Topological Invariants of Nanocones and Fullerenes

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Topological Invariants of Nanocones and Fullerenes Fatemeh Koorepazan-Moftakhar1,2, Ali Reza Ashrafi1,2,*, Ottorino Ori3,4 and Mihai V. Putz4 1 Department of Nanocomputing, Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 8731751167, I. R. Iran; 2Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I. R. Iran; 3Actinium Chemical Research, Via Casilina 1626/A, 00133 Rome, Italy; 4Laboratory of Computational and Structural Physical-Chemistry for Nanosciences and QSAR, Department of Biology-Chemistry, West University of Timioara, Pestalozzi Str. No. 16, Timioara 300115, Romania

Abstract: The Timisoara-eccentricity (TM-EC) index of a molecular graph is defined as the sum of i ii over all atoms i in , where i, i and i are the degree, eccentricity and the number of atoms at distance i from atom i. The topological efficiency index of is defined as = 2W / N w , where W denotes the Wiener index, w is the minimal vertex contribution and N is the number of carbon atoms. This paper is devoted to the study of nanocones and fullerenes by these new graph invariants. It is proved that the TM-EC index of a fullerene can be bounded by a polynomial of degree 2, for twelve infinite series of fullerenes. It is also shown that in one pentagonal carbon nanocone with exactly 5n2 + 10n + 5 carbon atoms, we have 1.24 and TM EC = 280n3 + 385n2 + 195n + 40. Finally, we examine the dual of this nanocone and prove that we have 1.24 and TM - EC = 70n3 + 20n2 - 5n.

Keywords: TM-eccentricity index, topological efficiency index, eccentric connectivity index, fullerene, nanocone. 1. INTRODUCTION A graph is a pair (V, E) of vertices and edges with E() P2(V()), where P2(V()) is the set of all 2-element subsets of V(). A molecular graph is a graph in which vertices are atoms of a given molecule and edges are its chemical bonds. Since the capacity of carbon is four, it is natural to consider all graphs with maximum degree 4, as a molecular graph. The graph is said to be 3connected if it does not have two vertices whose removal disconnects the graph. It is called a cubic graph, if all degrees are equal to 3. A fullerene graph is a cubic and planar 3-connected graph that its faces are pentagons and hexagons. These graphs are the best mathematical model for fullerenes molecules, which are polyhedral carbon molecules in which atoms are arranged in pentagons and hexagons. The fullerenes can be embedded into a sphere. Euler's formula implies that each n-vertex fullerene contains exactly twelve pentagons and n/2 – 10 hexagons [1]. The spherical fullerenes are also called buckyballs. The buckminsterfullerene C60 is the most important buckyballs discovered by Kroto and his team [2]. The importance of fullerene chemistry in technology causes that several researchers devoted their time to develop mathematical techniques for studying fullerenes. Myrvold et al. [3], prepared the FuiGui software for working with fullerenes. Schwerdtfeger et al. [4] designed the Program Fullerene software as second software for discovering mathematical properties of fullerene graphs. Fowler and his co-authors in several leading papers [5-7], considered the symmetry perception of fullerenes into account and in [8-11], the authors continued the pioneering works of Fowler and settled some new problems. Suppose is a molecular graph and x and y are vertices in . The length of a minimal path connecting x and y is called the *Address correspondence to this author at the Department of Nanocomputing, Institute of Nanoscience and Nanotechnology, University of Kashan, Kashan 87317-51167, I. R. Iran; and Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-51167, I. R. Iran; Tel: 0098 31 55912367; Fax: 0098 31 55912332; E-mail: [email protected] 1385-2728/15 $58.00+.00

