Computation of the different topological indices of nanostructures

In this research study, several topological indices have been investigated for linear [n]-Tetracene, VTetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori. The calculated indices are first, second, third and modified second Zagreb indices. In addition, the first and second Zagreb coindices of these nanostructures were calculated. The explicit formulae for connectivity indices of various families of Tetracenic nanotubes and nanotori are presented in this manuscript. These formulae correlate the chemical structure of nanostructures to the information about their physical features.


IntroduCtIon
A graph is a collection of points and lines connecting a subset of them.The points and lines of a graph are also called the vertices and edges of the graph, respectively.A simple graph is an unweighted, undirected graph without loops or multiple edges.All of the graphs in this paper are simple.A molecular graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds (hydrogen atoms are often omitted).In the past years, nanostructures involving carbon have been the focus of intense research, which is driven to a large extent by the quest for new materials with specific applications.A topological index is a numeric quantity of a molecule that is mathematically derived in an unambiguous way from the structural graph of a molecule.In theoretical chemistry, topological indices are used for modelling physical, pharmacological, biological and other properties of chemical compounds.Some exact formulae have been computed for topological indices of nanostructures and some graphs have been plotted.(Eliasi & Taeri, 2007;Heydari & Taeri, 2007;Mahmiani et al., 2008;Nikmehr et al., 2014).The use of topological indices as structural descriptors is important in proper and optimal nanostructure design.
The main goal of this paper is to compute some topological indices and polynomials for a family of linear [n]-Tetracene, lattice of V-Tetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori.

Methodology
This section presents some notations as well as preliminary notions, which will be needed for the rest of the paper.A graph G consists of a set of vertices V(G) and a set of edges E(G).The vertices in G are connected by an edge if there exists an edge uv ∈ E(G) is the number of vertices of G adjacent to u.There are several topological indices already defined.

Zagreb indices and coindices
One of the oldest graph invariants is the well-known Zagreb indices first introduced by Gutman and Trinajstić (1972) where they have examined the dependence of total �-electron energy on molecular structure and elaborated (Gutman et al., 1975).For a (molecular) graph G, the first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices.In fact, they are defined as: Journal of the National Science Foundation of Sri Lanka 43 (2) Some connectivity indices Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry.The product-connectivity index, also called Randić index of a graph G is defined such as: Dear AbhimaniRanatunga, Thank you so much in advances for your time and kind of attention.
Article entitled: "Computation of the Different Topological Indicesof Nanostructures" Please make the following corrections: This topological index was first proposed by Randić (1975).The concept of atom-bond connectivity index was introduced in the chemical graph theory by Estrada et al. (1998).The atom-bond connectivity index of a graph G is defined as follows: Zhou and Trinajstić (2009) proposed another connectivity index, named the sum-connectivity index.This index is defined as follows: The geometric-arithmetic index is another topological index based on degrees of vertices defined by Vukičević and Furtula (2009):

results and dIsCussIon
The use of topological and connectivity indices as structural descriptors is important in proper and optimal nanostructure design.The combinatorial design approach appears to be a useful platform for numerical experimentation in the design of nanostructures.The acenes or polyacenes are a class of organic compounds and polycyclic aromatic hydrocarbons made up of linearly fused benzene rings.The larger representatives have potential interest in optoelectronic applications and are actively researched in Chemistry and Electrical Engineering.Tetracene, also called naphthacene, is a polycyclic aromatic hydrocarbon.It has the appearance of a pale orange powder.Tetracene is the four-ringed member of the series of acenes.Figure 1 shows the linear [n]-Tetracene. respectively.
A recently proposed variant of the second Zagreb index, which is defined as (Nikolić et al., 2003): is known under the name modified second Zagreb index.
The third Zagreb index was first introduced by Fath-Tabar (2011).This index is defined as follows: The first and second Zagreb coindices were first introduced by Ashrafi et al. (2010).They are defined as follows:

