Short communication
On Multiple Zagreb Indices of TiO2 Nanotubes
Mehar Ali Malik* and Muhammad Imran
School of Natural Sciences, National University of Sciences and Technology, Sector H-12, P.O. 44000, Islamabad, Pakistan
* Corresponding author: E-mail: alies.camp@gmail.com, imrandhab@gmail.com Phone: +92 51 90855606, Fax: +92 51 90855552.
Received: 09-06-2015
Abstract
The first and second Zagreb indices were first introduced by I. Gutman and N. Trinajstic in 1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second multiple Zagreb indices of a graph G were introduced by Ghorbani and Azimi in 2012. In this paper, we calculate the Zagreb indices and the multiplicative versions of the Zagreb indices of an infinite class of titania nanotubes TiO2[m,n].
Keywords: TiO2 nanotubes, Topological indices, Zagreb index, Multiple Zagreb index
1. Introduction
Mathematical chemistry is a branch of theoretical chemistry in which we discuss and predict the chemical structure by using mathematical tools. Chemical graph theory is a branch of mathematical chemistry in which we apply tools from graph theory to model the chemical phenomenon mathematically. This theory plays a prominent role in the fields of chemical sciences.
A molecular graph is a simple graph in which the vertices denote atoms and the edges represent chemical bonds between these atoms. The hydrogen atoms are often omitted in a molecular graph. Let G be a molecular graph with vertex set V(G) = {v1, v2, ..., vn} and edge set E(G). We denote the order and size of G by |V|(G)| and |E(G)|, respectively. An edge in E(G) with end vertices u and v is denoted by uv. Two vertices u and v are said to be adjacent if there is an edge between them. The set of all vertices adjacent to a vertex u is said to be the neighbourhood of u, denoted as N(u). The number of vertices in N(u) is said to be the degree of u, denoted by d(u). The maximum and minimum vertex degrees in a graph G, respectively denoted by A(G) and 8(G), are defined as max {d(u)|ue V(G)} and min {d(u)| ue V(G)}, respectively. A (v1, vn) -path on n vertices is denoted by Pn and is defined as a graph with vertex set {v1:1< i <n} and edge set vlvi+1, for 1 < i < n-1. The length of a path Pn is the number of edges in it, that is, n-1.
A topological index is a molecular graph invariant which correlates the physico-chemical properties of a
molecular graph with a number.1 The first topological index was introduced by a chemist Harold Wiener in 1947 to calculate the boiling points of paraffins.2 This numerical representation of a molecular graph has shown to be very useful quantity to use in the quantitative structure-property relationship.3 It also has many applications in communication, facility location, cryptography, etc., that are effectively modeled by a connected graph G with some restrictions.4 This index was originally defined for trees to correlate certain physico-chemical properties of alkanes, alcohols, amines and their compounds. Hosoya5 defined the notion of Wiener index for any graph G as
A pair of molecular descriptors known as the first Zagreb index M1(G) and the second Zagreb index M2(G) were first appeared in the topological formula for total я-energy of conjugated molecules that was derived in 1972 by Gutman and Trinajstic.6 Afterwards, these indices have been used as branching indices7. Later on, the Zagreb indices found applications in QSPR and QSAR studies.8,9 Many topological indices have been studied in the literature for different chemical structures.10-13
The first Zagreb index M1(G) and the second Zagreb index M2(G) of a molecular graph G are respectively defined as
M,(G)= £ d(v)2= £ (rf(«) + rf(v>:
vef(C)

A/2(G)= ^ d(u)d(v).
(2)
(3)
ui'-H(O)
The new multiplicative versions of M1(G) and M2(G) indices, denoted by PM1(G) and PM2(G) (respectively), were first defined by Ghorbani and Azimi.14 These indices are defined as follows.
PMt(G)= П Ч+4Д
uv*E(G)
PM2(G)= П d(u)d(v).