distance between x and y. We write d (x,y) to denote this number. It is easy to see that the two parameters function d(–,–) satisfies all properties of a meter on the set of all atoms. The sum of all d(x,y) overall 2–element subsets {x,y} V() is called the Wiener index of [12]. Harold Wiener in his seminal paper used this number to obtain a formula for computing boiling point of paraffin. After pioneering work of Wiener, several chemists proposed thousands of such numbers for predicting physico–chemical properties of molecules. Hosoya [13], used the name topological index for this numerical values. Some of these numbers can be defined by distance function d (x,y) and so they are called distance based topological indices of molecules. Diudea and his co–workers [14, 15], were the first scientists considered the topology of nanotubes into account. They computed exact expressions for the Wiener index of armchair and zig–zag polyhex nanotubes. These papers were the starting point for several researches on distance properties of nanostructures. The aim of this paper is to consider some new distance-based topological indices and compute them for the one–pentagonal carbon nanocones and fullerenes. These materials originally discovered by Ge and Sattler in 1994 [16]. The mathematical properties of one–pentagonal nanocones are topic of some recent papers. In [17], two distance–based topological indices of one–pentagonal carbon nanocones named “PI and Szeged” were calculated, and [18] is devoted to the methods for computing “eccentric connectivity index” of this nanomaterial. Klavar et al. [19, 20] presented a method that later called “cut method”. They applied cut method to compute the Wiener index of some hexagonal systems. To explain, we assume that G is a molecular graph. We first partition the edge set of G into classes F1, ... , Fk such that graphs G - Fi, 1 i k, consists of two or more connected components. Then we use properties of the components of the graphs G – Fi to derive a required property of G. We call the sets Fi,1i k, cuts of G, see [21] for details. In [22, 23], the authors extended the pioneering work of Klavar et al. to the systems with a few number of pentagons, heptagons and so on. Using this method they calculated exact formulas for the Wiener index of nanocones. © 2015 Bentham Science Publishers

2 Current Organic Chemistry, 2015, Vol. 19, No. ??

Koorepazan-Moftakhar et al.

Ori and his co-workers [23-28] introduced an important parameter for evaluation of sphericality in fullerenes. To define, we assume that is a connected molecular graph with vertex set V() = {v1, v2, …, vn}. The status of the i-th vertex of is defined as wi = jdij. One can easily prove that

W( ) =

1 i=n w 2 i=1 i

(1)

are done with the aid of HyperChem [36], TopoCluj [37] and GAP [38]. 2. TOPOLOGICAL INVARIANTS OF A NANOCONE AND ITS DUAL The aim of this section is to calculate three parameters TM-EC, and E, for a series of nanocones and their duals, Figs. (1 and 2). The main result of this section is as follows:

The topological index wi expresses the contribution to W() coming from the vertex vi. Assume that w = Min{wi } and w = Max{wi } The topological efficiency index () is defined as

() =

2W () . nw

(2)

This topological index expresses the efficiency of a given graph in filling the space when compared to its minimal vertex, the most deeply embedded node in the graph. We also define:

E() =

w . w

(3)

In our tests E seems working a bit better of especially in fullerenes, providing a good ranking of isomer energies. These hollow molecules are probably very sensitive to the regions with maximum deformations. Notice that the fullerenes can be constructed from pentagonal nanocones and so we expect this parameter is also useful for predicting physical properties of nanocones. Suppose is a molecular graph and v V(). The eccentricity of v, (v), is the maximum length of a shortest path connecting v to other vertices of . The maximum and minimum eccentricities of are called diameter and radius of , respectively. The eccentric connectivity index, EC(), is a topological index proposed by Sharma et al. [29]. It is defined as the sum of all (v)(v) overall vertices of , where (v) denotes the degree of v in . In [30], the authors considered some extremal properties of this topological index into account. Recently, Putz and Ori [31] proposed a new version of this topological index, named Timisoara-eccentricity (TM-EC) index. This topological index can be defined as the summation of (v)(v)(v) over all vertices v, where (v) is the number of atoms at distance (v) from atom v:

TM EC( ) = i i i i=n i=1

(4)

The present authors [32], found closed formulas for sphericality index of four infinite series of fullerenes. In [33, 34], Klavar and his co-workers presented for the first time an interpolation method to obtain exact formulas for topological indices of an infinite family of fullerenes and general nanocones. Their method is efficient for 1- and 2-parametric families of graphs. By the mentioned method, closed formulas for fullerenes and nanocones were presented and several exceptions are pointed out for small values of k. In this paper we use simple interpolation by this assumption that our considering topological indices can be computed by polynomials. It is possible to check validity of our result by above interpolation method. Suppose is a planar graph. The dual of , denoted by Du(), is a graph in which V(Du()) is the set of all faces of . Two faces are adjacent if and only if they have a common edge. Throughout this paper our notation is standard and can be taken from the standard books on this topic [35]. Our calculations

Fig. (1). The Molecular Graph of Nanocone Hn, n 1.

Fig. (2). The Molecular Graph of Dual Nanocone Hn, n 1.

