View of Computation of Entire Zagreb Index Based on Several Nanotubes

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Computation of Entire Zagreb Index Based on Several Nanotubes

N. Parvathi

¹

, S.RajeswariSaminathan

²

1,2

Department of Mathematics, College of Engineering and Technology,

SRMIST, Kattankulathur, Chennai – 603203, India.

Corresponding author: [email protected]

Abstract:

A topological index is a molecular descriptor. From this index, it is possible to search mathematical values and further investigate from physicochemical properties of a molecule. There are different types of these indices like based on degrees, distance and counting related topological indices. Carbon nanotubes (CNTs) are types of nanostructures having a cylindrical shape. We also compute First Zagreb index and Second Zagreb index of nanotubes.We determine Hyper Zagreb index, First and Second Zagreb polynomial, and Sum-Connectivity index of carbon nanotubes.

Keywords:

Topological index, carbon nanotubes, Zagreb index, Hyper Zagreb index, Augmented Zagreb index, Zagreb polynomial, Sum-Connectivity index.

Introduction:

A topological index, also known as a connectivity index, is a type of molecular descriptor calculated based on the molecular graph of a chemical compound in the fields of chemical graph theory, molecular, topology, and mathematical chemistry. Topological indices are graph invariants that are used to establish Quantitative Structure-Activity Relationships (QSAR)[13]. In this situation, thevertices and edges of a chemical graph correspond to the atoms and bonds, respectively[2]. These indices can be thought of as molecular structure descriptors because they represent the degree of branching of the molecular carbon-atom skeleton.

The use of matter on an atomic, molecular, and supra-molecular scale for industrial purposes is known as

"

nanotechnology^". Nanotechnology is the analysis of nanostructures with sizes ranging from 1 to 100 nanometers[15]. Richard Feynman, a physicist, is known as the "Father of Nanotechnology." Many new materials and devices with a broad range of applications in medicine, electronics, and computing are being developed with the aid of nanotechnology.

Carbon Nanotubes (CNTs) are a form of nanostructure that is made up of carbon allotropes and has a cylindrical shape. SumioIijima was the first to implement it in 1991[15]. Fullerence materials, such as carbon nanotubes (CNTs), have applications in electronics, optics, materials science, and architecture. CNT is made up of a layer of carbon atoms that are bound together in a hexagonal (honeycomb) pattern.Graphene is a one-atom thick film of carbon that is wrapped around a cylinder and bonded together to form a CNT. Single walled carbon nanotubes (SWCNT) and multi walled carbon nanotubes (MWCNT) are two types of carbon nanotubes. The diameter (SWCNT) are 0.8 to 2nm and (MWCNT) are 5nm to 20 nm respectively, form a rope like structure. In addition to the two different possible structures there are three different possible types of (CNTs). These three types of CNTs are Armchair, Zig-Zag and Chiral.

A Lattice is a periodic sequence of points in which each point is indistinguishable from every other point and has the same surroundings. The two dimensional lattice graph also known as grid graph is the cartesian product of path graph of m and n vertices named as 2-D lattice

“ID(G) = ¹

𝑑_𝑖 𝑛

𝑖=1 is the inverse degree of a graph G with no isolated vertices.

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Where d_i is the vertex's degree 𝑣_𝑖 ∈ V (G). The inverse degree first drew attention thanks to conjectures made by the Graffiti computer programme [8]. In the 1980s, Narumi and Katayama considered the product

N K = N K(G) = 𝑛 𝑑𝑖 𝑖=1

And it named as “simple topological index”[9]. In more recent works on this graph invariant [7] the name

“Narumi- Katayama index” has been used. Some properties of the Narumi-Katayama index wereestablished in [10]”.

An important topological index introduced more 30 years ago by I. Gutman and N. Trinajistic is the first and second Zagreb indices(𝑀₁ 𝐺 𝑎𝑛𝑑 𝑀₂(𝐺)) 3 .

The Hyper Zagreb index (𝐻𝑀(𝐺)) being introduced by Shirdel et al. It is a distance based Zagreb index in 1913. 5 .

The Augmented Zagreb index (𝐴𝑍𝐼 (𝐺)) was introduced by Furtula et al. 5 .

