Topological properties of some nanostructures Текст научной статьи по специальности «Математика»

Topological properties of some nanostructures

S. Mondal1, A. Bhosale1, N. De2, A. Pal1

department of mathematics, NIT Durgapur, India, 2Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India

[email protected], [email protected], [email protected], [email protected]

DOI 10.17586/2220-8054-2020-11-1-14-24

Topological indices are numerical values associated with chemical constitution describing the structures of chemical compounds and helping to predict different physicochemical properties. In this report, some newly designed topological descriptors, namely, neighborhood Zagreb index (Mn ), neighborhood version of Forgotten topological index (Fn ), modified neighborhood version of Forgotten topological index (FN), neighborhood version of second Zagreb index (M|), neighborhood version of hyper Zagreb index (HMn) are obtained for the TURC4C8 (S), armchair nanotube TUAC6, V-phenylenic nanotube VPHX[m, n], and V-phenylenic nanotori VPHY[m,n].

Keywords: Topological indices, TURC4C8(S), armchair nanotube (TUAC6), V-phenylenic nanotube (VPHX[m, n]), V-phenylenic nanotori (VPHY [m, n]). Received: 30 December 2019 Revised: 3 January 2020

1. Introduction

We consider only molecular graphs throughout this article. By molecular graph [1-3], we mean a simple connected graph in which nodes are supposed to be atoms and edges are chemical bonds. The vertex and edge sets of a graph G are represented here by V(G) and E(G), respectively. The degree of a vertex v on a graph G, denoted by dG(v), is the total number of edges associated with v. Moreover, we define

Sg(v) = degG(u),

ueNa(v)

where

NG(v) = {u € V(G) : uv € E(G)}.

The chemical graph theory has a significant impact on the chemical science development. Chemical graph theory is a part of mathematical chemistry that uses graph theory for mathematically modeling chemical phenomena. In this field, a leading tool is topological index. A real valued mapping considering graphs as arguments is called a graph invariant if it gives same value to isomorphic graphs. In chemical graph theory, the graph invariants are named as topological indices. Topological indices play key role in QSPR/QSAR study. Topological indices interpret chemical compound structures and help to predict certain physicochemical properties such as entropy, boiling point, acentric factor, vaporization enthalpy, etc. Among different types of topological indices, degree based topological indices have prominent role in this research area. For some well-known degree based topological indices, readers are referred to [4-8]. In [9,10], some new neighborhood degree based indices are presented having good correlations with entropy and acentric factor. They are defined as follows.

The neighborhood Zagreb index is denoted by MN (G) and is defined as:

Mn (G)= ^ Sg (v)2.

vev (G)

Neighborhood version of Forgotten topological index is denoted by FN (G) and is defined as:

Fn (G)= ^ Sg(V)3. vev (G)

Modified neighborhood version of Forgotten topological index is denoted by FN (G) and is defined as:

FN (G)= E [Sg(U)2 + Sg(V)2].

uveE(G)

Neighborhood version of second Zagreb index is denoted by M| (G) and is defined by:

M2*(G)= £ [SG(u)SG(v)].

uveE(G)

Neighborhood version of hyper Zagreb index is denoted by HMN (G) and is defined by:

HMN (G) = £ Mu) + Mv)]2.

uveE(G)

A nanostructure is an intermediate object between microscopic and molecular structures. It is a molecular-scale product obtained from engineering. The most important class of such materials is the carbon nanotubes. Carbon nanotubes (CNTs) are carbon allotropes with cylindrical molecular structures, having diameters ranging from a few nanometers and lengths to several millimeters. Nanotubes are categorized as single-walled (SWNTs) and multi-walled (MWNTs) nanotubes. Researchers have found topological descriptors for various nanotube and nanotori. In [11], topological properties of TURC4C8(S) are investigated. Jiang et al. determined topological index of V-phenylenic nanotubes and V-phenylenic nanotori in [12]. Topological properties of armchair polyhex nanotube are discussed in [13]. For more discussion on this topic, readers are referred to [14-21]. Inspired by these works, we have derived MN, FN, F*N, M|, and HMN indices for TURC4C8(S), armchair polyhex nanotube TUAC6, V-phenylenic nanotube VPHX [m, n], and V-phenylenic nanotori VPHY[m, n]. Moreover, we have compared these indices graphically.