Main Theorem 1: Suppose Hn is a nanocone depicted in Fig. (1), where n denotes the number layers. Then the quantities TM-EC, and E, can be computed as follows: 62 310 4 1205 3 1135 2 2 n5 + n + n + n + 86n +15 3 3 6 6 1. ( H n ) = ; n 11, 20 3 29 2 299 2 n + 28 5n +10n + 5 n + n + 2 6 3

(

)

2.

34 3 57 2 139 n + n + n+6 2 6 ( Hn ) = 3 ; 20 3 29 2 299 n + n + n + 28 2 6 3

3.

TM EC( H n ) = 280n3 + 385n2 +195n + 40,

4.

TM EC( Du( H n )) = 70n3 + 20n2 5n,

E

5. ( Du( H n )) =

n 11,

31 5 155 4 40 3 85 2 3 n + n + n + n + n 12 3 12 2 6 ; n 19, 5 2 5 31 5 155 4 20 3 85 2 3 n + n + n + n n + n +1 n + 2 24 3 24 4 2 12

Topological Invariants of Nanocones and Fullerenes

Current Organic Chemistry, 2015, Vol. 19, No. ??

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7 17 3 n + 3n2 + n 6 6. ( Du( H n )) = 6 n 19 . 5 3 1 2 137 n + n + n 2 6 3 For the sake of completeness, we mention here two results which are crucial throughout this paper. Theorem 1 (See [39, 40]): The Wiener index of the nanocone Hn can be computed by the following formula: E

62 5 310 4 1205 3 1135 2 n + n + n + n + 86n +15. 3 3 6 6 Theorem 2 (See [18]): The eccentric connectivity index of onepentagonal carbon nanocone Hn can be computed by the following formula: W(H n ) =

215 2 155 n + n + 20. 2 2 To prove our main theorem, we first apply our mentioned computer packages to calculate and E for some exceptional cases that n 10. In Table 1, our calculations are recorded. EC( H n ) = 50n3 +

Table 1. The Quantities w , (H n ) and

E (H n ) , 1 n 10.

n

w

(H n )

E (H n )

1

49

1230 = 1.255 980

69 = 1.408 49

2

169

9730 = 1.279 7605

257 = 1.521 169

3

406

41580 = 1.28 32480

638 = 1.571 406

4

800

127710 = 1.277 100000

1280 = 1.6 800

5

1391

318890 = 1.274 250380

2251 = 1.618 1391

6

2219

690690 = 1.27 543655

3619 = 1.631 2219

7

3320

1348440 = 1.269 1062400

5452 = 1.642 3320

8

4724

2432190 = 1.271 1913220

7818 = 1.655 4724

9

6481

4121670 = 1.272 3240500

10785 = 1.664 6481

10

8631

6641250 = 1.272 5221755

14421 = 1.671 8631

From now on, we assume that n 11. By an inductive argument, it is easy to prove that Hn has exactly 5n2 + 10n + 5 vertices. By [18, Lemma 1], the diameter and radius of Hn are 4n + 2 and 2n + 2, respectively. From Fig. (3), it is clear that ten bold vertices have the maximum status. These vertices are having the maximum status in the nanocone. In Fig. (3), the vertices with minimum status are shown by big black and in Fig. (4), the vertices with minimum eccentricity are shown by big black. By considering a vertex with minimum status and methods given in [23, 39], we can see that 20 29 299 w = n3 + n 2 + n + 28. On the hand, if we choose a vertex 3 2 6 with maximum eccentricity, then w =

34 3 57 2 139 n + n + n+6 3 2 6

Fig. (3). Vertices with Minimum Status (Big Black), n 11, and Maximum Status (Black) in a Nanocone.

By calculations given above and definitions of and E, we have: 62 310 4 1205 3 1135 2 n + n + n + 86n +15 2 n5 + 3 6 6 3 ( Hn ) = , 29 299 20 2 3 2 n + 28 5n +10n + 5 n + n + 2 6 3

(

)

34 3 57 2 139 n + n + n+6 2 6 ( Hn ) = 3 . 20 3 29 2 299 n + n + n + 28 2 6 3 E

Fig. (4). Vertices with Minimum Eccentricity (Big Black) and Maximum Eccentricity (Black) in a Nanocone.