The first and second Zagreb polynomial(𝑍𝑔₁ 𝐺, 𝑥 𝑎𝑛𝑑 𝑍_𝑔₂(𝐺, 𝑥)) was introduced by Gutman et al. 3 The Sum Connectivity Index (𝑆𝐶𝐼 𝐺 )being introduced by Zhou and Trinajstic 7

Let G = (V, E) be a simple graph with n vertices and m edges and a vertex set V (G) = v_1,v₂,….. vnThe number of edges in the shortest path connecting a and b, denoted by d (a, b), is defined as the distance between two vertices, say a and b of G. [1, 2].

PRELIMINARIES:

Definition1:

“Let G be a graph. The First and Second Zagreb index of G is defined as M1 (G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎+ 𝑑𝑏)

M2 (G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎. 𝑑𝑏) Definition 2:

Let G be a graph. The Hyper-zagreb index is defined as HM(G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎+ 𝑑𝑏)² Definition 3:

For a graph G the augmented Zagreb index is defined as AZI(G) = ^𝑑^𝑎^.𝑑^𝑏

𝑑_𝑎+𝑑_𝑏−2

𝑒=𝑎𝑏 ∈𝐸(𝐺) 3

Definition 4:

Let G be a graph. The first and second Zagreb polynomial is defined as Zg1(G, x) = _{𝑒=𝑎𝑏 ∈𝐸(𝐺)}𝑥^(𝑑^𝑎^+𝑑^𝑏⁾

Zg2(G, x) = _{𝑒=𝑎𝑏 ∈𝐸(𝐺)}𝑥^(𝑑^𝑎^.𝑑^𝑏⁾ Definition 5:

Let G be a graph. The sum-connectivity index is defined as”

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SCI(G) = ¹

𝑑𝑎+𝑑_𝑏 𝑒=𝑎𝑏 ∈𝐸(𝐺)

2.MAIN RESULTS:

The First and Second Zagreb index, Hyper-zagreb index, Augmented Zagreb index, First and Second Zagreb polynomials, and Sum-Connectivity index of H-Naphthalenic nanotubes and TUC₄(p, q) nanotubes are all investigated in this paper. These topological indices for H-Naphthalenic nanotubes and TUC4 (p, q) nanotubes are computed in the following section.

2.1 H- Naphthalenic Nanotubes Results:

The entire lattice in this section is a plane tiling of C₄, C₆, and C₈, which can be used to cover a cylinder or a torus. As shown in figure 1, the H- Naphthalenic nanotube is denoted by NPHX [p, q], where p denotes the number of pairs of hexagons in a row and q denotes the number of hexagons in a column.

In fig.1. A 2D-lattice of H-Naphtalenic nanotube.

The edges ab with d_a= 2 and d_b = 3 are shown in red, while the edges ab with d_a = d_b = 3 are shown in black.

Lemma 1:

Let NPHX [p, q] represent the graph of H- Naphthalenic nanotubes with (p,q> 1). Then V (NPHX [p, q]) = 10pq is its vertex set.

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Lemma 2:

Consider the graph of H- Naphthalenic nanotubes (p, q ˃ 1) then its edge set is E (NPHX [p, q]) = 15pq – 2p

Now we can see that the two partitions of the edges in 2D- lattice from this nanotube.

i.e., E_{2, 3} = {e = ab ∈ E(G) ⁄da = 2, d_b = 3}

E_{3, 3} = {e = ab ∈ E (G) ⁄ da = 3, d_b = 3}

The number of edges of the given sets are given as below E_{2, 3} = 8pand E_{3, 3} = 15pq – 10p

Theorem 1:

Let's take a look at the graph of H-Naphtalenic nanotubes. The First and Second Zagreb indexes are then equivalent

M₁(G) = 90pq – 20p

M₂(G) = 135pq – 42p

Proof:

Let NPHX[p, q] be a graph. By using the edge partition of number of edges. Using lemma 2, then we compute First and Second Zagreb index

M₁(G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎+ 𝑑𝑏)

