optical and chemical properties of titania nanostructure.[4]. Titanium dioxide
(T iO2) has been used since the starting of the twentieth century and mainly
it is used as a commercial unit in sensors, solar cells, sun-blockers and in
pharmaceutical science mainly in drug delivery.[4][16-22]. The major aim of
2
(T iO2) is to produce photogenerated electron-hole pairs under lighting irradiation, which aids in breaking down the components of water into oxygen
and hydrogen, thereby solving the energy issue in the future as the most
potential fuel. Fujishima and colleagues were the first to demonstrate photocatalytic water splitting on a (TiO2) electrode under UV light [4][23-25]],
and titanium dioxide has since become one of the most researched components in material science. t has the broadest range of functional qualities
of all transition metal oxides, including chemical inertness, corrosion resistance, and stability, with an emphasis on improving biocompatibility [4][26]
and electrical and optical properties [4] (1). Since Iijima’s discovery of carbon nanotubes in 1991[4][27], demonstrated a unique combination of shape
and functionality, where properties can be influenced directly by geometry,
massive efforts have been made in the field of nanotechnology, primarily in
the chemical, physical, and biomedical, material science. Although carbon
remains the most studied nanomaterial, another class of nanotubular materials, mainly based on transition metal oxides, has piqued researchers curiosity
in the last two decades. Assefpour-Dezfuly[4] [28]was the first to try to make
anodized titania nanotubes who employed an alkaline peroxide treatment
followed by electrochemical anodization with chromic acid in an electrolyte
Since Zwilling et al. published the first self-organized nanotube layers on
Ti substrate by electrochemical anodization in chromic acid electrolytes containing fluorine ions in 1999, the field has rapidly expanded[4] [14].
In the last 20 years, the synthesis of titania nanotube arrays has contributed
to a diversification of prospective applications in areas such as anti-corrosion,
self-cleaning coatings, paints to sensors [31-33], dye-sensitized and solid state
bulk heterojunction solar cells [34-36] photocatalysis [37,38] electrocatalysis
and water photoelectrolysis[39]. These nanotubes outperform in biomedical applications such as biomedical coatings with improved osseointegration,
medication delivery systems, and advanced tissue engineering as biocompatible materials. [29,30,31,32].
Hosoya index is the first recognized topological index in the chemical graph
theory and it is popularly known as the topological index. To estimate the
boiling point of certain alkane isomers Harold Wiener in 1947 used the topological indices hence the name Wiener Index. [42]. Several topological indices
came into existence after his research and the other topological indices include Balban Index, Randic Index, Estada Index, Zagreb Index, Gutman
Index etc.
Randic Index is a well-known topological index invented in 1976 by Milan
3
Randic[43].
Definition 1.1. Let G be a graph. Then the Wiener Index of G is defined
as W(G) = 1
2
P
(u,v)∈E(G)
dG(u, v) here (u, v) is any ordered pair of vertices
in G and dG(a, b) is a − b geodesic (which is the shortest path) [42].
Definition 1.2. Let G be a graph, the first and second Zagreb index is
defined as
M1(G) = X
uv∈E(G)
[deg(u) + deg(v)]
M2(G) = X
uv∈E(G)
[deg(u) × deg(v)]
The generalized Zagreb index [45] of a connected graph G based on degree of
vertices of G for all p, q ∈ N is defined as Mp,q(G) = P
uv∈E(G)
(d
p
ud
q
v + d
p
vd
q
u
).
Definition 1.3. The very first and oldest degree based topological index is
Randic index denoted by Rα(G) introduced by Milan Randic in 1975 [43].
