THE F-INDEX AND COINDEX OF V-PHENYLENIC NANOTUBES AND NANOTORUS AND THEIR MOLECULAR COMPLEMENT GRAPHS Текст научной статьи по специальности «Математика»

The F-index and coindex of V-Phenylenic Nanotubes and Nanotorus and their molecular complement graphs

Mohammed S. Y. Alsharafi1, Abdu Q. Alameri2 1 Graduate School of Natural and Applied Sciences, Yildiz Technical University, Istanbul, Turkey 2Department of Biomedical Engineering, University of Science and Technology, Sana'a, Yemen alsharafi205010@gmail.com, a.alameri2222@gmail.com

DOI 10.17586/2220-8054-2021-12-3-263-270

The forgotten topological index was defined to be used in the analysis of chemical structures which often appear in drug molecular graphs. In this paper, we studied the F-index and F-coindex for certain important physico chemical structures such as V-Phenylenic Nanotube VPHX [m, n] and V-Phenylenic Nanotorus VPHY [m, n] and their molecular complement graph. Moreover, we computed F-polynomial of the V-Phenylenic Nanotubes and Nanotorus. These explicit formulae can correlate the chemical structure of molecular graphs of Nanotubes and Nanotorus to information about their physicochemical structure.

Keywords: F-index, F-coindex, V-Phenylenic Nanotubes and Nanotori, molecular graph, molecular complement graph. Received: 10 April 2021 Revised: 18 April 2021

1. Introduction

Chemical graph theory is a part of mathematical chemistry that uses graph theory for mathematically modeling chemical phenomena. Chemical graphs are models of molecules in which atoms are represented by vertices and chemical bonds by edges of a graph. The topological indices explain chemical compound structures and help to predict certain physicochemical properties such as entropy, boiling point, acentric factor, vaporization enthalpy, etc [1]. We denote the V-Phenylenic nanotubes and nanotorus by VPHX [m, n], and VPHY[m, n] respectively, where m and n are the number of atoms in rows and columns. Many well-known topological indices of the V-Phenylenic nanotubes and nanotorus have been computed. The forgotten index one of the most important topological indices which preserve the symmetry of molecular structures and provide a mathematical formulation to predict their physical and chemical properties [2]. In this article, in view of structure analysis and mathematical derivation, we find the F-index and coindex of certain molecular graphs nanotubes and nanotorus that are interesting molecular graphs and nano-structures. Since the forgotten topological index and coindex are considered among the most effective topological indices in analysis the QSPR/QSAR with high accuracy, so we intend to compute the F-index and F-coindex for some nanostructures such as V-Phenylenic Nanotube VPHX [m, n] and V-Phenylenic Nanotorus VPHY[m, n] and their polynomials which are useful for description of some characteristics of nanostructures. Topological indices are the molecular descriptors that describe the structures of chemical compounds and they help us to predict certain physicochemical properties [3]. The first and second Zagreb indices can be regarded as one of the oldest graph invariants which was defined in 1972 by Gutman and Trinajsti [4,5]. The first and second Zagreb indices defined for a molecular graph G as:

Mi(G)= £ M«0 + Sg(*)], M2(G)= £ <SG(u) M*).

uveE(G) uveE(G)

The first and second Zagreb coindices have been introduced by A. R. Ashrafi, T. Doslic, and A. Hamzeh in 2010 [6]. They are respectively defined as:

Mi(G) = £ Mu) + M«)], M2(G) = £ SG(U) Mv).

uv / E(G) uv / E(G)

Furtula and Gutman in 2015 introduced forgotten index (F-index) [7] which defined as:

F (G) = £ Sg 3(*)= £ (¿G2W+ ¿G2(*))

v e V (G) uv e E(G)

N. De, S. M. A. Nayeem and A. Pal. in 2016 defined forgotten coindex (F-coindex) [8], which defined as:

F (G) = £ SG3(*)= £ (SG2(U)+ SG2(*)).

v /V(G) uv /E(G)

