Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1631–1638
© Research India Publications
http://www.ripublication.com/gjpam.htm
Computation of Topological Indices of Triglyceride
M. R. Rajesh Kanna
Post Graduate Department of Mathematics,
Maharani’s Science College for Women,
J. L. B. Road, Mysore - 570 005, Karnataka, India.
D. Mamta
Department of Mathematics,
The National Institute of Engineering,
Mysuru-570008, Karnataka, India.
R. S. Indumathi
Maharaja institute of technology,
Belawadi, Srirangapatna taluk,
Mandya-571438, Karnataka, India.
Abstract
In this paper, we compute ABC index, Randic connectivity index, Sum connectivity index, GA index, Harmonic index, General Randic index of Triglyceride.
AMS subject classification:
Keywords: ABC index, Randic connectivity index, Sum connectivity index, GA
index, Harmonic index, General Randic index and Triglyceride.
1.
Introduction
A triglyceride is an ester formed by three fatty acids to a single glycerol molecule.
Triglycerides are mainly circulated in the body to provide cells for energy, it is the main
constituents of body fat in humans and animals, as well as vegetable fat. After the body
consumes a meal with fats, the unused portions are transported to fat cells and stored as
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M. R. Rajesh kanna, D. Mamta and R. S. Indumathi
triglycerides. Triglycerides are fats, and they are used to produce the energy currency of
a cell called adenosine triphosphate (ATP). When energy is required by cells, the fat is
removed and sent to cells via cholesterol transport. Triglycerides are also used in the cell
membrane to control permeability of the cell. It is the most common form of fat in foods
and in the body and it is needed for good health and they are a rich energy source, as
they provide more than twice as much energy for the body as carbohydrates and protein.
However, high triglyceride levels increase the risk of heart disease, according to the
American Heart Association.
Triglyceride fatty acid tails can be saturated or unsaturated. Saturated fatty acid tails
are all single bond carbons. This means that for each carbon, there are two hydrogens
and two carbons attached. There are no double bonds in a saturated molecule. Unsaturated fatty acids have at least one double bond. Single bond molecules are called
monosaturated. A molecule that contains more double bonds is called polyunsaturated.
Here we consider carbon atoms of saturated fatty acid.
Topological indices are the molecular descriptors that describe the structures of chemical compounds and they help us to predict certain physico-chemical properties like
boiling point, enthalpy of vaporization, stability, etc. Topological indices are introduced
to test the medicinal properties of new drugs which is widely welcomed in developing
areas. All molecular graphs considered in this paper are finite, connected, loopless, and
without multiple edges. Let G = (V , E) be a graph with vertex set V and edge set E .
The degree of a vertex u ∈ E(G) is denoted by du and is the number of vertices that are
adjacent to u. The edge connecting the vertices u and v is denoted by uv. Recently Sridhar and his co-authors [13] determined ABC index, ABC4 index, Randic connectivity
index, Sum connectivity index, GA index and GA5 index of Graphene.
The ABC index is one of the degree based molecular descripter, which was introduced
by Estrada et al. [8] in late 1990’s and it can be used for modelling thermodynamic
properties of organic chemical compounds, it is also used as a tool for explaining the
stability of branched alkanes [9]. For further results on ABC index see the papers
[3, 4, 5, 6, 10, 11, 17, 18, 19] and the references cited there in.
Definition 1.1. Let G=(V , E) be a molecular graph and du is the degree of the vertex
u, then ABC index of G is defined as,
du + dv − 2
.
ABC(G) =
du dv
uv∈E(G)
The first and oldest degree based topological index is Randic index [12] denoted by
χ (G) and was introduced by Milan Randic in 1975. It provides a quantitative assessment
of branching of molecules.
Definition 1.2. Randic index was defined as follows
χ(G) =
uv∈E(G)
√
1
.
du dv
Computation of Topological Indices of Triglyceride
1633
Sum connectivity index belongs to a family of Randic like indices and it was introduced by Zhou and Trinajstic [21]. Further studies on Sum connectivity index can be
found in [22, 23].
Definition 1.3. For a simple connected graph G, its sum connectivity index S(G) is
defined as,
1
.
S(G) =
√
d
+
d
u
v
uv∈E(G)
The GA index of G was introduced by D. Vukicevic et al. [14]. Further studies on
GA index can be found in [2, 7, 16].
Definition 1.4. Let G be a graph and e = uv be an edge of G then,
√
2 du dv
.
GA(G) =
du + dv
e=uv∈E(G)
The Harmonic index was introduced by Zhong [20]. It has been found that the
harmonic index correlates well with the Randic index and with the π-electronic energy
of benzenoid hydrocarbons.
Definition 1.5. Let G = (V , E) be a graph and du be the degree of a vertex u then
Harmonic index is defined as
2
.
H (G) =
du + dv
e=uv∈E(G)
Further studies on H (G) can be found in [15, 22].
There are several degree based indices introduced to test the properties of compounds
and drugs, which have been widely used in chemical and pharmacy engineering. Bollobas
and Erdos [1] introduced the General Randic index.
Definition 1.6. Let G = (V , E) be a graph and du be the degree of a vertex u then the
General Randic index is defined as
Rk (G) =
(du dv )k .
e=uv∈E(G)
where k is a real number.