Let G be a graph. Then the Randic Index of G is defined as
Rα(G) = X
uv∈E(G)
1
√
SuSv
Definition 1.4. One of the well-known connectivity topological index is
atom-bond connectivity (ABC) index introduced by Estrada et. al. in [47].
For a graph G, ABC Index is defined as
ABC(G) = X
uv∈E(G)
r
Su + Sv − 2
SuSv
Definition 1.5. Another topological index, Geometric-Arithmetic index is
introduced by Vukicevic and Furtula [44].
Consider a graph, then its Geometric index popularly known as GA index is
defined as
GA(G) = X
uv∈E(G)
2
√
SuSv
(Su + Sv)
4
Definition 1.6. The Sum Connectivity Index SCI(G) was invented by Zhou
and Trinajstic [46].
The Sum Connectivity Index SCI(G) is a topological index of a molecular
graph G is defined as
SCI(G) = X
uv∈E(G)
1
√
Su + Sv
In this article edge versions of important degree based topological indices are
computed.
The figure shows the Titania structure of the graph T iO2[m, n].
Figure 1: The graph of T iO2[m, n] Nanotubes with m = 6 and n = 4
2 Main Results
Though Carbon nanotube, so far explored in a massive way another class of
nanomaterial which is based on transition metal oxide attracted considerable
interests in the last two decades. The carbon nanotube, when combined with
titania nanotube, helps in the enhancement of electrical properties and it is a
well-known semiconductor that has mind blowing technological applications
in various fields.
Titania nanotubes are applied in material science and it has more technological applications. Moreover, titanium nanotubes form an interesting class
in the nanomaterial and experimental research shows that it is not only restricted to a single field but it has endless applications. Titania nanotubes
have been synthesized and investigated as potential technology materials
over the last two decades utilizing a variety of approaches. Since the growth
5
mechanism for T iO2 nanotubes is still not well defined, their comprehensive
theoretical studies attract enhanced attention. T iO2 sheets with a thickness
of a few atomic layers were found to be remarkably stable [40][41].
The structure of Titania nanotube T iO2[m, n] is represented in the above figure and we represent two variables in the structure to define the nanotubes
where m denotes the number of octagons in a row and n denotes the number
of octagons in a column of the titania nanotube.
The partition of the vertex set and edge set in Titania nanotube T iO2 are
discussed below. Let us consider the simple connected graph with the vertex
set V (G) and edge set E(G). Here minimum degree is represented by δ(G)
and maximum degree is represented by ∆(G) and the values are restricted
between i, j and k. Also Su is defined as follows:
Su =
X
v∈NG(u)
d(v) where NG(u) = {v ∈ V (G)|uv ∈ E(G)}
δ(G) ≤ k ≤ ∆(G)
2δ(G) ≤ i ≤ 2∆(G)
δ(G)
2 ≤ j ≤ ∆(G)
2
Vk = {v ∈ V (G)|d(u) = 2}
Ei = {e = uv ∈ E(G)|d(u) + d(v) = i}
In the molecular graph of T iO2 nanotube, we observe that 2 ≤ d(G) ≤ 5,
the vertex partition of the structure is given below.
V2 = {v ∈ V (G)|d(u) = 2}
V3 = {v ∈ V (G)|d(u) = 3}
V4 = {v ∈ V (G)|d(u) = 4}
V5 = {v ∈ V (G)|d(u) = 5}
It is easy to see that |V2| = 2mn + 4n.
|V3| = 2mn
|V4| = 2n
|V5| = 2mn
6
and hence we have |V (T iO2)| = 6n(m + 1).
The edge partition of the titanium nanotube is given as follows.
E6 = {e = uv ∈ E(G)|d(u) = 2 and d(v) = 4}
E7 = {e = uv ∈ E(G)|d(u) = 2 and d(v) = 5} ∪ {e = uv ∈ E(G)|d(u) = 3 and d(v) = 4}
Ei = {e = uv ∈ E(G)|d(u) = 3 and d(v) = 5}
The vertex partition Vk and edge partition Ei are collectively exhaustive that
is
∆(
[
G)
k=δ(G)
Vk = V (G)
2∆(
[
G)−2
i=3δ(G)
Ek = E(G)
Table 1: The edge partitions based on degree of end vertices
Edge partition E6 E7 Es
Cardinality 6n 4mn + 4n 6mn − 2n
