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 TiO_{2}[m, n].
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Manuscript submitted on 04-03-2016 |
Original Manuscript | The Redefined First, Second and Third Zagreb Indices of Titania Nanotubes TiO2[m,n] |
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 [1Harary F. Graph Theory. Reading: Addison-Wesley 1969., 2West DB. Introduction to Graph Theory. Upper Saddle River, USA: Prentice Hall 1996.].
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. [3Gao 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.
[http://dx.doi.org/dx.doi.org/10.1155/2016/3216327] , 4Gao W, Farahani MR, Shi L. Forgotten topological index of some drug structures. Acta Med Mediter 2016; 32: 579-85.].
The first and second Zagreb indices which were introduced by Gutman and Trinajstić [5Gutman I, Trinajstic N. Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem Phys Lett 1972; 17(4): 535-8.
[http://dx.doi.org/10.1016/0009-2614(72)85099-1] ] in 1972 are the oldest topological indices of graphs. They are degree based indices and expressed as follows:
In 2012, Ghorbani and Azimi [6Ghorbani M, Azimi N. Note on multiple Zagreb indices. Iran J Math Chem 2012; 3(2): 137-43.] proposed the multiple versions of Zagreb indices of a graph G. These new indices are first multiple Zagreb index PM_{1}(G), second multiple Zagreb index PM_{2}(G) and defined as:
The reader can find more information about multiple versions of Zagreb indices of some molecular graphs and Nanotubes in [7Gao 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.
[http://dx.doi.org/10.17344/acsi.2016.2378] [PMID: 27333562] -10Farahani MR, Gao W. On multiple zagreb indices of polycyclic aromatic hydrocarbons PAH. J Chem Pharm Res 2015; 7(10): 535-9.].
In 2013, Shirdel et al. [11Shirdel GH, Pour HR, Sayadi AM. The hyper-Zagreb index of graph operations. Iran J Math Chem 2013; 4(2): 213-20.] 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 [12Farahani MR. The hyper-zagreb index of benzenoid series. Front Math Appl 2015; 2(1): 1-5.-16Gao W, Shi L, Farahani MR. Distance-based indices for some families of dendrimer nanostars. IAENG Int J Appl Math 2016; 46(2): 168-86.].
In 2004, Gutman and Das [17Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Commun Math Comput Chem 2004; 50: 83-92.] defined the first and second Zagreb Polynomial in the following way:
The properties of M_{1}(G, x), M_{2}(G, x) polynomials for some chemical structures have been studied in the literature [17Gutman I, Das KC. The first Zagreb index 30 years after. MATCH Commun Math Comput Chem 2004; 50: 83-92., 18Gutman I. New bounds on zagreb indices and the zagreb co-indices. Bol Soc Paran Mat 2013; 31(1): 51-5.].
Ranjini et al. [19Ranjini PS, Lokesha V, Usha A. Relation between phenylene and hexagonal squeeze using harmonic index. Int J Graph Theory 2013; 1: 116-21.] 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.
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 (TiO_{2}) 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 TiO_{2} sheets with a thickness of a few atomic layers were found to be remarkably stable [20Ramazani M, Farahmandjou M, Firoozabadi TP. Effect of nitric acid on particle morphology of the TiO2. J Nanosci Nanotechnol 2015; 11(1): 59-62.-28Farahani MR, Jamil MK, Imran M. Vertex PIv topological index of Titania Nanotubes. Appl Math Nonlinear Sci 2016; 1(1): 170-5.]. The graph of the Titania Nanotubes TiO_{2}[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 [25Malik MA, Imran M. On multiple Zagreb indices of TiO2 Nanotubes. Acta Chim Slov 2015; 62(4): 973-6.
[http://dx.doi.org/10.17344/acsi.2015.1746] [PMID: 26680727] ] computed the first and second Zagreb indices, first and second multiple Zagreb index for an infinite class of Titania Nanotubes TiO_{2}[m, n].
In this paper, we computed the redefined Zagreb indices of Titania Nanotubes TiO_{2}[m, n], for this initially we perform some necessary calculations.
Fig. (1) For m = 4 and n = 6, the graph of TiO_{2}[m, n]-Nanotubes. |
Let us define the partitions for the vertex set and edge set of Titania Nanotubes TiO_{2}[m, n], for δ(G) ≤ a ≤ Δ(G), 2δ(G) ≤ b ≤ 2Δ(G) and δ(G)^{2} ≤ c ≤ Δ(G)^{2}, then we have [25Malik MA, Imran M. On multiple Zagreb indices of TiO2 Nanotubes. Acta Chim Slov 2015; 62(4): 973-6.
[http://dx.doi.org/10.17344/acsi.2015.1746] [PMID: 26680727] , 29Farahani MR. Some connectivity indices and zagreb index of polyhex nanotubes. Acta Chim Slov 2012; 59(4): 779-83.
[PMID: 24061358] , 30Imrana M, Baigb AQ, Ali H. On molecular topological properties of hex-derived networks. J Chemometr 2016; 30(3): 121-9.
[http://dx.doi.org/10.1002/cem.2785] ]:
From [25Malik MA, Imran M. On multiple Zagreb indices of TiO2 Nanotubes. Acta Chim Slov 2015; 62(4): 973-6.
[http://dx.doi.org/10.17344/acsi.2015.1746] [PMID: 26680727] , 29Farahani MR. Some connectivity indices and zagreb index of polyhex nanotubes. Acta Chim Slov 2012; 59(4): 779-83.
[PMID: 24061358] , 30Imrana M, Baigb AQ, Ali H. On molecular topological properties of hex-derived networks. J Chemometr 2016; 30(3): 121-9.
[http://dx.doi.org/10.1002/cem.2785] ], we can see that for all vertex/atom v in the molecular graph of TiO_{2} Nanotubes 2 ≤ d(v) ≤ 5, thus five vertex partitions of TiO_{2} with their cardinalities are as follows (see Table 1):
The edge partitions of TiO_{2} Nanotubes with their cardinalities (see Table 2) are stated as follows.
For every vertex v ϵ V (G), d(v) belongs to exactly one class V_{a} 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 E_{b} (resp. E_{c}^{*}) for 2δ(G) ≤ b ≤ 2Δ(G) and δ(G)^{2} ≤ c ≤ Δ(G)^{2}. So, the vertex partitions V_{a} and the edge partitions E_{b} and E_{c}^{*} are collectively exhaustive, that is:
Now, we compute the redefined first, second and third Zagreb indices of Titania Nanotubes TiO_{2}[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 TiO_{2}[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.
The authors confirm that this article content has no conflict of interest.
We thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by NSFC (11401519).
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