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INERTIA, NULLITY AND SIGNATURE OF CNC4 [n] NANOCONEi.

Byline: Sakander Hayat and Mehar Ali Malik

ABSTRACT

Spectrum related topological indices play a vital role in theoretical chemistry and nanotechnology as well. Inertia indices, nullity and signature are studied for different chemical structure and nanostructures by considering their chemical graphs. They relate the stability of various organic compounds.In this paper, inertia indices, nullity and signature of CNC4 [n] nanocone are studied.

Keywords: Inertia indices, nullity, signature, carbon nanocones

1 INTRODUCTION

Carbon nanocones have been observed, since 1968 or even earlier [6], on the surface of naturally occurring graphite. Their bases are attached to the graphite and their height varies between 1 and 40 micrometers. Their walls are often curved and are less regular than those of the laboratory made nanocones.

Let G be an n -vertex molecular graph with vertex set V (G) = {v1 , v2 ,, vn } and edge set E(G) . The vertices of G correspond to atoms and an edge between two vertices corresponds to the chemical bond between these vertices.

The adjacency matrix A(G) = [aij ]nn(usually denoted by A ) of the graph G is defined as:

Equations

The characteristic polynomial of G is a polynomial of degree n , defined as (G, ) = det (In Eqs. A) , where I n denotes the identity matrix of order n . The zeros of (G, ) are eigenvalues of A and multiset of eigenvalues of A is called the spectrum of A .

The eigenvalues and spectrum of A are respectively called the eigenvalues and spectrum of the graph G . As G is a simple graph, the matrix A is real, symmetric with zero trace. Thus all eigenvalues of A are real and their sum is zero [4]. The notations used in this article are mainly taken from book [14]. The positive (resp., negative) inertia index of a graph G , denoted by p(G) resp., n(G) ), is defined to be the number of positive (resp., negative) eigenvalues of its adjacency matrix. The signature of G , denoted by s(G) , is defined as the difference between positive and negative eigenvalues of G . The nullity of G , symbolized as (G) is defined as the multiplicity of eigenvalue ze ro in adjacency spectrum of G . Obviously, p(G) Eqs. n(G) (G) =| V (G) | .

These parameters attract much attention of the researchers in the field of mathematical chemi stry, theoretical and computational chemistry due to their direct applications in chemistry [3]. Nullity of a chemical graph is related to the stability of saturated hydrocarbons [2].

For further study of these parameters in different perspectives

2 RESULTS FOR CNC4 [n] NANOCONE

In this section, we study certain spectrum based topological hexagons on its conical surface. If there are n layers of hexagons on the conical surface around square, then we represent the graph of that nanocones as CNC4 [n] in which 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 in Fig. 2. CNC4 [2] nanocone is shown

Table 1: The inertia indices, nullity and signature of CNC4 [n] with 1 n 11.

###T = CNC3[n]###p (T )###n (T )###(T )###s (T )

###1###8###8###0###0

###2###18###18###0###0

###3###32###32###0###0

###4###50###50###0###0

###5###72###72###0###0

###6###98###98###0###0

###7###128###128###0###0

###8###162###162###0###0

###9###200###200###0###0

###10###242###243###0###0

###11###288###288###0###0

Table 2: The quadratic curves fitted of the curves presented in Table 1.

###S = CNC 4 [n]###p(S)###n(S)###(S)###s(S)

###n 1mod(2)###2(n 1)###2

###2(n 1)###2

###0###0

###n 0 mod(2) with n greater than 0###2(n 2 2n 1)###2(n 2 2n 1)###0###0

We denote S as the graph of ] [ 4 n CNC nanocone. The molecules of S are drawn in HyperChem [15] for each value of n , 11 1 n . The adjacency matrices of these molecular graphs are constructed with the help of TopoCluj [5]. Then the inertia indices, signature and nullity are calculated using MATLAB. By using "cftoolbox" of MATLAB, a quadratic polynomial is fitted to the exact values of inertia indices of T for 11 1 n . The obtained data is arranged in Table 1.

Using the data given by Table 1, a non-linear polynomial is fitted. The inertia of this nanocone is plotted using MATLAB as shown in Figure 3. The results are displayed in Table 2. There is an important conclusion drawn about ] [ 4 n CNC nanocone.

Equations

3 CONCLUDING REMARKS

The study of spectrum based topological indices play an important role in QSPR/QSAR studies in which they relate the stability of different organic compounds especially carbon nanotubes and nanocones. In this paper, certain spectrum based topological indices of ] [ 4 n CNC nanocone are strong-minded. We used different software like Hyperchem to draw nanocones, TopoCluj to compute their adjacency matrices and MATLAB to compute its spectrum. These results theoretically provide a basis to study various physico-chemical properties like stability of these nanostructures.

REFERENCES

[1] A. R. Ashrafi, Experimental results on the energy and Estrada index of ,8] [4 7 5 p C HC nanotubes, Optoelectron. Adv. Mater. Rapid Comm. 4 (1), 48 (2010).

[2] P. W. Atkins, J. de Paula, Physical Chemistry, eighthed., Oxford University Press, 2006.

[3] L. Collatz,U. Sinogowitz, Spektren endlicher Grafen, Abh. Math. Semin. Univ. Hambg., 21(8), 63 (1957).

[4] D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs-Theory and Application, third ed., Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995.

[5] M. V. Diudea, O. Ursu, Cs. L. Nagy, TOPOCLUJ, Babes-Bolyai University, Cluj 2002.

[6] J. Gillot, W. Bollmann, B. Lux, Cigar-shaped conical crystals of graphite, Carbon 6, 381-7 (1968).

[7] S. Hayat, Valency based topological indices of carbon nanocones, Sci. Int. 27, 863 (2015).

[8] S. Hayat, M. Imran, On topological properties of nanocones ] [n CNC k , Studia UBB Chemia, LIX, 4, 113 (2014).

[9] S. Hayat, A. Khan, F. Yousafzai, M. Imran, Computational results on inertia indiecs, signature and nullity of ) ( 8 4 R C C nanotube, Optoelectron. Adv.

[10] H. Ma, W. Yang, Sh. Li, Positive and negative inertia index of a graph, Linear Algebra Appl. 438, 331 (2013).

[11] M. A. Malik, R. Farooq, Computational results on energy and Estrada index of ] , [ 8 4 q p C TUC nanotube, Optoelectron. Adv. Mater. Rapid Comm. 9, 311 (2015).

[12] M. A. Malik, R. Farooq, On conjectures on energy and Estrada index of ] [n CNCk nanocones, Optoelectron. Adv. Mater. Rapid Comm. 9, 415 (2015).

[13] G. R. Omidi, On the nullity of bipartite graphs, Graphs Combin. 25, 111 (2009).

[14] N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992.

[15] HyperChem package Release 7.5 for Windows, Hypercube Inc., 1115 NW 4th Street, Gainesville, Florida 32601, USA 2002. Mater. Rapid Comm. 9, 478 (2015).
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Publication:Science International
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Date:Jun 30, 2015
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