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VALENCY BASED TOPOLOGICAL INDICES OF CARBON NANOCONES.

Byline: Sakander Hayati

ABSTRACT

Topological indices are the global parameters defined for the simple graphs such that they give the same numerical value if the graphs are isomorphic. These numbers are of much importance because of their chemical importance, they correlate certain physico-chemical properties of certain organic compounds such hydrocarbons etc. A chemical graph is a graph which is created from some molecular structure by applying some graphical operations. Valency/degree is a local graph parameter, which is defined for every vertex as the number of connections with other vertices in a graph, just like for an atom in a molecule. Degree based topological indices play a vital role in QSAR/QSPR studies and correlate various physico-chemical properties of chemical compound. Carbon nanocones are conical structures which are allotropes of carbon having at least one dimension of the order one micrometer or smaller.

In this paper, valency based topological indices of carbon nanocones are strong-minded.

2010 Mathematics Subject Classification: 05C12, 05C90

Keywords:General Randi c index, Zagreb index, Harmonic index, Sum-connectivity index, Carbon nanocone

1 INTRODUCTION

Mathematical chemistry is a branch of theoretical chemistry in which we discuss and predict the chemical structure by using mathematical tools and does not necessarily refer to the quantum mechanics. Chemical graph theory is a branch of mathematical chemistry in which we apply tools of graph theory to model the chemical phenomenon mathematically.

This theory plays a prominent role in the fields of chemical sciences. A moleculer/chemical graph is a simple finite graph in which vertices denote the atoms and edges denote the chemical bonds in underlying chemical structure. This is more important to say that the hydrogen atoms are often omitted in any molecular graph.

Nanobiotechnology is a rapidly advancing area of scientific and technological opportunity that applies the tools and processes of nanofabrication to build devices for studying biosystems. Chemical graph theory found a prominent place in this prestigious area of research. In the last few decades there is a lot of research which has been done in this field. This theory contributes a main role in the fields of chemical sciences.

A moleculer/chemical graph is a simple finite hydrogen depleted graph in which vertices denote the atoms and edges denote the chemical bonds in underlying chemical structure.

A nanostructure is an object of intermediate size between microscopic and molecular structures. It is product derived through engineering at molecular scale. Carbon nanocones are conical structures which are allotropes of carbon having at least one dimension of the order one micrometer or smaller.

A topological index is a function Top " from to the set of real numbers, where is the set of finite simple graphs with the property that Top(G) = Top(H ) if both G and H are isomorphic. Obviously, the number of edges and vertices of a graph are topological indices also. A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or a matrix which represents the whole graph, and these representations are aimed to be uniquely Throughout in this article, G is considered to be connected graph with vertex set V (G) and edge set E(G) and du is the degree of vertex u V (G) .

The notations are standard and mainly taken from books [12, 21]. The very first and oldest degree based topological index is Randi c index [20] denoted by (H ) and introduced by Milan Randi c in 1975 . The Randi c index of graph G is defined as

Equations

The general Randi c index was proposed by Bollobas and ErdAls [5] and Amic et al. [3] independently, in 1998 . Then it has been extensively studied by both mathematicians and theoretical chemists [13]. Many important mathematical properties have been established. For a survey of results, we refer to the new book by Li and Gutman [18].

The general Randi c index

Equations

An important topological index introduced about forty years ago by Ivan Gutman and Trinajsti c is the Zagreb index or more precisely first zagreb index denoted by M1 (G) and was defined as the sum of degrees of end vertices of all edges of G . The first zagreb index is defined as

Equations

The reduced version of second zagreb index is defined [7] as follows:

With motivation from the Randi c index, a closely related variant of the Randi c connectivity index called the sum-connectivity index was recently proposed by Zhou and Trinajsti c [22] in 2009. The sum- connectivity index (G) was defined as follows:

Equations

Another variant of the Randi c index named the harmonic index which first appeared in [6]. For a graph G , the harmonic index H (G) is defined as

Equations

Harmonic index is extensively studied for unicyclic and bicyclic graphs in [11].

2 MOTIVATIONS, RESULTS AND DISCUSSION

In this paper, we study carbon nanocones having triangle and pentagon as their cores and having hexagonal layers on its conical surface. A type of pentagonal nanocone is shown in Fig. 1. It has a pentagon on its top which acts as its core and is encompassing the hexagonal layers on its conical surface. The study of physico-chemical properties of these carbon containing chemical compounds is a respected problem in nanotechnology and theoretical chemistry. The study of topological descriptors of these chemical compounds supports this phenomenon in a productive way. Because of importance of topological indices in these organic compounds, we intend to study them for two important classes of carbon nanocones. These results provide a basis to understand the topology of these chemical compounds.

Wiener index is one of the most studied topological index in the literature. Alipour et al. studied the Wiener index of one-heptagonal nanocone [1]. They gave the idea to compute this so-called index numerically.

The vertex PI, Szeged and omega polynomials of carbon nanocones CNC 4[n] have been studied by Ghorbani at el. in [8]. For further study of topological indices of various nanostructures, networks and graphs see [2, 4,9,10, 14, 15, 16, 17,19].