On the other hand, from Fig. (5), we can see that a nanocone has three types of vertices; those with (v) = 2 (black), (v) = 11 (ordinary) and (v) = 20 (big black). Using these calculations, we have TM EC( H n ) = 280n3 + 385n2 +195n + 40. We now consider the dual of a nanocone, Fig. (2). Notice that in Hn, two faces are disjoint or has exactly one common edge and so Du(Hn) has exactly 2.5n2 + 2.5n +1 vertices and 7.5n2 + 2.5n edges. From Fig. (6), there are five bold vertices have the maximum status. A simple calculation shows that Du(Hn) has radius n and diameter 2n. Since all coefficients in the equation given minimum and maximum are rational numbers, they are easily computable by interpolation. We have:



Table 2. The Values of w, and E, for n 18.

Fig. (5). Vertices with (v) = 2 (black), (v) = 11 (ordinary) and (v) = 20 (big black).

n

w

E

1

5

1.333333

1.4

2

25

1.275

1.48

3

70

1.258065

1.528571

4

149

1.259376

1.57047

5

271

1.26578

1.605166

6

447

1.26757

1.626398

7

687

1.267511

1.640466

8

998

1.270497

1.655311

9

1391

1.272848

1.667146

10

1877

1.273791

1.675546

11

2466

1.273917

1.681671

12

3168

1.273565

1.686237

13

3993

1.272935

1.689707

14

4951

1.272148

1.692385

15

6052

1.271278

1.694481

16

7298

1.271766

1.697999

17

8706

1.271942

1.700781

18

10286

1.271894

1.702994

Fig. (6). Vertices with Minimum Status (big black), n 19, and Maximum Status (black) in a Dual Nanocone.

17 3 7 n + 3n2 + n, 6 6 5 3 1 2 137 w( Du( H n )) = n + n + n ; n 19. . 3 2 6

w( Du( H n )) =

and Fig. (7). Vertices with Minimum Eccentricity (big black) and Maximum Eccentricity (black) in a Dual of Nanocone.

In Table 2, the values of Du(Hn), n 18, for w, and E are recorded. By a simple interpolation and Figs. (7 and 8), one can see 31 155 4 20 3 85 2 3 that W ( Du( H n )) = n5 + n + n + n + n, 12 24 3 24 4

EC( Du( H n )) = 25n3 +10n2

and

TM EC( Du( H n ))

= 70n + 20n 5n. 3

2

By substituting these quantities on Equations (2) and (3), the Cases 5 and 6 of our main theorem can be proved. 3.TOPOLOGICAL INVARIANTS OF FULLERENES Let be a connected n-vertex graph. Doli et al. [30], proved that the maximum possible value of the eccentric connectivity index of is given by

EC() =

4 3 n + o(n2 ). 27

Fig. (8). Vertices with (v) = 1 (black), (v) = 10 (ordinary) and (v) = 45 (black).



Here, f (n) = o(g(n)) means that for all c > 0 there exists some k>0 such that 0 f(n) < cg(n), for each natural number n, n k. The value of k must not depend on n, but may depend on c. In this work we will do the same for twelve sequences A[n], B[n], C[n], D[n], E[n], F[n], G[n], H[n], I[n], J[n], K[n] and L[n] of fullerene graphs of orders 50 + 10n, 60 + 12n, 10n, 12n, 12n + 2, 12n + 4, 12n + 6, 12n + 8, 12n + 14, 12n + 18, 24n and 24n + 12, respectively. The fullerene graphs A[n] and B[n] are firstly introduced in [8], but other classes of fullerene graphs introduced in [35]. The main theorem of this section is as follows: Main Theorem 2: The EC and TM-EC of A[n], B[n], C[n], D[n], E[n], F[n], G[n], H[n], I[n], J[n], K[n] and L[n] fullerene graph can be evaluated by polynomials of degree 2. We first consider the fullerene series A[n] into account. The fullerene A[n] (an example in Fig. 1) has exactly 50 + 10n vertices, n 2. Some of the present authors [10] proved that for the fullerene series A[n], we have EC( A[2]) = 1920, EC( A[3]) = 2460,

EC( A[4]) = 2850,

EC( A[5]) = 3360,

5

We now consider the fullerene series B[n], n 1, containing 60 +12n carbon atoms, an example in Fig. (10). In [10], it is reported that there are eight exceptional cases for EC values of B[n] as follows: EC( B[1]) = 2088, EC( B[2]) = 2772, EC( B[3]) = 3168, EC( B[4]) = 3960, EC( B[5]) = 4464, EC( B[6]) = 5184, EC( B[7]) = 5832, EC( B[8]) = 6696.