Then, Farahani et al. [9-11] computed the first and second Zagreb, first and second Hyper-Zagreb and multiplicative and Redefined Zagreb indices of V-Phenylenic Nanotube VPHX [m, n] and V-Phenylenic Nanotorus VPHY [m, n] and their polynomials. Ashrafi et al. [12] studied the computing Sadhana polynomial of V-phenylenic nanotubes and nanotori. Alamian et al. [13] studied PI Polynomial of V-Phenylenic Nanotubes and Nanotori, Z. Ahmad et al. [14] presented new results on eccentric connectivity indices of V-Phenylenic nanotube, and there are a lot of researchers who have studied some topological indices on V-Phenylenic Nanotube VPHX [m,n] and V-Phenylenic Nanotorus VPHY[m, n] that cannot be all mentioned here. B. Furtula et al. [7] and De, Nilanjan et al. [8] defined the F-index and F-coindex and studied their of some special graph and graph operation. Nanotubes and Nanotorus play an important role in many applications such as Energy storage, Bioelectronics and Optoelectronics. Because of their unique structural, electrical, optical, and mechanical properties, graphene nanosheets drew dramatic attention of academic and industrial research [15] and as nanotubes introduced into graphene could be extremely useful and exploited to generate novel, innovative, and useful materials and devices. Here, we present the F-index and F-coindex and their topological polynomials of V-Phenylenic Nanotube VPHX [m, n] and V-Phenylenic Nanotorus VPHY [m, n] which are useful for surveying structure of nanotubes and nanotorus. Any unexplained terminology is standard, typically as in [16-21].

2. Preliminaries

In this section, we give some basic and preliminary concepts which we shall use later.

Proposition 2.1 [2,8] Let G be a simple graph on n vertices and m edges. Then:

F(G) = n(n - 1)3 - 6m(n - 1)2 + 3(n - 1)Mi(G) - F(G), F(G) = (n - 1)M1(G) - F(G), F(G) = 2m(n - 1)2 - 2(n - 1)M1(G) + F(G).

Theorem 2.2 [9,10] The first and second Zagreb and VPHX [m, n] and V-Phenylenic Nanotorus VPHY[m, n](Vi

M1(VPHX [m, n]) = 54mn - 10m,

M2(VPHX[m, n]) = 81mn + 3m,

HM(VPHX[m, n]) = 4m(81n - 20),

HM2(VPHX[m,n]) = 9m(81n - 29),

Hyper-Zagreb indices of the V-Phenylenic Nanotubes

n, n € N - {1}) (Fig. 1,2) are given by:

M1 (VPHY [m, n]) = 54mn, M2(VPHY[m, n]) = 81mn, HM(VPHY[m, n]) = 324mn, HM2(VPHY [m, n]) = 729mn.

3. Main results

In this section, we compute the forgotten topological index and coindex for certain important chemical structures such as line graphs of the V-Phenylenic Nanotubes VPHX [m, n] and V-Phenylenic Nanotorus VPHY [m, n] (Vm, n € N - {1}) and their molecular complement graph. Here, we study also F-polynomial of V-Phenylenic Nanotubes and Nanotorus.

3.1. F-index and coindex of the V-Phenylenic Nanotubes VPHX [m, n](Vm, n € N - {1})

Theorem 3.1.1 The F-index of the V-Phenylenic Nanotubes VPHX[m,n](Vm, n € N - {1}) (Fig. 1) is given by:

F(VPHX[m, n]) = 162mn - 38m. Proof. By definition of the F-index F(G) = ^ [¿G (u) + ¿G (v)], and by replacing each G with VPHX [m,n],

uveE(G)

which yield:

F (VPHX [m,n])= ^

uveE(VPHX[m,n]

And the partitions of the vertex set and edge set V(VPHX [m, n]), E(VPHX [m, n]), of V-Phenylenic nanotube are given in Table 1,2 respectively [9].

The edge set of VPHX [m, n] is divided into two edge partitions based on the sum of degrees of the end vertices

as:

E5(VPHX[m,n]) = Eg = {e = uv € E(VPHX[m,n]) : ¿(u) = 2,J(v) = 3}, E6(VPHX[m, n]) = Eg = {e = uv € E(VPHX[m,n]) : ¿(u) = 3,J(v) = 3}.

¿2PHX[m,n] (u) + ¿2PHX[m,n] (v) .