2. Motivation

The correlation coefficient (r) of topological indices with different physicochemical properties for a benchmark data set is determined to check the utility of the indices in QSPR/QSAR analysis. According to the International Academy of Mathematical chemistry, an index is considered to be useful if r2 > 0.8. In [9, 10], the chemical applicability of the indices MN, FN, FN, M2*, and HMN are studied taking octane isomers as data set. The r2 values of those indices with entropy are 0.907, 0.88, 0.868, 0.899 and 0.88, respectively. The r2 values of those indices with acentric factor are 0.989, 0.989, 0.952, 0.971 and 0.961, respectively. The aforesaid indices are therefore effective in QSPR/QSAR analysis with powerful accuracy. In addition, their isomer discrimination ability also remarkable [9,10] in comparison with the other degree based indices. With the help of nanotechnology, many new materials and devices are in progress with a wide range of applications in medicine, electronics and computers. Motivated by the importance of topological indices and the nanotechnology, we intend to compute the aforementioned indices for some nano-structures which are described in the next section.

3. Preliminaries

The 2D and 3D lattice of TURC4C8(S) nanotube are shown in Fig. 1. We consider mn numbers of C8 and C4 cycles in the 2D lattice of TURC4C8(S) nanotube. We denote this graph by TUC4C8 [m, n]. From Fig. 1, it is clear that this graph has 8mn + 2m nodes and 12mn + m edges.

□ n n

V / \

n —[mn-m] [iv i-m I I r

.i.^iCFvX

\_/ \_/ w

(a)

FIG. 1. (a) The 2D and (b)the 3D lattice of TUC4C8 [m, n] nanotube

We consider a class of armchair polyhex nano tubes TUAC6 [m, n] having m and n numbers of hexagons in each rows and columns, respectively. The molecular graph of TUAC6 [m, n] is depicted in the Fig. 2. We can say from Fig. 2, that m is even for all n G N. This nanotube has 2mn + 2m and 3mn + 2m numbers of nodes and edges respectively.

Fig. 2. (a) The 2D and (b)the 3D lattice of Armchair polyhex nanotubes TUAC6

Also we consider V-phenylenic nanotube and V-phenylenic nanotori whose 2D lattices are depicted in Figs. 3 and 4. Phenylenes are polycyclic conjugated molecules, made of C4 and C6 such that every C4 is adjacent to two C6 and lies between two C6. No two C6 are mutually adjacent. Each C6 is adjacent to only two C4 cycles.

Fig. 3. The molecular graph of V-phenylenic nanotube VPHX [m, n]

Fig. 4. The molecular graph of V-phenylenic nanotori VPHY[m, n]

4. Main results

In this section, our goal is to compute aforesaid indices for TUC4C8[m, n], armchair polyhex nanotube TUAC6, V-phenylenic nanotube VPHX[m, n], and V-phenylenic nanotori VPHY[m, n]. Following Figs. 1, 2, 3, and 4, first we obtain vertex and edge partitions of nanotubes and nanotori discussed above and then proceed for main theorems.

The vertex and edge partitions for TUC4C8[m, n] nanotube are given in Table 1 and 2, respectively. The vertex and edge partitions for TU AC6 [m, n] nanotube are given in Table 3 and 4, respectively. The vertex and edge partitions for VPHX [m, n] nanotube are given in Table 5 and 6, respectively.

Table 1. Vertex partition of TUC4C8[m, n]

Sg(u) 5 8 9

Frequency 4m 4m 8mn - 6m

Table 2. Edge partition of TUC4C8 [m, n]

(Sg (u), Sg(v)) (5, 5) (5, 8) (8, 8) (8, 9) (9, 9)

Frequency 2m 4m 2m 4m 12mn - 11m

Table 3. Vertex partition of TUAC6 [m, n]

Sg(u) 5 8 9

Frequency 2m 2m 2m(n - 1)

Table 4. Edge partition of TUAC6 [m, n]

(Sg(u), Sg(v)) (5, 5) (5, 8) (8, 8) (8, 9) (9, 9)

Frequency m 2m m 2m m(3n - 4m)

Table 5. Vertex partition of VPHX[m, n]