(4)
(5)
... ■ I : ( i
In this paper, we study the Zagreb and multiplicative versions of Zagreb indices of titania TiO2 nanotubes. As a well-known semiconductor with numerous technological applications, Titania nanotubes are comprehensively studied in materials science. Titania nanotubes were systematically synthesized during the last 10-15 years using different methods and carefully studied as prospective technological materials. Since the growth mechanism for TiO2 nanotubes is still not well defined, their comprehensive theoretical studies attract enhanced attention. The TiO2 sheets with a thickness of a few atomic layers were found to be remarkably stable.15
Recently, the multiple Zagreb indices of circumcoro-nene homologous series of benzenoids were studied by Fa-rahani.16 The Zagreb indices of some other nanotubes and benzenoid graphs can also be found in the literature.1718
gons in a column and n denotes the number of octagons in a row of the Titania nanotube.
In the following, we perform some necessary calculations for computing the Zagreb indices and multiplicative versions of Zagreb indices defined in the previous section.
Let us define the partitions for the vertex set and edge set of the titania nanotube TiO2, for 8(G) < k < A(G), 28(G) < i < 2A(G), and 8(G)2 < j < A (G)2, then we have

(6)
(7)
(8)
In the molecular graph of TiO2 nanotubes, we can see that 2 < d(v) < 5. So, we have the vertex partitions as follows.
V^{uzV(G)\d{u) = 2}, V3 = iuenG)\d(u) = 3}, V4 = {ueV(G)\d(u) = 4}, Vs= {usV(G) d(u) = 5}.
(9) (10) (11) (12)
Similarly, the edge partitions of the graph of TiO 2 nanotubes are as follows.

(13)
2. Main Results
The graph of the titania nanotube TiO2 [m,n] is presented in Figure 1, where m denotes the number of octa-
u {e = Mv e E{G) \ d(u) = 3 & d(v) = 4},
(14)
(15)
Figure 1: The graph of TiO2 [m,n] -nanotubes, for m = 4 and m = 6.
(16)
Since for every vertex ve V(G), d(v) belongs to exactly one class Vk for 2 < к < 5 and for every edge uve -E(G), d(u) + d(v) (resp. d(u)d(v)) belongs to exactly one class Ei (resp. E*) for 28(G) < i < 2A (G), and 8(G)2< i < A(G)2. So, the vertex partitions Vk and the edge partitions Ei and Ej* are collectively exhaustive, that is
A<G)
2Л(СГ|
\JK = r(G\ и = E<
(17)
When m = 1, the number of vertices of degree 2 in TiO2[1,n] are 2n + n + 1 x (2n) + n. Thus for TiO2[m,n], we have |V2| = 2n + n + m(2n) + n = 2mn + 4n. Similarly, the cardinalities of all the vertex and edge partitions can be obtained, which are presented in Table 1.
Theorem 2.3 The first multiple Zagreb index of the
TiO2 nanotube is given by
PM, (770, ) = 677376m2«4 +1 12896mn* -
(20)
Proof. From equation 4, we have PM1(TiO2) = 6x |E6| x7x |E7| x8x |E8|. Using cardinalities of the edge partitions given in Table 1, we get PM1(TiO2) = 677376m2n4 + 112896mn4 - 112896n4, as required.
Theorem 2.4 The second multiple Zagreb index of the (TiO2) nanotube is given by
РМ2(ТЮг) = 4147200m-«4 + 691200m«4 --691200л4.
(21)
Proof. Using equation 5, we have M2(TiO2) = 8x |E8* | x10x |E1*0 | x12x |E12*| x15x |E1*5|. By using cardinalities of the edge partitions from Table 1, we get PM2(TiO2) = 4147200m2n4 + 691200mn4 - 691200n4, as required.
3. Conclusion
Table 1: The vertex and edge partitions of the TiO2 nanotubes along with their cardinalities.
Vertex partition
Cardinality
V2 V3 V4 V5
E = E *
8
E7
E = E *
8 15 E *
E * 12
2mn + 4n 2mn 2n 2mn 6n
4mn + 4n 6mn - 2n 2mn + 2n 2n
Theorem 2.1 The first Zagreb index of the TiO2 na-notube is given by
Mt(Ti02) = 16mn + 48n.
(18)
Proof. From equation 2, we have M1(TiO2) = 6x |E6| + 7x |E7 | + 8x |E8|. Using cardinalities of the edge partitions from Table 1, we get M1(TiO2) = 76mn + 48n, as required.