Fig. 1. The molecular graph of VPHX[m, n] nanotube

Table 1. The edge partition of VPH X [m, n] nanotubes

Edge partition E5 = E6 E6 = E9

Cardinality 4m 9mn — 5m

Table 2. The vertex partition of VPHX[m, n] nanotubes

Vertex partition V2 V3

Cardinality m + m 6mn — 2m

Thus:

VPHX[m,n] (u) + 6VPHX[m,n] (v)

F (VPHX [m,n]) = £ 6-

uv e E(V PHX [m,n])

= X/ 62PHX[m,n] (u) + 6VPHX[m,n] (v)

uve E*(VPHX[m,n])

+ £ 6VPHX[m,n] 6VPHX[m,n] (v)

uve E*(VPHX[m,n])

= 13|Eg(VPHX [m,n])| + 18|Eg (VPHX [m,n])| = 52m + 18[9mn — 5m] = 162mn — 38m. □

Theorem 3.1.2 The F-polynomial of VPHX [m, n] nanotube (Fig. 1) is given by:

F(VPHX[m, n], x) = m 4x13 + [9n — 5]

Proof. Since the F-polynomial of graph G is

F (G,x)

£ x[4 (u) + ÄG (v)] v e E(G)

F (VPHX [m, n], x)

And, as Theorem 3.1.1, the partitions of the vertex set and edge set V(VPHX[m, n]), E(VPHX[m, n]), of V-Phenylenic nanotube are given in Table 1,2 respectively, we have:

E x^2

uveE(VPHX[m,n]

£ x[i? uveE*(VPHX [m,n])

E

uveE*(VPHX [m,n])

|Efi(VPHX [m,n])|x13 + |E"(VPHX [m,n])|x18 4mx13 + [9mn — 5m]x18

+

Xi"VPHX[m,n]iU) + SV PHX[m,n](v)]

Xl"VPHX[m,n](u) + 6V PHX[m,n] (v)]

X[SV PHX[m,n](u) + SV P H X [m,n] (v)]

4x + [9n - 5];

„18

. □

We can also get the F-index of VPHX[m, n] nanotube by derivating the relation F-polynomial of VPHX [m, n] nanotube above as:

F(VPHX[m,n]) = (VPHX[m, n], x) ^ = ^ + [9n - 5]x

18

dx

= 162mn — 38m.

dx

|x=1

Corollary 3.1.3 The F-index of complement VPHX [m, n] nanotube (Fig. 1) is given by:

F (VPHX [m,n]) = 6mn

6mn 1

- 6(9mn - m)

6mn 1

+ 3

6mn 1

(54mn — 10m) — (162mn — 38m).

Proof. By Proposition 2.1 we have

F (G) = n(n — 1)3 — 6m(n — 1)2 + 3(n — 1)M1 (G) — F (G).

And F(VPHX[m, n]) = 162mn — 38m given in Theorem 3.1.1 above. M1(VPHX[m,n]) = 54mn — 10m and the partitions of the vertex set and edge set of (VPHX [m, n]) nanotubes are given in [9]:

£ |V(VPHX[m, n])| = 6mn, £ |E(VPHX[m,n])| = 9mn — m

Thus:

F(VPHX[m, n]) = Y^ |V(VPHX[m,n])|^£|V(VPHX[m,n])| — 1

— 6£ |E(VPHX[m,n])|[£ |V(VPHX[m,n])| — 1 2 + ^ E |V(VPHX[m, n])| — 1 M1(VPHX[m,n]) — F((VPHX[m,n])

6mn 6mn 1

— 6(9mn — m)

6mn 1

+ 3 6mn — 1 (54mn — 10m) — (162mn — 38m). □

Corollary 3.1.4 The F-coindex of VPHX [m, n] nanotube (Fig. 1) is given by:

F(VPHX[m, n]) = 12m2n(27n — 5) — 24m(9n — 2).

Proof. By Proposition 2.1, we have F(G) = (n — 1)M1(G) — F(G), F(VPHX[m,n]) = 162mn 38m given in Theorem 3.1.1 and M1(VPHX [m, n]) = 54mn — 10m given in Theorem 2.2 above and since n £ |V(VPHX[m, n])| = 6mn. Then:

F(VPHX[m, n]) = [£ |V(VPHX[m,n])| — 1 M1(VPHX[m,n])

— F (VPHX [m, n])

= (6mn — 1)(54mn — 10m) — (162mn — 38m)

= 12m2n(27n — 5) — 24m(9n — 2). □

=m

3

2

3

2

Table 3. Some topological indices values of H = VPHX [m, n] nanotubes

m n Mi(H ) M2(H ) F (H ) HM (H) HM2(H ) F (H)