EC( A[6]) = 3870 . For

other values of n, we have

45 2 2175 n + 330n + 2 2 EC( A[n]) = 45 n2 + 330n +1050 2

n is odd & n 7 . n is even & n 8

Fig. (10). The Fullerene B[9].

On the other hand, we have:

27n2 + 432n +1413 n is odd & n 9 EC( B[n]) = . 27n2 + 432n +1476 n is even & n 10 For TM-EC of B[n], there are ten exceptional values as follows: TM EC( B[1]) = 3384, TM EC( B[2]) = 2772, TM EC( B[3]) = 9504, TM EC( B[4]) = 4824, TM EC( B[5]) = 10728, TM EC( B[6]) = 12312, TM EC( B[7]) = 17064, TM EC( B[8]) = 16920, TM EC( B[9]) = 20520, TM EC( B[10]) = 18648.

Fig. (9). The Fullerene A[9].

We now draw the molecular graphs of A[2], A[3], A[3], A[4], A[5] and A[6] by HyperChem and then apply the software TopoCluj and Computer Algebra System GAP for computing EC and TM-EC of fullerenes. We mention here that the EC of A[n], B[n], C[n] and D[n] reported in [10] and we refer to this paper for more information on EC value of fullerenes. The calculation of EC number of E[n] and F[n] together with the TM-EC values of sequences A[n], B[n], C[n], D[n], E[n] and F[n] of fullerenes are the aim of our paper. To compute the TM-EC values of series A[n], Fig. (9), we first notice that there are seven exceptional cases as TM EC TM EC( A[2]) = 6270, TM EC( A[3]) = 3060,

( A[4]) = 7620,

TM EC( A[5]) = 7740,

TM EC( A[6]) =

11700, TM EC( A[7]) = 10500 and TM EC( A[8]) = 13860 . In other cases, we have:

45n2 + 600n +1695 n is odd & n 9 TM EC( A[n]) = . 2 45n + 810n + 3300 n is even & n 10 To compare the EC and TM-EC of the fullerenes in A[n], we compute the limit of EC(A[n])/TM-EC(A[n]), when n tends to infinity. We have:

1 EC( A[n]) Limn = . TM EC( A[n]) 2

In general, we have the following formula: 54n2 + 936n + 3402 n is odd & n 11 TM EC( B[n]) = . 54n2 + 828n + 2808 n is even & n 12 Again, we compare the EC and TM-EC of B[n] by computing EC(B[n]) / TM-EC(B[n]), when n tends to infinity. We have:

Limn

1 EC( B[n]) = . TM EC( B[n]) 2

We now present a sequence C[n] of fullerenes in which the limit of EC/TM-EC is equal to 1/5. The fullerene C[n] (an example in Fig. 11) has exactly 10n carbon atoms, n 2. In [41], it is reported that EC(C[n]) = 45n2 15n, n2. There are eight exceptional cases for TM-EC of C[n]. These are: TM EC(C[2]) = 300, TM EC(C[3]) = 720, TM EC(C[4]) = 1230, TM EC(C[5]) = 3900, TM EC(C[6]) = 7320, TM EC(C[7]) = 10440, TM EC(C[8]) = 14790, TM EC(C[9]) = 17820.

On the other values of m for other values of n, n 10, TM EC(C[n]) = 225n2 75n. Therefore,

Limn

1 EC(C[n]) = . TM EC(C[n]) 5

We now consider the fullerene series D[n] (an example in Fig. 12) with exactly 12n carbon atoms, n 2. By the main result of [41], EC( D[n]) = 45n2 18n. On the other hand, a tedious calculation show that



Fig. (11). The Fullerene C[9].

Fig. (12). The Fullerene D[9].

Fig. (13). The Fullerene E[9].

Fig. (14). The Fullerene F[9]. TM EC( D[2]) = 900, TM EC( D[3]) = 756, TM EC( D[4]) = 1656, TM EC( D[5]) = 1908, TM EC( D[6]) = 7164, TM EC( D[7]) = 12384, TM EC( D[8]) = 18648, TM EC( D[9]) = 24804, TM EC( D[10]) = 32796, TM EC( D[11]) = 38412.