2.
Main results
Consider a two-dimensional structure of triglyceride as shown in the figure-1. Let |Ei,j |
denotes the number of edges connecting the vertices of degrees di and dj . The figure-1
contains the edges of the type E2,2 , E2,3 , E1,3 and E2,1 which are colored in green, red,
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M. R. Rajesh kanna, D. Mamta and R. S. Indumathi
black and blue respectively. ∴ Triglyceride contains |E2,2 | = 41, |E2,3 | = 9, |E1,3 | = 3
and |E2,1 | = 3 edges.
Theorem 2.1. The atom bond connectivity index of triglyceride is given by, ABC(G)=
39.926149.
Proof. The atom-bond connectivity index of triglyceride is
du + dv − 2
ABC(G) =
.
du dv
uv∈E(G)
2+2−2
2+3−2
1+3−2
+ |E2,3 |
+ |E1,3 |
= |E2,2 |
2.3
1.3
2.2
2+1−2
+ |E2,1 |
.
2.1
2
1
1
1
= |E2,2 | √ + |E2,3 | √ + |E1,3 |
+ |E2,1 | √
3
2
2
2
1
1
2
1
= 41 √ + 9 √ + 3
+ 3√
3
2
2
2
ABC(G) = 39.926149.
Computation of Topological Indices of Triglyceride
1635
Theorem 2.2. The randic connectivity index of triglyceride is given by,
χ (G) = 28.02760577.
Proof. The randic connectivity index of triglyceride is
χ(G) =
√
e=uv∈E(G)
1
.
du dv
1
1
1
1
= |E2,2 | √
+ |E1,3 | √
+ |E2,1 | √ .
+ |E2,3 | √
2.3
1.3
2.1
2.2
1
1
1
1
= |E2,2 | √ + |E2,3 | √ + |E1,3 | √ + |E2,1 | √ .
6
3
2
4
1
1
1
1
= 41 + 9 √ + 3 √ + 3 √ .
2
6
3
2
χ(G) = 28.02760577.
Theorem 2.3. The sum connectivity index of triglyceride is given by,
S(G) = 27.75697317.
Proof. The sum connectivity index of triglyceride is
S(G) =
1
.
√
d
+
d
u
v
uv∈E(G)
1
1
+ |E1,3 | √
+ |E2,1 | √
.
2+2
2+3
1+3
2+1
1
1
1
1
= |E2,2 | √ + |E2,3 | √ + |E1,3 | √ + |E2,1 | √ .
4
4
3
5
1
1
1
1
= 41 + 9 √ + 3 + 3 √ .
2
2
3
5
S(G) = 27.75697317.
= |E2,2 | √
1
+ |E2,3 | √
1
Theorem 2.4. The geometric-arithmetic index of triglyceride is given by, GA(G) =
28.24466641.
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M. R. Rajesh kanna, D. Mamta and R. S. Indumathi
Proof. The geometric-arithmetic index of triglyceride is
√
2 du dv
.
GA(G) =
du + dv
e=uv∈E(G)
√2.1 √2.3 √1.3 √2.2 = |E2,2 | 2
.
+ |E1,3 | 2
+ |E2,1 | 2
+ |E2,3 | 2
2+1
2+3
1+3
2+2
√6 √3 √2 √4 + |E1,3 | 2
+ |E2,1 | 2
.
+ |E2,3 | 2
= |E2,2 | 2
5
4
3
4
√
√2 √6 3
+3
+3 2
.
= 41 + 9 2
5
2
3
GA(G) = 28.24466641.
Theorem 2.5. The harmonic index of triglyceride is given by, H (G) = 27.6.
Proof. The harmonic index of triglyceride is
2
H (G) =
.
du + dv
e=uv∈E(G)
2 2 2 2 + |E2,3 |
+ |E1,3 |
+ |E2,1 |
.
= |E2,2 |
2+2
2+3
1+3
2+1
2
2
2
2
= |E2,2 |
+ |E2,3 |
+ |E1,3 |
+ |E2,1 |
.
4
5
4
3
2
2
2
2
= 41
+9
+3
+3
.
4
5
4
3
H (G) = 27.6.
Theorem 2.6. The general randic index of triglyceride is given by,
Rk (G)=41(2)2k + 9(3k )(2k ) + 3(3k ) + 3(2k ).
Proof. The general randic index of triglyceride is
(du dv )k .
Rk (G) =
e=uv∈E(G)
= |E2,2 |(2.2)k + |E2,3 |(2.3)k + |E1,3 |(1.3)k + |E2,1 |(2.1)k .
= |E2,2 |(4k ) + |E2,3 |(6k ) + |E1,3 |(3k ) + |E2,1 |(2k ).
= 41(4k ) + 9(6k ) + 3(3k ) + 3(2k ).
Rk (G) = 41(22k ) + 9(3k )(2k ) + 3(3k ) + 3(2k ).
Computation of Topological Indices of Triglyceride
3.
1637
Conclusion
The problem of finding the general formula for ABC index, Randic connectivity index,
Sum connectivity index, GA index, Harmonic index, General Randic index of triglyceride is solved here analytically without using computers.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this
paper.
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