Sg(u) 6 8 9

Frequency 2m 4m 6mn - 6m

Table 6. Edge partition of VPHX[m, n]

(Sg(u), Sg(v)) (6, 8) (8, 8) (8, 9) (9, 9)

Frequency 4m 2m 2m 9m(n - 1)

Theorem 1. The neighborhood Zagreb index MN of TU C4C8 [m, n] (m, n > 2), TU AC6 [m, n], VPHX [m, n], and VPHY[m, n] nanotubes are given by:

(i) MN(TUC4C8[m, n]) = 648mn - 130m,

(ii) MN(TUAC6[m,n]) = 162mn - 16m,

(iii) MN(VPHX[m,n]) = 486mn - 158m,

(iv) MN(VPHY[m, n]) = 486mn.

Proof. The general formula of neighborhood Zagreb index MN is given by:

Mn (G) = £ Sg (v)2.

vev (G)

(i) Let G be the TUC4C8[m,n] nanotube for (m,n > 2). Then applying the Table 1 on the definition of neighborhood Zagreb index, we obtain:

Mn(G) = £ Sg(V)2 + £ Sg(V)2 + £ Sg(V)2

veV5 veVs veVg

= |V5|(52) + |V8|(82) + |VQ|(92) = 4m(52)+ 4m(82) + (8mn - 6m)(92) = 648mn 130m.

(ii) Let G be the V-phenylenic nanotube (TUAC6[m, n]). Then applying the Table 3 on the general formula of neighborhood Zagreb index, we have:

MN(G) = £ SG(v)2 + £ SG(v)2 + £ SG(v)2 vev5 vevs veVg

= \V5\(52) + |V8|(82) + \V9\(92) = 2m(62) + 2m(82) + 2m(n — 1)(92) = 486mn — 158m.

(iii) Let G be the V-phenylenic nanotube (VPHX[m, n]). Then applying the Table 5 on the general formula of neighborhood Zagreb index, we have:

Mn(G) = £ Sg(V)2 + £ Sg(V)2 + £ Sg(V)2 vev6 vevs vevg

= \Ve\(62) + \V8\(82) + \V9\(92) = 2m(62) + 4m(82) + (6mn — 6m)(92) = 486mn — 158m.

(iv) Let G be the V-phenylenic nanotori (VPHY[m, n]). Its clear from Fig. 3 that V(G) = V9 and \ V9 \ = 6mn. The required result follows clearly from the definition of neighborhood Zagreb index.

□

Fig. 5. Topological indices for TUC4C8 [m, n] nanotube

Theorem 2. The neighborhood version of Forgotten topological index FN of TU C4C8 [m, n] (m, n > 2), TU AC6 [m, n], VPHX [m, n], and VPHY[m, n] nanotubes are given by:

(i) Fn(TUC4C8[m, n]) = 5832mn — 1826m,

(ii) Fn(TUAC6[m, n]) = 1458mn — 184m,

(iii) Fn(VPHX[m, n]) = 4374mn — 1894m,

(iv) Fn(VPHY[m, n]) = 4374mn,

Proof. The general formula of neighborhood version of Forgotten topological index FN is given by:

Fn (G)= £ Sg(V)3. vev (g)

(i) Let G be the TUC4C8[m,n] nanotube for (m,n > 2). Then applying the Table 1 on the definition of neighborhood version of Forgotten topological index, we obtain:

Fn(G) = £ Sg(V)3 + £ Sg(V)3 + £ Sg(V)3 vev5 vevs vev9

= \V5\(53) + \V8\(83) + \Vq\(93) = 4m(53) + 4m(83) + (8mn — 6m)(93) = 5832mn — 1826m.

(ii) Let G be the V-phenylenic nanotube (TUAC6[m, n]). Then applying the Table 3 on the general formula of neighborhood version of Forgotten topological index, we have:

Fn(G) = £ ¿G(v)3 + £ Mv)3 + £ Mv)3

veV5 veVs veVg

= V |(53) + |V8|(S3) + |Vg|(93) = 2m(53) + 2m(S3) + 2m(n - 1)(93) = 145Smn — 1S4m.