If n 12 then TM EC( D[n]) = 324n2 108n and so,

Limn

5 EC( D[n]) = . TM EC( D[n]) 36

We now present a series of fullerenes with exactly 12n + 2 carbon atoms, Fig. (5). With the best of our knowledge, even the EC of this series is not considered in other papers. So, we first compute the EC of this series. We first name this series E[n], n 3. Then, EC( E[3]) = 798, EC( E[4]) = 1176, EC( E[5]) = 1644,

EC( E[6]) = 2160, EC( E[7]) = 2820, EC( E[8]) = 3588, On the other hand, for n 10, EC(E[n]) = 54n 2 +12n. A tedious calculation on TM-EC of E[n] (an example in Fig. 13) show that, TM EC( E[3]) = 1008, TM EC( E[4]) = 2238, TM EC( E[5]) = 3306, TM EC( E[6]) = 7014, TM EC( E[7]) = 11808, TM EC( E[8]) = 14508, TM EC( E[9]) = 17946, TM EC( E[10]) = 23058, TM EC( E[11]) = 27126.

12,

1 EC( E[n]) = . TM EC( E[n]) 4

The fullerene series F[n] with 12n + 4 carbon atoms, n 2, is our next series of fullerenes that its eccentric connectivity index is considered into account (an example in Fig. 14). Our calculation shows that EC( F[2]) = 456, EC( F[3]) = 840, EC( F[4]) = 1242,

EC( F[5]) = 1743, EC( F[6]) = 2307,

EC( F[6]) = 3015,

and for n 8, EC( F[n]) = 54n + 42n + 21. On the other hand, TM EC( F[n]) = 189n2 + 27n + 36, n 11. For exceptional cases, we have: 2

TM EC( F[2]) = 1476, TM EC( F[3]) = 1512,

TM EC( F[4]) = 2772,

TM EC( F[5]) = 3429, TM EC( F[6]) = 6651,

TM EC( F[7]) = 9135,

TM EC( F[8]) = 12843, TM EC( F[9]) = 16326, TM EC( F[10]) = 19404.

EC( E[9]) = 4500.

For other values of n, n TM EC(E[n]) = 216n2 + 72n. Therefore,

Limn

we

have

This implies that

Limn

2 EC( F[n]) = . TM EC( F[n]) 7

Our next series of fullerenes, G[n], is containing fullerenes with 12n + 6 carbon atoms, Fig. (15). The eccentric connectivity indices of the first six terms of this series can be computed as follows: EC(G[3]) = 882, EC(G[4]) = 1296, EC(G[5]) = 1806, EC(G[6]) = 2382, EC(G[7]) = 3102, EC(G[8]) = 3930.



7

Fig. (15). The Fullerene G[9].

Fig. (16). The Fullerene H[9].

Fig. (17). The Fullerene Graph I[9].

For n 9, we can see that EC(G[n]) = 54n2 + 54n + 24. On the other hand, for the TM-EC, we have eight exceptional cases as follows:

For other values of n, we have EC(I[n]) = 54n2 + 120n + 66 and TM-EC(I[n]) = 108n2 + 252n + 144. Therefore,

TM EC(G[3]) = 1092, TM EC(G[4]) = 1839, TM EC(G[5]) = 2430,

Limn

TM EC(G[6]) = 4338,TM EC(G[7]) = 5688, TM EC(G[8]) = 7875, TM EC(G[9]) = 9864, TM EC(G[10]) = 11688.

For other values of n, we have TM-EC(G[n]) = 108n2 + 72n +36. This implies that,

1 EC(G[n]) Limn = . TM EC(G[n]) 2 The fullerene series H[n] is the 8th series of fullerenes that contains 12n + 8, n 3, carbon atoms, Fig. (16). There are three exceptional cases for EC and five exceptional cases for TM-EC as follows: EC( H[3]) = 990, EC( H[4]) = 1398, EC( H[5]) = 1950, TM EC( H[3]) = 693, TM EC( H[4]) = 1617, TM EC( H[5]) = 1884, TM EC( H[6]) = 2358, TM EC( H[7]) = 2112.