,
...( 7) It should be noted that Zagreb coindices of G are not Zagreb indices of ; the defining sums run over , but the degrees are with respect to G.
Journal of the National Science Foundation of Sri Lanka 43 (2) June 2015 Before proceeding to the main results, the next section will prove two simple lemmas, which will be useful later.
Lemma 3.1 It holds that:   ii.M 2 (T,x) = (7n -4)x 9 + (16n -4)x 6 + 6x Proof.By definitions of the first and second Zagreb polynomials and partition of edges described in Table 1 of Lemma 3.1, we can see that: i. ii.
Next we calculate the first and second Zagreb indices for a linear [n]-Tetracene.ii.M 2 (T) = 159n -36.
Proof.We know the first and second Zagreb indices will be the first derivative of M 1 (T,x) and M 2 (T,x) evaluated at x = 1, respectively.Thus, i. ii.
June 2015 Journal of the National Science Foundation of Sri Lanka 43( 2) ii. F 972p 2 q 2 -144p 2 q -72pq 2 -216pq + 48p + 24q G 972p 2 q 2 -144p 2 q -216pq + 48p K 972p 2 q 2 -72pq 2 -216pq + 24q L 972p 2 q 2 -216pq Theorem 3. 7 The first and second Zagreb coindices of nanostructures are computed as: Proof.We have just applied the Lemma 3. (L) = 60480 and Finally, we calculate the Randić index, Sum-connectivity index, geometric-arithmetic index and atom-bond connectivity index of nanostructures by using an algebraic method.In general we have the following theorem without proof:  Theorem 3.9 The atom-bond connectivity index of nanostructures are computed as:

ConClusIon
Chemical graph theory is an important tool for studying molecular structures and has an important effect on the development of chemical sciences.The study of topological indices is currently one of the most active research fields in chemical graph theory.We have presented here some theoretical results on the Zagreb and connectivity indices of the linear [n]-Tetracene, V-Tetracenic nanotube, H-Tetracenic nanotube and Tetracenic nanotori.These formulae make it possible to correlate the chemical structure of nanostructures with a large amount of information about their physical features.

Figure 2 :
Figure 2: Basic structure of a Tetracene

Example 3 . 4 Example 3 . 5
4] be a nanotube with atoms and 200 chemical bonds.Then one can see that χ(F) = 71.6632and X(F) = 84.3093.Let G = G[2,4] be a nanotube with atoms and 208 chemical bonds.Then one can see that ABC(G) = 139.9607.Let K = K[2,3] be a nanotube with atoms and 156 chemical bonds.Then one can see that GA(K) = 155.7575.Nanostructure x (in the first table  is a Randić index but in the second table  is a Sum-connectivity index.)The connectivity indices of Nanostructures are computed as: 11. Page 128, column 1, line -1, change "χG = ∑

Table 1 :
Computing the number of vertices and edges for a linear [n]-Tetracene

Table 2 :
Computing the number of vertices and edgesThe following Lemma is crucial for our main results.We apply a similar reasoning as in the proof of Lemma 3.1 to calculate the quantities of |V|, |E 1 |, |E 2 | and |E 3 | of the nanostructures F, G, K and L.
2 -404n + 54 Theorem 3.4 The first and second Zagreb coindices of a linear [n]-Tetracene are computed as: Proof.By applying Lemma 3.1 and Lemma 3.2, we are done.Journal of the National Science Foundation of Sri Lanka 43(2) June 2015 0 Theorem 3.6 The first, second, modified second and third Zagreb indices of nanostructures are computed as:Proof.Proof.The proof follows from Theorem 3.1.Proof.The proof is clear, by using Lemma 3.3 and proof of Theorem 3.3.Example 3.1 Let F = F[3,10] be a lattice with 540 atoms and 778 chemical bonds.Then one can see that M 1 (F) = 4540, M 2 (F) = 6642, M 2 * (F) = 94.2222 and M 3 (F) = 80.June 2015 Journal of the National Science Foundation of Sri Lanka 43(2) 2, Lemma 3.3 and using the Theorem 3.6.
Randić index but in the second table  is a Sum-connectivity index.)connectivity indices of Nanostructures are computed as: ∈ , " 26.Page 132, in Theorem 3.8, in second table: (in the first table  is a Randić index but in the second table  is a Sum-connectivity index.)Theorem 3.8 The connectivity indices of Nanostructures are computed as: