The Redefined First, Second and Third Zagreb Indices of Titania Nanotubes TiO2[m,n]

Abstract

The first and the second Zagreb indices are two of the most thoroughly studied and oldest topological indices. Recently in 2013, Ranjini et al. re-defined the Zagreb indices, i.e., the redefined first, second and third Zagreb indices of a graph G are defined as , and , respectively. In this research paper, we compute the redefined Zagreb indices of the Titania Nanotubes TiO2[m, n].

Keywords: Carbon nanotube, Degree of vertex, Molecular graphs, Nano structures, Redefined zagreb indices, Titania nanotubes, Topological indices, Zagreb indices.

INTRODUCTION

Let G(V (G), E( G)) be a simple connected graph. In the setting of chemical graph theory, we use a graph G to model a chemical structures. Namely, we use the vertices and edges in G to represent respectively the atoms and the bonds in chemical structures. The vertex set and edge set of G are denoted by V (G) and E (G) respectively and for u, v ϵ V (G); e = uv is an edge of G(e ϵ E (G)). In a simple connected molecular graph G as order n, d(v) be the vertex degrees of vertices/atom v in G. Then 0 ≤ δ(G) ≤ d(v) ≤ Δ(G) ≤ n - 1, where δ(G) and Δ(G) are the minimum and maximum of degrees d(v) for all v ϵ V (G). The notations and terminologies that were used but were undefined in this paper can be found in [1, 2].

A topological index is a real number associated with a graph which characterizes the topology of the graph and is invariant under graph isomorphism. There are many distance or degree based topological indices. Degree based topological indices are of great importance and play a vital role in chemical graph theory. Some recent results on topological indices of chemical graphs have been studied by Gao et al. [3, 4].

The first and second Zagreb indices which were introduced by Gutman and Trinajstić [5] in 1972 are the oldest topological indices of graphs. They are degree based indices and expressed as follows:

In 2012, Ghorbani and Azimi [6] proposed the multiple versions of Zagreb indices of a graph G. These new indices are first multiple Zagreb index PM1(G), second multiple Zagreb index PM2(G) and defined as:

The reader can find more information about multiple versions of Zagreb indices of some molecular graphs and Nanotubes in [7-10].

In 2013, Shirdel et al. [11] introduced another degree based version of topological index named Hyper-Zagreb index and it is defined as:

For more study about some properties of hyper Zagreb indices, see [12-16].

In 2004, Gutman and Das [17] defined the first and second Zagreb Polynomial in the following way:

The properties of M1(G, x), M2(G, x) polynomials for some chemical structures have been studied in the literature [17, 18].

Ranjini et al. [19] re-defined the Zagreb indices, i.e. the redefined first, second and third Zagreb indices for a graph G and these are manifested as

and

respectively.

MAIN RESULTS

Titania Nanotubes are studied comprehensively in materials science. Carbon nanotube composites have attracted much attention due to their unique properties and promising applications. Titanium dioxide (TiO2) is an important semiconductor material, and it has been applied as white pigment, cosmetic, catalyst and carrier owing to its excellent physical and chemical properties. The TiO2 sheets with a thickness of a few atomic layers were found to be remarkably stable [20-28]. The graph of the Titania Nanotubes TiO2[m, n] is presented in Fig. (1), where m denotes the number of octagons in a column and n denotes the number of octagons in a row of the Titania Nanotubes. Malik and Imran [25] computed the first and second Zagreb indices, first and second multiple Zagreb index for an infinite class of Titania Nanotubes TiO2[m, n].

In this paper, we computed the redefined Zagreb indices of Titania Nanotubes TiO2[m, n], for this initially we perform some necessary calculations.

Fig. (1). For m = 4 and n = 6, the graph of TiO2[m, n]-Nanotubes.

Let us define the partitions for the vertex set and edge set of Titania Nanotubes TiO2[m, n], for δ(G) ≤ a ≤ Δ(G), 2δ(G) ≤ b ≤ 2Δ(G) and δ(G)2c ≤ Δ(G)2, then we have [25, 29, 30]:

From [25, 29, 30], we can see that for all vertex/atom v in the molecular graph of TiO2 Nanotubes 2 ≤ d(v) ≤ 5, thus five vertex partitions of TiO2 with their cardinalities are as follows (see Table 1):

Table 1.
The vertex partitions of the TiO2 Nanotubes along with their cardinalities.
Vertex partition V2 V3 V4 V5
Cardinality 2mn+4n 2mn 2n 2mn

The edge partitions of TiO2 Nanotubes with their cardinalities (see Table 2) are stated as follows.

Table 2.
The edge partitions of the TiO2 Nanotubes along with their cardinalities.
Vertex partition E6 = E8* E7 E8 = E15* E10* E12*
Cardinality 6n 4mn+4n 6mn-2n 4mn+2n 2n

For every vertex v ϵ V (G), d(v) belongs to exactly one class Va for 2 ≤ a ≤ 5 and for every edge uv ϵ E (G), d(u)+d(v) (resp. d(u)d(v)) belongs to exactly one class Eb (resp. Ec*) for 2δ(G) ≤ b ≤ 2Δ(G) and δ(G)2c ≤ Δ(G)2. So, the vertex partitions Va and the edge partitions Eb and Ec* are collectively exhaustive, that is:

Now, we compute the redefined first, second and third Zagreb indices of Titania Nanotubes TiO2[m, n] in the following theorems.