(iii) Let G be the V-phenylenic nanotube (VPHX[m, n]). Then applying the Table 5 on the general formula of neighborhood version of Forgotten topological index, we have:

FN(G) = £ ¿G(v)3 + £ Mv)3 + £ Mv)3

veVe veVs veVg

= V1(63) + |V8|(S3) + |Vg|(93) = 2m(63) + 4m(S3 ) + (6mn — 6m)(93) = 4374mn — 1S94m.

(iv) Let G be the V-phenylenic nanotori (VPHY[m, n]). From Fig. 3, we have, V(G) = Vg and |Vg| = 6mn. Using the definition of neighborhood version of Forgotten topological index, the desired result can be obtained easily.

□

Theorem 3. The modified neighborhood version of Forgotten topological index FN of TUC4C8[m, n] (m, n > 2), TUAC6[m, n], VPHX [m, n], and VPHY[m, n] nanotubes are given by:

(i) FN(TUC4C8[m, n]) = 1944mn - 490m,

(ii) FN (TUAC6 [m, n]) = 486mn - 2m,

(iii) F;^(VPHX[m,n]) = 1458mn - 512m,

(iv) F^ (VPHY[m, n]) = 1458mn.

Proof. The general formula of modified neighborhood version of Forgotten topological index FN is given by:

F^ (G) = £ [Sg(U)2 + Sg(V)2].

uveE(G)

(i) Let G be the TU C4C8 [m, n] nanotube for (m, n > 2). Then applying the Table 2 on the definition of modified neighborhood version of Forgotten topological index, we obtain:

FN (G) = £ Mu)2 + ¿g (v)2]+ £ (u)2 + ¿g(v)2] + £ Mu)2 + ¿g(v)2]

Mv£E(5,5) MveE(5,8) «veE(B,s)

+ £ Mu)2 + ¿g(v)2] + £ Mu)2 + ¿g (v)2]

«v£E(8,9) «v£E(9,9)

= |E(5,5)|(52 + 52) + |E(5,8)|(52 + 82) + |E(8,8)|(82 + 82) + |E(8,9)|(82 + 92) + |E(9,9)|(92 + 92)

= 2m(52 + 52) + 4m(52 + 82) + 2m(82 + 82) + 4m(82 + 92) + m(12n - 11)(92 + 92).

After simplification the desired result can be obtained easily.

(ii) Let G be the TU AG6 [m, n] nanotube. Then applying the Table 4 on the definition of modified neighborhood version of Forgotten topological index, we obtain:

FN (G) = £ [¿g(u)2 + ¿G (v)2] + £ [¿G (u)2 + ¿g(v)2]+ £ [¿g(u)2 + ¿g(v)2]

Uv£E(5,5) Mv£E(5,8) U«eE(8,8)

+ £ [¿g(u)2 + ¿g(v)2]+ £ [¿g(u)2 + ¿G (v)2]

= |E(5,5)|(52 + 52) + |E(5,8)|(52 + 82) + |E(8,8)|(82 + 82) + |E(8,9)|(82 + 92) + |E(9,9)|(92 + 92)

= m(52 + 52) + 2m(52 + 82) + m(82 + 82) + 2m(82 + 92) + m(3n - 4)(92 + 92).

After simplification the required result can be obtained easily.

(iii) Let G be the VPHX [m, n] nanotube. Then applying the Table 6 on the definition of modified neighborhood version of Forgotten topological index, we obtain:

FN (G) = £ [¿G(u)2 + ¿g(v)2]+ £ [¿g (u)2 + ¿g (v)2]

u»EE(6,8) u«eE(8,8)

+ £ [¿g(u)2 + ¿G (v)2] + £ [¿g(u)2 + ¿g(v)2]

= |E(6,8)|(62 + 82) + |E(8,8)|(82 + 82) + |E(8,9)|(82 + 92) + |E(9,9)|(92 + 92) = 4m(62 + 82) + 2m(82 + 82) + 2m(82 + 92) + 9m(n - 1)(92 + 92).

After simplification the required result can be obtained easily. (iv) Let G be the VPHY[m, n] nanotube. From Fig. 3, it is clear that E(G) = E(9,9) and |E(9,9) | = 9mn. Thus, we have FN (G) = 9mn(92 + 92) = 1458mn

Hence the proof.