Table 2: The edge partitions based on degree sum of neighbors of end vertices
(Su, Sv) where uv ∈
E(T iO2[m, n])
(10,5) (7,5) (7,9) (8,9) (10,9) (11,9)
Number of edges 2 2 2n 4n 2n 6m
Table 3: The edge partitions based on degree sum of neighbor of end vertices
(Su, Sv) where uv ∈
E(T iO2[m, n])
(13,9) (7,13) (10,13) (11,13) (13,13)
Number of edges 3n 2n 4mn + 2n 2mn − 2n 6mn − 4n
7
Theorem 2.1. The generalized Zagreb index GZ of the Titania nanotube T iO2[m, n] is given by
Mp,q(T iO2) =6n(d
2
ud
4
v + d
4
vd
2
u
) + (4mn + 4n)(d
2
ud
5
v + d
5
vd
2
u
)
+ (6mn − 2n)(d
3
ud
5
v + d
5
vd
3
u
)
Proof:
To compute the generalized Zagreb index of the Titania nanotube T iO2[m, n],
we need an edge partition of the Titania nanotube T iO2[m, n] based on the
degree sum of neighbors of end vertices of each edge. The edge partitions are
represented with their corresponding cardinalities in table 2.
Now with the help of the formula of Zagreb Index, we get the require result.
Mp,q(T iO2) = X
uv∈E(G)
(d
p
ud
q
v + d
p
vd
q
u
)
=
X
uv∈E6(G)
(d
p
ud
q
v + d
p
vd
q
u
) + X
uv∈E7(G)
(d
p
ud
q
v + d
p
vd
q
u
)
X
uv∈E8(G)
(d
p
ud
q
v + d
p
vd
q
u
)
=|E6(G)(d
2
ud
4
v + d
4
vd
2
u
) + |E7(G)(d
2
ud
5
v + d
5
vd
2
u
)|E8(G)(d
3
ud
5
v + d
5
vd
3
u
)
=6n(d
2
ud
4
v + d
4
vd
2
u
) + (4mn + 4n)(d
2
ud
5
v + d
5
vd
2
u
)(6mn − 2n)(d
3
ud
5
v + d
5
vd
3
u
)
Theorem 2.2. The Sum - Connectivity index of titania nanotube
T iO2[m, n] is given by
2
√
15
+
2
√
12
+
n
2
+
4n
√
17
+
2n
√
19
+
6m
√
20
+
3m
√
22
+
n
√
5
+
4mn + 2n
√
23
+
mn − n
√
6
+
6mn − 4n
√
26
Proof:
To compute the Sum -Connectivity index of the Titania nanotube T iO2[m, n],
we need an edge partition of the Titania nanotube T iO2[m, n] based on the
degree sum of neighbors of end vertices of each edge. The edge partitions are
represented with their corresponding cardinalities in tables 2 and 2.
Now with the help of the formula of Sum-Connectivity Index, we get the
8
require result.
SCI(G) = X
uv∈E(G)
1
√
Su + Sv
=
2
√
10 + 5
+
2
√
7 + 5
+
2n
√
7 + 9
+
4n
√
8 + 9
+
2n
√
10 + 9
+
6m
√
11 + 9
+
3m
√
13 + 9
+
2n
√
7 + 13
+
4mn + 2n
√
10 + 13
+
2mn − 2n
√
11 + 13
+
6mn − 4n
√
13 + 13
=
2
√
15
+
2
√
12
+
n
2
+
4n
√
17
+
2n
√
19
+
6m
√
20
+
3m
√
22
+
n
√
5
+
4mn + 2n
√
23
+
mn − n
√
6
+
6mn − 4n
√
26
Theorem 2.3. The Randic index of titania nanotube T iO2[m, n] is
given by
√
2
5
+
2
√
35
+
2n
3
√
7
+
2n
3
√
2
+
2n
3
√
10
+
2m
√
11
+
m
√
13
+
2n
√
91
+
4mn + 2n
√
130
+
2mn − 2n
√
143
+
6mn − 4n
13
Proof:
To compute the Randic index of the Titania nanotube T iO2[m, n], we need an
edge partition of the Titania nanotube T iO2[m, n] based on the degree sum
of neighbors of end vertices of each edge. The edge partitions are represented
with their corresponding cardinalities in table 2 and 2.
Now with the help of the formula of Randic Index,we get the require result.
Rα =
X
u,v∈E(G)
1
√
Su.Sv
=
2
√
10 × 5
+
2
√
7 × 5
+
2n
√
7 × 9
+
4n
√
8 × 9
+
2n
√
10 × 9
+
6n
√
11 × 9
+
3n
√
13 × 9
+
2n
√
7 × 13
+
4mn + 2n
√
10 × 13
+
2mn − 2n
√
11 × 13
+
6mn − 4n
√
13 × 13
=
√
2
5
+
2
√
35
+
2n
3
√
7
+
2n
3
√
2
+
2n
3
√
10
+
2m
√
11
+
m
√
13
+
2n
√
91
+
4mn + 2n
√
130
+
2mn − 2n
√
143
+
6mn − 4n
13
9
Theorem 2.4. The Harmonic index of titania nanotube T iO2[m, n] is
given by
3
5
+
n
4
+
8n
17
+
4n
19
+
3m
5
+
3m
11
+
n
5
+
4(2mn + n)
23
+
mn − n
6
+
2(3mn − 2n)
13
Proof:
To compute the Harmonic index of the Titania nanotube T iO2[m, n], we need
an edge partition of the Titania nanotube T iO2[m, n] based on degree sum of
neighbors of end vertices of each edge. The edge partitions are represented
with their corresponding cardinalities in tables 2 and 2.
Now with the help of the formula of Harmonic Index, we obtain the required
result as follows.
H(G) = X
u,v∈E(G)
1
√
SuSv
=2 
2
10 + 5
+ 2 
2
7 + 5
+ 2n