Theorem 1: Let be the Titania Nanotubes, then the redefined first Zagreb indices is:

Proof. In terms of the definition of the revised first Zagreb index, we have:

Form Table 2 we get:

which is the required result.

Theorem 2. Let be the Titania Nanotubes, then the redefined second Zagreb indices is:

Proof. By means of the definition of the revised second Zagreb index, we infer:

From Table 2 we deduce:

which is the expected result.

Theorem 3. Let TiO2[m, n] be the Titania Nanotubes, then the redefined third Zagreb indices is:

Proof. From the definition of revised third Zagreb index, we yield:

From Table 2 we obtain:

which is the expected result.

CONFLICT OF INTEREST

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

ACKNOWLEDGEMENTS

We thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by NSFC (11401519).

REFERENCES

1
Harary F. Graph Theory. Reading: Addison-Wesley 1969.
2
West DB. Introduction to Graph Theory. Upper Saddle River, USA: Prentice Hall 1996.
3
Gao W, Wang W F, Farahani M R. Topological indices study of molecular structure in anticancer drugs. J Chem 2016; 2016 Article ID 3216327, 8 page.
4
Gao W, Farahani MR, Shi L. Forgotten topological index of some drug structures. Acta Med Mediter 2016; 32: 579-85.
5
Gutman I, Trinajstic N. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem Phys Lett 1972; 17(4): 535-8.
6
Ghorbani M, Azimi N. Note on multiple Zagreb indices. Iran J Math Chem 2012; 3(2): 137-43.
7
Gao W, Farahani MR, Kamran Jamil M. The eccentricity version of atom-bond connectivity index of linear polycene parallelogram benzenoid ABC5(P(n,n)). Acta Chim Slov 2016; 63(2): 376-9.
8
Farahani MR, Rajesh Kanna MR. On multiple zagreb indices of armchair polyhex nanotubes. Phy Sci Int J 2016; 9(1): 1-5.
9
Farahani MR. On multiple zagreb indices of dendrimer nanostars. Int Lett Chem Phy Astron 2015; 52: 147-51.
10
Farahani MR, Gao W. On multiple zagreb indices of polycyclic aromatic hydrocarbons PAH. J Chem Pharm Res 2015; 7(10): 535-9.
11
Shirdel GH, Pour HR, Sayadi AM. The hyper-Zagreb index of graph operations. Iran J Math Chem 2013; 4(2): 213-20.
12
Farahani MR. The hyper-zagreb index of benzenoid series. Front Math Appl 2015; 2(1): 1-5.
13
Farahani MR. Computing the hyper-zagreb index of hexagonal nanotubes. J Chem Mater Res 2015; 2(1): 16-8.
14
Gao W, Siddiqui MK, Imran M, Jamil MK, Farahani MR. Forgotten topological index of chemical structure in drugs. Saudi Pharm J 2016; 24(3): 258-64.
15
Gao W, Wang WF. Revised Szeged index and revised edge-szeged index of special chemical molecular structures. J Interdisciplinary Math 2016; 19(3): 495-516.
16
Gao W, Shi L, Farahani MR. Distance-based indices for some families of dendrimer nanostars. IAENG Int J Appl Math 2016; 46(2): 168-86.
17
Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Commun Math Comput Chem 2004; 50: 83-92.
18
Gutman I. New bounds on zagreb indices and the zagreb co-indices. Bol Soc Paran Mat 2013; 31(1): 51-5.
19
Ranjini PS, Lokesha V, Usha A. Relation between phenylene and hexagonal squeeze using harmonic index. Int J Graph Theory 2013; 1: 116-21.
20
Ramazani M, Farahmandjou M, Firoozabadi TP. Effect of nitric acid on particle morphology of the TiO2. J Nanosci Nanotechnol 2015; 11(1): 59-62.
21
Evarestoy RA, Zhukovskii YF, Bandura AV, Piskunov S. Symmetry and models of single-walled TiO2 Nanotubes with rectangular morphology. Cent Eur J Phys 2011; 9(2): 492-501.
22
Evarestov RA, Zhukovskii YF, Bandura AV, Piskunov S, Losev MV. Symmetry and models of double-wall BN and TiO2 Nanotubes with hexagonal morphology. J Phys Chem 2011; 115(29): 14067-76.
23
Evarestov RA, Zhukovskii YF, Bandura AV, Piskunov S. Symmetry and models of single-wall BN and TiO2 Nanotubes with hexagonal morphology. J Phys Chem 2010; 114(49): 21061-9.
24
Imran M, Hayat S, Mailk MY. On topological indices of certain interconnection networks. Appl Math Comput 2014; 244: 936-51.
25
Malik MA, Imran M. On multiple Zagreb indices of TiO2 Nanotubes. Acta Chim Slov 2015; 62(4): 973-6.
26
Gao W, Farahani MR. Degree-based indices computation for special chemical molecular structures using edge dividing method. Appl Math Nonlinear Sci 2016; 1(1): 99-122.
27
Gao W, Wang WF. Degree-based indices of polyhex nanotubes and dendrimer nanostar. J Comput Theor Nanosci 2016; 13(3): 1577-83.
28
Farahani MR, Jamil MK, Imran M. Vertex PIv topological index of Titania Nanotubes. Appl Math Nonlinear Sci 2016; 1(1): 170-5.
29
Farahani MR. Some connectivity indices and zagreb index of polyhex nanotubes. Acta Chim Slov 2012; 59(4): 779-83.
30
Imrana M, Baigb AQ, Ali H. On molecular topological properties of hex-derived networks. J Chemometr 2016; 30(3): 121-9.