For other values of n, we have EC(H[n]) = 54n2 + 102n + 54 and TM-EC(H[n]) = 27n2 + 51n + 27. Therefore,

Limn

EC( H[n]) = 2. TM EC( H[n])

Our 9th series of fullerenes has exactly 12n + 14 carbon atoms, n 2. We use the notation I[n] for this series of fullerenes, Fig. (17). The eccentric connectivity index of the first seven terms of this series is as follows: EC(I[2]) = 798, EC(I[3]) = 1176, EC(I[4]) = 1644, EC(I[5]) = 2160, EC(I[6]) = 2820, EC(I[7]) = 3588, EC(I[8]) = 4500.

The first nine values of TM-EC of these fullerenes are: TM EC( I[2]) = 504, TM EC( I[3]) = 1119, TM EC( I[4]) = 1653, TM EC( I[5]) = 3507, TM EC( I[6]) = 5094,TM EC( I[7]) = 7254, TM EC( I[8]) = 8973, TM EC( I[9]) = 11529, TM EC( I[10]) = 13563.

1 EC( I[n]) = . TM EC( I[n]) 2

Our next series of fullerenes, J[n], has exactly 12n + 18 carbon atoms, n 2. An example of these fullerenes is depicted in Fig. (18). The exceptional cases for EC and TM-EC are as follows: EC(J[2]) = 882, EC(J[3]) = 1296, EC(J[4]) = 1806, EC(J[5]) = 2382, EC(J[6]) = 3102, EC(J[7]) = 3930. TM EC( J[2]) = 1092, TM EC( J[3]) = 1839, TM EC( J[4]) = 2430, TM EC( J[5]) = 4338, TM EC( J[6]) = 5688, TM EC( J[7]) = 7875, TM EC( J[8]) = 9864, TM EC( J[9]) = 11688.

For other values of n, EC and TM-EC can be computed by the following formulas: EC(I[n]) = 54n2 + 162n + 132 and TM-EC(I[n]) = 108n2 + 288n + 216. Therefore,

Limn

1 EC( J[n]) = . TM EC( J[n]) 2

We are now consider a fullerene series, K[n], that its general term has exactly 24n carbon atoms, n 2. In Fig. (19), the fullerene K[9] is depicted. The exceptional cases for EC and TM-EC are as follows: EC( K[2]) = 1080, EC( K[3]) = 2088, EC( K[4]) = 3168, EC( K[5]) = 4464, EC( K[6]) = 5832.

TM EC( K[2]) = 1044, TM EC( K[3]) = 1692, TM EC( K[4]) = 4752, TM EC( K[5]) = 5364, TM EC( K[6]) = 8532, TM EC( K[7]) = 10260.

For other values of n, EC and TM-EC can be computed by the following formulas: EC(I[n]) = 108n2 + 324n - 72 and TM-EC(I[n]) = 108n2 + 396n + 36.



Fig. (18). The Fullerene Graph J[9].

Fig. (19). The Fullerene Graph K[9].

Therefore,

Limn

EC( K[n]) = 1. TM EC( K[n])

Our final series of fullerenes, L[n], has exactly 24n + 12 carbon atoms, n 2. In Fig. (20), the fullerene L[9] is depicted. The exceptional cases for EC and TM-EC are as follows: EC(L[2]) = 1620, EC(L[3]) = 2772, EC(L[4]) = 3960, EC(L[5]) = 5184, EC(L[6]) = 6696, TM EC( L[2]) = 972, TM EC( L[3]) = 1386, TM EC( L[4]) = 2412, TM EC( L[5]) = 6156, TM EC( L[6]) = 8460, TM EC( L[7]) = 9324.

Apart from these exceptional cases, one can compute polynomials of degree two to compute EC and TM-EC of these fullerenes as follows: EC(I[n]) = 108n2 + 432n + 180 and TM-EC(I[n]) = 108n2 + 396n + 180. Therefore,

Limn

EC( L[n]) = 1. TM EC( L[n])