2
7 + 9
+ 4n

2
8 + 9
+ 2n

2
10 + 9
+ 6m

2
11 + 9
+ 3m

2
13 + 9
+ 2m

2
7 + 13
+ (4mn + 2m)

2
10 + 13
+ (2mn − 2n)

2
11 + 13
+ (6mn − 4m)

2
13 + 13
=
3
5
+
n
4
+
8n
17
+
4n
19
+
3m
5
+
3m
11
+
n
5
+
4(2mn + n)
23
+
mn − n
6
+
2(3mn − 2n)
13
Theorem 2.5. The Inverse Sum index of titania nanotube T iO2[m, n]
is given by
2

50
15
+ 2 
35
12
+ 2n

63
16
+ 4n

72
17
+ 2n

90
19
+ 6m

99
20
+ 3m

117
22 
+ 2n

91
20
+ (4mn + 2n)

130
23 
+ (2mn − 2m)

143
24 
+ (6mn − 4n)

169
26 
Proof:
To compute the Inverse sum index of the Titania nanotube T iO2[m, n], we
need an edge partition of the Titania nanotube T iO2[m, n] based on degree
10
sum of neighbors of end vertices of each edge. The edge partitions are represented with their corresponding cardinalities in tables 2 and 2.
Now with the help of the formula of Inverse sum Index, we get the require
result.
Inverse Sum index = X
u,v∈E(G)
2
√
SuSv
(Su + Sv)
= 2 
10 × 5
10 + 5
+ 2 
7 × 5
7 + 5
+ 2n

7 × 9
7 + 9
+ 4n

8 × 9
8 + 9
+ 2n

10 × 9
10 + 9
+ 6m

11 × 9
11 + 9
+ 3m

13 × 9
13 + 9
= 2 
50
15
+ 2 
35
12
+ 2n

63
16
+ 4n

72
17
+ 2n

90
19
+ 6m

99
20
+ 3m

117
22 
+ 2n

91
20
+ (4mn + 2n)

130
23 
+ (2mn − 2m)

143
24 
+ (6mn − 4n)

169
26 
3 Conclusion
In this paper, degree-based topological indices such as Sum-Connectivity
Index, Randic index, Harmonic Index, inverse Sum Index of titania nanotube are discussed. Though topological indices connect various fields such
as Biology, Pharma Industry, Informatics, material science, the most critical usage in recent years implies to Quantity Structure-Activity Relationship
and Quantity Structure-Property Relationship. The steady progress in the
application demonstrates that nanotubes continue to play a very big role
and shines even better in multiple areas. Moreover, the progress brings great
change in the upcoming years.
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