2 2 196 330 572 1136 2394 3.936 x 103

2 3 304 492 896 1784 3852 9.744 x 103

3 2 294 495 858 1704 3591 9.432 x 103

3 3 456 738 1344 2676 5778 22.824 x 103

4 4 824 1308 2440 4864 10620 75.840 x 103

5 5 1300 2040 3860 7700 16920 189.840 x 103

Table 4. The edge and vertex partitions of VPHY [m, n] nanotorus

Edge partition — Eg

Cardinality 9mn

Vertex partition V3

Cardinality 6mn

Corollary 3.1.5 The F-coindex of complement VPHX [m, n] nanotube (Fig. 1) is given by:

F(VPHX[m,n]) = 2(9mn - m)

+ 162mn — 38m. Proof. By Proposition 2.1 we have

6mn — 1

- 2

6mn 1

(54mn — 10m)

F(G) = 2m(n - 1)2 - 2(n - 1)Mi(G) + F(G),

F(VPHX[m, n]) = 162mn — 38m given in Theorem 3.1.1 and and M1(VPHX[m, n]) = 54mn — 10m given in Theorem 2.2 above and as Corollary 3.1.3 the partitions of the vertex set and edge set of (VPHX [m, n]) nanotubes. Then:

F (VPHX [m, n]) = 2 £ |E(VPHX [m,n])||"£ |V(VPHX [m,n])|- 12

— 2Σ |V(VPHX[m,n])| — 1 Mi(VPHX[m,n]) + F(VPHX[m,n])

2(9mn — m)

6mn 1

2

6mn 1

(54mn — 10m)

+ 162mn — 38m. □

In Table 3 some index and coindex values of VPHX [m, n] nanotubes. formulas reported in Theorem 2.2, Theorem 3.1.1 and Corollary 3.1.4 for the VPHX [m, n] nanotube. In the table, it shows that values of first and second Zagreb indices, first and second Hyper-Zagreb indices, F-index and F-coindex are in increasing order as the values of m, n increase.

3.2. F-index and coindex of the V-Phenylenic Nanotorus VPHY [m, n](Vm, n G N — {1})

Theorem 3.2.1 The F-index of the V-Phenylenic Nanotorus VPHY[m, n](Vm,n G N — {1}) (Fig. 2) is given by:

F(VPHY[m, n]) = 162mn. Proof. By definition of the F-index F(G) = J2 [^G (u) + ^G (v)], and by replacing each G with VPHY[m, n],

uv e E(G)

which yield to F(VPHY[m,n]) = Y.uve E(VPHY[m,n] PHY[m n] (u) + PHY[m n] (v) , and the partitions of

the vertex set and edge set V(VPHY[m, n]), E(VPHY[m, n]), of V-Phenylenic nanotorus are given in Table 4 respectively [9-11].

The edge set of VPHY[m, n] have only one type of edges: E6(VPHY[m, n]) = E = {e = uv G E(VPHY[m, n]) : J(u) = 3, J(v) = 3},

2

2

FIG. 2. The molecular graph of VPHY [m, n] nanotorus

Thus:

F (VPHY [m, n]) = Y

uveE(VPHY [m,n])

2PHY[m,n] (u) + 3

VPHY [m,n](v)

VPHY[m,n] (u) + 3VPHY[m,n] (v)

= E

uveE*(VPHY [m,n])

= 18|E"(VPHY [m,n])| = 162mn. □

Theorem 3.2.2 The F-polynomial of VPHY[m, n] nanotorus (Fig. 2) is given by:

F (VPHY [m, n], x) = 9mnx18 Proof. Since the F-polynomial of graph G

F (G, x) = £

x

[¿à (u)+S2a (v)]

uveE(G)

And as Theorem 3.2.1 the partitions of the vertex set and edge set V(VPHY[m, n]), E(VPHY[m, n]), of V-Phenylenic nanotorus are given in Table 4 we have:

F (VPHY [m, n], x) = E x[S22 PHY [m^i^l PHY [m,n](v)]

uv£E(V PHY [m,n]

= E x[5VPHY [m,n](u) + SVPHY [m,n](v)]

uveE*(VPHY [m,n])

= |E"(VPHY [m, n])|x18 =9mnx18. □

We can also get the F-index of VPHY[m, n] nanotorus by derivating the relation F-polynomial of VPHY[m, n] nanotorus above as:

F (VPHY [m, n]) =

dF (VPHY [m, n], x) d[9mnx18]

^ ix=1 ^

dx dx

Corollary 3.2.3 The F-index of complement VPHY[m, n] nanotorus (Fig. 2) is given by:

-|x=i = 162mn.