This implies that

M₁(G) = 𝑎𝑏 ∈𝐸_2,3(𝑑𝑎+ 𝑑𝑏) + 𝑎𝑏 ∈𝐸_3,3(𝑑𝑎+ 𝑑𝑏) = (8p) (2 + 3) + (15pq – 10p) (3 + 3) = 40p + 90pq – 60p

M₁(G) = 90pq – 20p Also M₂(G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎. 𝑑𝑏)

= 𝑎𝑏 ∈𝐸_2,3(𝑑𝑎. 𝑑𝑏) + 𝑎𝑏 ∈𝐸_3,3(𝑑𝑎. 𝑑𝑏) = (8p) (2.3) + (15pq – 10p) (3.3) = 48p + 135pq – 90p

M₂(G) = 135pq – 42p.

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Theorem 2:

Consider the NPHX [p, q] nanotube graph. And there's the Hyper-zagreb index.

HM(G) = 540pq – 160p

Proof:

Let the graph of NPHX[p, q]. Then we calculate the Hyper-Zagreb index is HM(G) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)(𝑑𝑎+ 𝑑𝑏)²

This implies that

HM(G) = _{𝑎𝑏 ∈∈𝐸}_2,3(𝑑_𝑎+ 𝑑_𝑏)² + _{𝑎𝑏 ∈∈𝐸}_3,3(𝑑_𝑎+ 𝑑_𝑏)² = (8p) (2 + 3)² + (15pq – 10p) (3 + 3)²

= (8p) (25) + (15pq -10p) (36) = 200p + 540pq – 360p HM(G) = 540pq – 160p This completes the proof.

Theorem 3:

Assume that the graph of NPHX[p, q] nanotube. Then its Augmented Zagreb index is AZI(G) = 21, 357.3pq – 14, 174.2p

Proof:

Let the graph of NPHX[p, q]. Then we compute the Augmented Zagreb index is AZI(G) = ^𝑑^𝑎^.𝑑^𝑏

𝑑_𝑎+𝑑_𝑏−2

𝑒=𝑎𝑏 ∈𝐸(𝐺) ³

= ^𝑑^{𝑎 .}^𝑑^𝑏

𝑑_𝑎+𝑑_{𝑏 −2}

𝑎𝑏 ∈𝐸_2,3 ³ + ^𝑑^{𝑎 .}^𝑑^𝑏

𝑑_𝑎+𝑑_{𝑏 −2}

𝑎𝑏 ∈𝐸_3,3 ³

= (8p) ^2.3

2+3−2

3 + (15pq – 10p) ^3.3

3+3−2 3

= (8p) (2)³ + (15pq – 10p) (11.25)³ = (8p) (8) + (15pq – 10p) (1423.82) AZI(G) = 21, 357.3pq – 14,174.2p.

Hence the proof.

2.2 Nanotubes Covered by C

4

:

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The 2D lattice of this family of nanotubes is a plane tiling of C₄ in this section. TUC₄[p, q], where p is the number of squares in a row and q is the number ofsquares in a column, denotes this family of nanotubes.

Figure 2: A C₄-coated TUC₄[p, q] nanotube

.

Lemma 3:

If TUC₄ [p, q] is the graph of nanotubes protected by C₄, then V (TUC₄ [p, q]) = (q + 1) (p + 1) is the number of vertices.

Lemma 4:

Consider the graph of TUC₄ [p, q] nanotubes; the edge set of this graph is E(TUC₄[p, q]) = (2q + 1)(p + 1) Now we can see that the partitions of the

i.e., E_{3, 3} = {e = ab ∈ E(G) / da = 3, d_b = 3}

E_{3, 4}= {e = ab ∈ E(G) / da = 3, d_b = 4}

E_{4, 4}= {e = ab ∈ E(G) / da = 4, d_b = 4}

The number of edges of the given sets are given as follows.