□

Theorem 4. The neighborhood version of second Zagreb index M| of TUG4G8[m,n] (m, n > 2), TUAC6[m,n], VPHX [m, n], and VPHY[m, n] nanotubes are given by:

(i) M|(TUG4G8[m,n]) = 972mn - 265m,

(ii) M| (TUAG6 [m, n]) = 243mn - 11m,

(iii) M|(vPHX [m,n]) = 729mn - 265m,

(iv) M|(vPHY[m, n]) = 729mn.

Proof. The general formula of neighborhood version of second Zagreb index M| is given by:

M2(G)= £ [¿g (u)¿G (v)].

uveE(G)

(i) Let G be the TUG4G8[m, n] nanotube for (m, n > 2). Then applying the Table 2 on the general form of

neighborhood version of second Zagreb index, we obtain the following computation.

M(G) = ]T MuMg(V)] + E [¿G(U)^G(V)] + [5G(u).5G(v)j

««££(5,5) uveE(5j8) u«£E(8,8)

+ E MuMc(v)] + E [¿G (u).^G(v)]

(M).0G(V)

««££(8,9) ««££(9,9)

= |E(5,5) |(5.5) + |E(5,8) |(5.8) + |E(8,8) | (8.8) + |E(8,9) | (8.9) + |E(9,9) | (9.9) = 2m(5.5) + 4m(5.8) + 2m(8.8) + 4m(8.9) + (12mn - 11m)(9.9).

After simplification, the desired result can be easily obtained.

(ii) Let G be the TUAC6 [m, n] nanotube. Then, applying the Table 4 on the definition of neighborhood version of second Zagreb index, we get the following derivation:

M(G) = E [¿g(u).M*)]+ E [Mu).M*)]+ E Mu)^g(v)]

««££(5,5) ««££(5,8) ««££(8,8)

+ E [¿g(u)^g(v)] + E [¿G (u).^G(v)]

««££(8,9) ««££(9,9)

= |E(5,5) |(5.5) + |E(5,8) |(5.8) + |E(8,8) | (8.8) + |E(S,9) | (8.9) + |E(9,9) | (9.9) = 4m(6.8) + 2m(8.8) + 2m(8.9) + 9m(n - 1)(9.9) = 243mn — 11m.

(iii) Let G be the VPHX [m, n] nanotube. Then, applying the Table 6 on the definition of neighborhood version of second Zagreb index, we get the following derivation:

M(G) = E [¿G (u) .¿G (v)] + E [¿G (u) .¿G (v)]

««££(6,8) ««££(8,8)

+ E [¿G (u).¿G(v)]+ E [¿G (u) .¿G (v)]

««££(8,9) ««££(9,9)

= |E(6,8) | (6.8) + |E(8,8) | (8.8) + |E(8,9) | (8.9) + |E(9,9) | (9.9) = 4m(6.8) + 2m(8.8) + 2m(8.9) + 9m(n — 1)(9.9) = 729mn — 265m.

(iv) Let G be the VPHY[m, n] nanotube. From Fig. 3, it is clear that E(G) = E(9,9) and |E(9,9) | = 9mn. Thus, we have FN(G) = 9mn(9.9) = 729mn.

Hence the proof. □

Fig. 7. Topological indices for VPHY[m, n] nanotori.

Theorem 5. The neighborhood version of hyper Zagreb index HMN of TU C4C8 [m, n] (m, n > 2), TU AC6 [m, n], VPHX [m, n], and VPHY[m, n] nanotubes are given by:

(i) HMN(TUC4C8[m,n]) = 3888mn - 1020m,

(ii) HMN(TUAC6[m,n]) = 972mn - 24m,

(iii) HMN(VPHX [m,n]) = 2916mn - 1042m,

(iv) HMN(vphy[m, n]) = 2916mn.

Proof. The general formula of neighborhood version of hyper Zagreb index HMN is given by:

hmn (G) = E m«0 + mv)]2.