F (VPHY [m, n]) = 6mn (6mn — 1)3 — 9(6mn — 1)2 + 27(6mn — 2) Proof. By Proposition 2.1 we have

F (G) = n(n — 1)3 — 6m(n — 1)2 + 3(n — 1)M1 (G) — F (G),

And F (VPHY [m, n]) = 162mn given in Theorem 3.2.1 above. M1(VPHY [m, n]) = 54mn and the partitions of the vertex set and edge set of (VPHY[m, n]) nanotorus are given in [9].

E |V(VPHY[m,n])| = 6mn, E |E(VPHY[m,n])| = 9mn

Table 5. Some topological indices values of G = VPHY[m, n] nanotorus

m n Mi(G) M2(G) F (G) HM (G) HM2(G) F (G)

2 2 216 324 648 1296 2916 4.320 x 103

2 3 324 486 972 1944 4374 10.368 x 103

3 2 324 486 972 1944 4374 10.368 x 103

3 3 486 729 1458 2916 6561 24.300 x 103

4 4 864 1296 2591 5184 11664 79.488 x 103

5 5 1350 2025 4050 8100 18225 197.100 x 103

Thus:

F(VPHY [m, n])

£|V (VPHY [m,n])|^£|V (VPHY [m,n])| — 1 6 £ |E(VPHY[m,n])|f£ |V(VPHY[m,n])| — 1

+ ^ £ IV(VPHY[m, n])| - 1 M1(VPHY[m,n]) - F((VPHY[m,n]) = 6mn |^(6mn - 1)3 - 9(6mn - 1)2 + 27(6mn - 2) . □

Corollary 3.2.4 The F-coindex of VPHY[m, n] nanotorus (Fig. 2) is given by:

F(VPHY[m,n]) = 54mn(6mn - 4).

Proof. By Proposition 2.1 we have F(G) = (n - 1)M1(G) - F(G), F(VPHY[m, n]) = 162mn given in Theorem 3.1.1 and M1(VPHY[m, n]) = 54mn given in Theorem 2.2 above and since n = J2 |V(VPHY[m, n])| = 6mn. Then:

F(VPHY[m,n]) = [£ |V(VPHY[m,n])|- 1 M1(VPHY[m,n]) - F(VPHY[m,n]) = 54mn(6mn - 4). □

Corollary 3.2.5 The F-coindex of complement VPHY[m, n] nanotorus (Fig. 2) is given by:

F (VPHY [m, n])

18mn

(6mn — 1)2 — 36mn + 15

Proof. By Proposition 2.1 we have

F(G) = 2m(n - 1)2 - 2(n - 1)M1(G) + F(G),

F(VPHY[m, n]) = 162mn given in Theorem 3.2.1 and and M1(VPHX[m, n]) = 54mn given in Theorem 2.2 above and as Corollary 3.2.3 the partitions of the vertex set and edge set of (VPHY[m, n]) nanotorus. Then:

F(VPHY[m, n]) = 2£ |E(VPHY[m,n])|^£|V(VPHY[m,n])| — 1

— 2^£ |V(VPHY[m, n])| — 1 Mi(VPHY[m,n]) + F(VPHY[m,n])

= 18mn 6mn — 1

— 108mn

6mn 1

+ 162mn

18mn

(6mn — 1)2 — 36mn + 15

□

In Table 5 some index and coindex values of VPHY [m, n] nanotorus formulas reported in Theorem 2.2, Theorem 3.1.1 and Corollary 3.2.5 for the VPHY[m, n] nanotorus. Table 5 shows that values of first and second Zagreb indices, first and second Hyper-Zagreb indices, F-index and F-coindex are in increasing order as the values of m, n increase.

2

2

4. Conclusion

The forgotten index is one of the most important topological indices which preserves the symmetry of molecular structures and provides a mathematical formulation to predict their physical and chemical properties. The present study has computed the F-index and F-coindex of a physico chemical structure of V-Phenylenic Nanotube VPHX [m, n] and V-Phenylenic Nanotorus VPHY[m, n] and their molecular complement graphs. The study also has computed F-polynomial of V-Phenylenic Nanotube and V-Phenylenic Nanotorus. As the F-index and coindex can been used in QSPR/QSAR study and play a crucial role in analyzing some physico-chemical properties, the results obtained in our paper illustrate the promising prospects of application for nanostructures.

Acknowledgements

The authors would like to thank the reviewer for his constructive suggestions and useful comments which resulted in an improved version of this paper.

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