E_{3, 3} = 2p + 2

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E_{3, 4}= 2P + 2

E_{4, 4}= (p + 1) (2q – 3)

Theorem 4:

Consider the graph of TUC₄[p,q] nanotubes covered by C₄, then its First and Second Zagreb polynomial is Z𝑔1 (TUC₄[p, q], x) = (p +1) 2x⁶ + (p +1)2x⁷ + (2pq – 3p + 2q – 3)x⁸

Z𝑔2 (TUC₄[p, q], x) = (p +1)2x⁹ + (p +1)2x¹² + (2pq – 3p +2q -3)x¹⁶

Proof:

The First Zagreb polynomial is defined as Zg₁(G, x) = 𝑒=𝑎𝑏 ∈𝐸(𝐺)𝑥^(𝑑^𝑎^+𝑑^𝑏⁾ This implies that

Zg₁(TUC₄[p, q], x) = 𝑎𝑏 ∈𝐸_3,3𝑥^(𝑑^𝑎^+𝑑^𝑏⁾ + 𝑎𝑏 ∈𝐸_3,4𝑥^(𝑑^𝑎^+𝑑^𝑏⁾ + 𝑥^(𝑑^𝑎^+𝑑^𝑏⁾

𝑎𝑏 ∈𝐸_3,4

= (2p + 2)x^{3 +3} + (2p + 2)x^{3 + 4}+ (p + 1)(2q – 3)x^{4 +4}

= (2p + 2) x⁶ + (2p +2) x⁷ + (2pq – 3p + 2q – 3) x⁸

The Second Zagreb polynomial is defined as Zg₂(G, x) = _{𝑒=𝑎𝑏 ∈𝐸(𝐺)}𝑥^(𝑑^𝑎^.𝑑^𝑏⁾

Zg₂(TUC₄[p, q], x) = _{𝑎𝑏 ∈𝐸(𝐺)}𝑥^(𝑑^𝑎^.𝑑^𝑏⁾

= _{𝑎𝑏 ∈𝐸}_3,3𝑥^(𝑑^𝑎^.𝑑^𝑏⁾ + _{𝑎𝑏 ∈𝐸}_3,4𝑥^(𝑑^𝑎^.𝑑^𝑏⁾ + _{𝑎𝑏 ∈𝐸}_3,4𝑥^(𝑑^𝑎^.𝑑^𝑏⁾

= (2p + 2) x^3.3 + (2p + 2) x^3.4 + (p + 1) (2q – 3) x^4.4

= (2p + 2) x⁹ + (2p + 2) x¹² + (p + 1) (2q – 3) x¹⁶

= (2p + 2) x⁹ + (2p + 2) x¹² + (2pq – 3p + 2q – 3) x¹⁶. Hence the proof.

Theorem 5:

Let the graph of TUC₄[p, q] nanotubes then its Sum-Connectivity index is SCI(G) = ^2(𝑝+1)

3 + ^(𝑝+1)

3 + 𝑝+1 (2𝑞−3) 4

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Proof:

By the definition of Sum-Connectivity index is

SCI(G) = ¹

𝑑_𝑎+𝑑_𝑏 𝑒=𝑎𝑏 ∈𝐸(𝐺)

This implies that

SCI(G) = ¹

𝑑_𝑎+𝑑_𝑏 𝑒=𝑎𝑏 ∈𝐸(𝐺)

= ¹

𝑑𝑎+ 𝑑_𝑏

𝑎𝑏 ∈𝐸_3,3 + ¹

𝑑𝑎+ 𝑑_𝑏

𝑎𝑏 ∈𝐸_3,4 + + ¹

𝑑𝑎+ 𝑑_𝑏 𝑎𝑏 ∈𝐸_4,4

= (2p + 2) ¹

3.3 + (2p + 2) ¹

3.4 + (p +1) (2q – 3) ¹

4.4

= (2p + 2) ¹

9 + (2p + 2) ¹

12 + (p +1) (2q – 3) ¹

16

= ^2(𝑝+1)

3 + ^{(𝑝 +1)}

3 + (𝑝+1)(2𝑞−3) 4

Hence the proof.

Conclusion:

Certain degree-based topological indices, such as the First and Second Zagreb index, the Hyper-Zagreb index, and the Augmented Zagreb index for nanotubes protected by C₄, C₆, and C₈, are discussed in this paper. We also looked at the Sum-Connectivity index and the First and Second Zagreb polynomials. For NPHX[p,q] and TUC₄[p, q] nanotubes, we found closed formulae for these topological indices.

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