Theorem 2.2 The second Zagreb index of the TiO2 nanotube is given by
М2(ТЮ2) = 130тн+б2я,
(19)
Proof. Using equation 3, we get M2(TiO2) = 8x |E8*| + 10x |E1*0 | + 12x |E1*2 | + 15x |Ej*5|. By cardinalities of the edge partitions from Table 1, we get M2(TiO 2) = 130mn + 62n, as required.
The Zagreb indices were first appeared in the topological formula for the total п-energy of conjugated molecules that has been derived in 1972 by Gutman and Trinajstic3. These indices have also been used as branching indices and have found applications in QSPR and QSAR studies. In this paper, we studied the Zagreb indices and the newly defined multiplicative versions of the Zagreb indices of an infinite class of Titania nanotubes 7iO2[m,n]. Closed form formulas have been derived for the above mentioned indices.
4. References
1.	C. Hansch, L. Leo, Exploring QSAR fundamentals and applicability in chemistry and biology, Amer. Chem. Soc., Washington DC, 1996.
2.	H. Wiener, J. Am. Chem. Soc. 1947, 69, 17-20. http://dx.doi.org/10.1021/ja01193a005
3.	J. Devillers, A. T. Balaban, Topological Indices and Related Descriptors in QSAR and QSPR, Gordon & Breach, Amsterdam, 1999.
4.	A. A. Dobrynin, R. Entringer, I. Gutman, Acta Applicandae Mathematicae, 2001, 66, 211-249. http://dx.doi.org/10.1023/A:1010767517079
5.	H. Hosoya, Bull. Chem. Soc. Jpn. 1971, 4, 2332-2339. http://dx.doi.org/10.1246/bcsj.44.2332
6.	I. Gutman, N. Trinajstic, Chem. Phys. Lett. 1972, 17, 535538. http://dx.doi.org/10.1016/0009-2614(72)85099-1
7.	M. V. Diudea, (Ed.), QSPR/QSAR Studies by Molecular Descriptors, NOVA, New York, 2001.
8.	K. Xu, K.Ch. Das, MATCH Commun. Math. Comput. Chem. 2012, 68, 257-272.
9. I. Gutman, O. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin, 1986. http://dx.doi.org/10.1007/978-3-642-70982-1
10.	M. Eliasi, A. Iranmanesh, I. Gutman, MATCH Commun. Math. Comput. Chem. 2012, 68, 217-230.
11.	A. Al-Kandari, P. Manuel, I. Rajasingh, J. Comb. Math. Comb. Comput. 2011, 79, 33-42.
12.	B. Zhou, D. Stevanovic, MATCH Commun. Math. Comput. Chem. 2006, 56, 571-578.
13.	M. Rostami, M. Shabanian, H. Moghanian, Digest J. Nano-mater. Biostruct. 2012, 7, 247-252.
14.	M. Ghorbani, N. Azimi, Iranian Journal of Mathematical Chemistry, 2012, 3, 137-143.
15.	R. A. Evarestov, Y. F. Zhukovskii, A. V. Bandura, S. Pisku-nov, Cent. Eur. J. Phys. 2011, 9, 492-501.
16.	M. R. Farahani, On Multiple Zagreb indices of Circumcoro-nene Homologous Series of Benzenoid, Chemical Physics Research Journal, 2014, 7(2), 277-281.
17.	M. R. Farahani, Some Connectivity Indices and Zagreb Index of Polyhex Nanotubes, Acta Chim. Slov. 2012, 59, 779783.
18.	M. R. Farahani, Zagreb indices and their polynomials of the linear parallelogram of benzenoid graph, Global Journal of Chemistry, 2015, 1(1), 16-19.
Povzetek
Prvi in drugi zagrebški indeks sta vpeljala I. Gutman in N. Trinajstic leta 1972. Izkazalo se je, da sta ta dva indeksa uporabna pri napovedih protivnetnih lastnosti nekaterih kemijskih spojin in drugih raziskavah. Ghorbani in Azimi sta prvi in drugi zagrebški indkes leta 2012 uporabila v grafu G. V tem delu smo zagrebška indeksa in njune večkratnike izračunali za nanocevke TiO2[m,n].