The maximum eccentricity in a fullerene graph F, diam(F), is called diameter of F. Andova et al. [42] proved that in each nvertex fullerene graph F, diam(F) n/5 + 1. So, EC(F) = vV(F) (v)(v) = 3 vV(F)(v) 3n(n/5 + 1). Thus we can assume that in fullerenes EC(F) A polynomial of degree 2 in terms of the number of vertices. (5) Our calculations given this paper are also confirmed (1). On the other hand, our computations with TM-EC suggest the following conjectures: Conjecture 1: In each fullerene F, i(F) 12, 1 i n. We could not find a fullerene F such that i(F) = k, when k = 10, 12. But for other values k, 1 k 12, we have a fullerene in which for at least a vertex i, i(F) = k. On the other hand, by this assumption that our conjecture is correct, we have: TM-EC(F) = vV(F) (v)(v)(v) = 3 vV(F)(v)(v) 3kn(n/5 + 1), where k is a fixed number not related to n. Our calculations given this paper are confirmed that TM-EC is a polynomial of degree 2 in terms of n. We record this observation as Conjecture 2. Conjecture 2: The TM-EC of each fullerene is a polynomial of degree 2 in terms of the number of vertices. 5. APPENDIX A: SOME GAP PROGRAM FOR CALCULATION EC AND TM-EC OF FULLERENES GAP Groups, Algorithm and Programming is an open source package for working with finite groups and its applications. In our work, we use the version GAP 4.r.7 for calculation of symmetry, EC and TM-EC of fullerenes. To do this, we first draw a fullerene by HyperChem and then we load the .hin file into TopoCluj software prepared by Mircea Diudea and his team. Finally, we prepare some GAP codes for calculating the symmetry, EC and TM-EC of fullerenes by their distance matrices. Finally, we apply some ordinary interpolation method to find formula for these topological indices. Our GAP programs used in this paper for calculating EC and TM-EC indices are given in this section. A Gap Program for Computing TM-EC

A Gap Program for Computing EC

ect:=function(D)

ME:=function(g,D)

4. CONCLUSION

local aa,a,b,deg,i,m,j,p,s,bb,ECT,h,k,tt;

local h,u,j,m,max,deg,A,EA,p,k,l,i,s;

In this paper, exact formulas for computing EC, TM-EC, and E for a pentagonal nanocones and its dual are computed. Our calculations show that in a one pentagonal carbon nanocone with exactly 5n2 + 10n + 5 carbon atoms, we have 1.24, E 1.7 and TM - EC = 280n3 + 385n2 + 195n + 40. We also examine the dual of this nanocone and prove that we have again 1.24, E 1.7, but TM - EC = 70n3 + 20n2 - 5n.

a:=[];b:=[];tt:=[];deg:=[];

h:=Elements(g);;A:=[];EA:=[];

for i in [1..Size(D)] do

for u in [1..Size(D)] do

m:=Maximum(D[i]);

for j in h do

Add(a,m);Add(tt,i);

m:=D[u][u^j];;

od;

Add(A,m);

for k in [1..Size(D)] do

od;

j:=0;p:=0;

max:=Maximum(A);;Add(EA,max);

Fig. (20). The Fullerene Graph L[9].


Current Organic Chemistry, 2015, Vol. 19, No. ?? [12]

for h in [1..Size(D)] do

od;

if D[k][h]=a[k] then j:=j+1; fi;

deg:=[];

if D[k][h]=1 then

for k in [1..Size(D)] do

p:=p+1;

p:=0;

[14]

fi;

for l in [1..Size(D)] do

[15]

od;

if D[k][l]=1 then

[16]

Add(b,j);Add(deg,p);

p:=p+1;

od;

fi;

s:=0;

od;


Add(deg,p);

s:=s+a[i]*b[i]*deg[i];

od;

od;

s:=0;

[20]

ECT:=s;bb:=Set(b);aa:=Set(a);


[21]

Print(b,"\n");

s:=s+EA[i]*deg[i];

return ECT;

od;

end;

return s;end;

[13]

[17] [18]

[19]

[22]

[23]

CONFLICT OF INTEREST [24]

The authors confirm that this article content has no conflict of interest.

[25]

ACKNOWLEDGEMENTS

[26]

The authors are indebted to the referees for their suggestions and helpful remarks. The first and second authors are partially supported by the University of Kashan under grant number 364988/3. OO kindly thank Biology-Chemical Department of West University of Timisoara for hospitality during his visit on spring of 2014 at Laboratory of Computational and Structural Physical-Chemistry for Nanosciences and QSAR of MVP by which the TM-EC idea was emerging.

[2] [3]

[4]

[5] [6]

[7] [8]

[9]

[10]

[11]

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Received: October 12, 2014

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Revised: November 17, 2014

Accepted: December 14, 2014