2.1 RESULTS FOR CNC3[n]

NANOCONE

In this section, we compute degree based topological indices of CNC3[n] nanocones. A CNC3[n] nanocones consists of a triangle as its core and encompassing the layers of hexagons on its conical surface. If there are n layers of hexagons on the conical surface around triangle, then we number n denotes the number of layers of hexagons and number in the subscript shows the sides of polygon which acts as the core of nanocones. The CNC3[2] nanocone is shown in Fig. 2.

Following theorem presents the analytically closed formula of general Randi c index for CNC3[n] nanocone.

Theorem2.1.1.Consider the CNC3[n] nanocone, then its general Randi c index is equal to Proof.Consider H be the CNC3[n] nanocone with defining parameter n . The number of vertices and edges in

Equations

There are three types of edges in H based on degrees of end vertices of each edge. Table 1 shows such an edge partition of H .

Table 1: Edge partition of CNC3[n] nanocone based on degrees of end vertices of each edge.

###For###=1

###(d u , d v )###Number of edges

###where uv E (G)

###(2,2)###3

###(2,3)###6n

###(3,3)###9 2 3

###n n

###2###2

CNC3[n] nanocone based on degrees of end vertices of each edge. For = 1

Equations

By using edge partition given in table 1, we get a this expression in parameter n ,

Equations

By using edge partition given in table 1, we get this expression

In the following theorem, we compute harmonic index for triangular nanocones CNC3[n] .

Theorem 2.1.4.For index is equal to CNC3[n] nanocone, the harmonic

Equations

Proof.Let CNC3[n] be the chemical graph of triangular nanocones. We prove it by using edge partition from table 1. We know

Equations

2.2 Results for CNC5[n] nanocone In this section, we determine valency based topological indices of nanocone. The vertex and edge cardinalities are

Equations

Nanocones are often called one pentagonal nanocones, and word pentagonal used for pentagon as its core and like other families of nanocones there are hexagonal layers on its conical surface (Fig. 3).

Equations

Table 2: Edge partition ofnanocone based on degrees of end vertices of each edge.

###For = 1

###Number of edges

(d u , d v )

whereuv E (G)

(2,2)###5

(2,3)###10n

(3,3)###15 2 5

###n n

###2###2

Equations

Now we apply the formula

Equations

In the following theorem, we compute reduced second Zagreb index for this nanocone.

Theorem 2.2.2.

Equations

In the following theorem, we compute sum-connectivity index for this class of nanocones.

Theorem 2.2.3

Equations

3 CONCLUDING REMARKS

Topological indices play a vital role in the study of physico-chemical properties of chemical compounds. Valency based topological indices have got a prominent place in this study due to prediction of various chemical properties such as stability, kovat's constant, enthalpy etc. with high predictive power. To compute and study these topological indices for various nanostructures like nanotubes, nanostar dendrimers and nanocones is a respected problem in nanotechnology. In this study, we compute various valency based topological indices of two important classes of carbon nanocones. These results give valuable information regarding chemical properties of these nanocones. In future, we are interested to study topological description of these chemical graphs by studying their distance based topological indices.

REFERENCES

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[2] Y. Alizadeh, S. Klavzar, M. A. Hosseinzadeh, Interpolation method and topological indices: 2 -parametric families of graphs, MATCH Commun. Math. Comput. Chem., 69(2013) 523 534 .

[3] D. Amic, D. Beslo, B. Lucic, S. Nikolic and N. Trinajsti c , The vertex-connectivity index revisited, J. Chem. Inf. Comput. Sci., 38(1998) 819822 .

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[7] B. Furtula, I. Gutman, S. Ediz, On diffrence of Zagreb indices,, Discrete Appl. Math., 178(2014) 83 88 .

[8] M. Ghorbani, M. Jalali, The vertex PI, Szeged and omega polynomials of carbon nanocones Commun. Math. Comput. Chem., 62(2009) 353 362 .

[9] S. Hayat, M. Imran, Computation of topological indices of certain networks, Appl. Math. Comput., 240(2014) 213-228.

[10] S. Hayat and M. Imran, Computation of topological indices of certain nanotubes J. Comput. Theor. Nanosci., 7(2015) 12-17.

[11] Y. Hu, X. Zhou, On the harmonic index of the unicyclic and bicyclic graphs, WSEAS Transaction on Mathematics, 6(12)(2013) 716 726 .

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[16] A. Khaksar, M. Ghorbani, H. R. Maimani, On atom bond connectivity and GA indices of nanocones, Optoelectronics and Advanced Materials-Rapid Communications, 4(2010) 1868 1870 .

[17] M. H. Khalifeh, M. R. Darafsheh, Hassan Jolany, The hyper-Wiener index of one pentagonal carbon nanocone, Journal of Current Nanoscience, 9(2013) .

[18] X. Li and I. Gutman, Mathematical aspects of Randic-type molecular structure descriptors, mathematical chemistry monographs No.1, University of Kragujevac, (2006).

[19] M. Saheli, H. Saati, A. R. Ashrafi, The eccentric connectivity index of one pentagonal carbon nanocones, Optoelectronics and Advanced Materials-Rapid Communications, 4(2010) 896 897 .

[20] M. Randi c , On Characterization of molecular branching J. Amer. Chem. Soc. 97(1975) 6609 6615 .

[21] N. Trinajsti c , Chemical graph theory, CRC Press, Boca Raton, FL1992 .

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