««££(G)

(i) Let G be the TUC4C8[m,n] nanotube for (m, n > 2). Then applying Table 2 on the general form of neighborhood version of hyper Zagreb index, we obtain the following computation:

hmn (G) = E M«0 + ¿g(v)]2 + E M«0 + ¿g(v)]2 + E M«0 +

««££(5,5) ««££(5,8) ««££(8,8)

¿g(v)]2 + E Mu) + ¿g(v)]2 + E Mu) + ¿g(v)]2

««££(8,9) ««££(9,9)

= |E(5,5)|(5 + 5)2 + |E(5,8)|(5 + 8)2 + |E(8,8) | (8 + 8)2 + |E(8,9)|(8 + 9)2 + |E(9,9) | (9 + 9)2

= 2m(5 + 5)2 + 4m(5 + 8)2 + 2m(8 + 8)2 + 4m(8 + 9)2 + (12mn - 11m)(9 + 9)2.

After simplification the desired result can be obtained easily.

(ii) Let G be the TUAC6 [m, n] nanotube. Then applying the Table 4 on the definition of neighborhood version of second Zagreb index, we get the following derivation:

hmn (G) = E Mu)+ ¿g (v)]2 + E Mu) + Mv)]2 + E Mu) + Mv)]2

««££(5,5) ««££(5,8) ««££(8,8)

+ E [¿g(u) + ¿g(v)]2 + E [¿g(u) + ¿g(v)]2

««££(8,9) ««££(9,9)

= |E(5,5) | (5 + 5)2 + |E(5,8)|(5 + 8)2 + |E(8,8) | (8 + 8)2 + |E(8,9)|(8 + 9)2 + |E(9,9)| (9 + 9)2

= 4m(5 + 5)2 + 4m(5 + 8)2 + 2m(8.8) + 2m(8 + 9)2 + 9m(n - 1)(9 + 9)2 = 972mn - 24m.

(iii) Let G be the VPHX [m, n] nanotube. Then applying the Table 6 on the definition of neighborhood version of second Zagreb index, we get the following derivation:

hmn (G) = E [¿g(u) + ¿g(v)]2 + E [¿g(u) + ¿g(v)]2 + E [¿g(u)

««££(6,8) ««££(8,8) ««££(8,9)

+^g(v)]2 + E [¿g(u)+ ¿g(v)]2

««££(9,9)

= |E(6,8)|(6 + 8)2 + |E(8,8) | (8 + 8)2 + |E(8,9)|(8 + 9)2 + |E(9,9)|(9 + 9)2 = 4m(6 + 8)2 + 2m(8.8) + 2m(8 + 9)2 + 9m(n - 1)(9 + 9)2 = 2916mn - 1042m.

(iv) Let G be the VPHY[m, n] nanotube. From Fig. 3, it is clear that E(G) = E(9,9) and |E(9,9) | = 9mn. Thus, we have FN(G) = 9mn(9 + 9)2 = 2916mn

Hence the proof.

□

3.5 x 10^ 3. x 10*^ 2.5 x 10^ 2. x l(fi 1.5 x 10^ 1. X lO6^ 5. x 10ri

Fig. 8. Topological indices for TUAC6[m, n] nanotube.

The surface plotting of topological indices for the nanotubes and nanotori are shown in the Figs. 5, 6, 7, and 8. We have built the figures using Maple 2015.1 software taking the parametric values (m, n) in [2, 50]. For different indices, different colors are used. We put cyan, blue, green, red and gold colors for MN, FN, FN, M2*, and HMN indices respectively.

5. Remarks and conclusion

In this article, the structures of TURC4C8(S), armchair polyhex nanotube TUAC6, V-phenylenic nanotube VPHX[m, n], and V-phenylenic nanotori VPHY[m,n] are discussed and explicit expressions of MN, FN, FN, M2*, and HMN are derived for them. In fact, comparison among these indices for the considered nanotubes and nanotori are shown in the Figs. 5,6,7, and 8. Clearly, the indices for different nanotubes and nanotori are growing in the following order.

TUAC6[m, n] < VPHY[m,n] < VPHX[m,n] < TURC4C8(S), where in each case, indices have following order.

mn < M| < FN < hmn < fn .

Thus, for each structure discussed above, the indices behave somewhat differently. The formulas obtained here enable the chemical structure of nano structures to be correlated with a large amount of information about their physicochemical characteristics.

Acknowledgments

The first author is very obliged to the Department of Science and Technology (DST), Government of India for the Inspire Fellowship [IF170148].

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