DOI: I0.i7344/acsi.20i6.2337_Acta Chim. Slov. 2016, 63, 363-368_ 363
Scientific paper
On Eccentric Connectivity Index of TiO2 Nanotubes
Imran Nadeem and Hani Shaker*
Department of Mathematics, COMSATS Institute of Information Technology, Defence Road, Off Raiwind Road, Lahore, Pakistan
* Corresponding author: E-mail: hani.uet@gmail.com, imran7355@gmail.com Phone: +92-42-111001007, +92-321-4120429
Received: 12-02-2016
Abstract
The eccentric connectivity index (ECI) is a distance based molecular structure descriptor that was recently used for mathematical modeling of biological activities of diverse nature. The ECI has been shown to give a high degree of predictability compare to Wiener index with regard to diuretic activity and anti-inflammatory activity. The prediction accuracy rate of ECI is better than the Zagreb indices in case of anticonvulsant activity. Titania nanotubular materials are of high interest metal oxide substances due to their widespread technological applications. The numerous studies on the use of this material also require theoretical studies on the other properties of such materials. Recently, the Zagreb indices were studied of an infinite class of titania (TiO2) nanotubes [32]. In this paper, we study the eccentric connectivity index of these nanotubes.
Keywords: TiO2 nanotubes, Topological indices, Eccentric connectivity index
1. Introduction
Cheminformatics is a new subject which is a combination of chemistry, mathematics and information science. It studies quantitative structure activity relationships (QSAR) and structure property relationships (QSPR) that are used to predict the biological activities and properties of chemical compounds. In the QSAR/QSPR study, physicochemical properties and topological indices are used to predict biological activity of the chemical compounds.
A topological index is a numerical descriptor of the molecular structure based on certain topological features of the corresponding molecular graph. Topological indices are graph invariant and are a convenient means of translating chemical constitution into numerical values which can be used for correlation with physical properties in QSPR/QSAR studies.1-3 Topological indices are also used as a measure of structural similarity or diversity and thus they may give a measure of the diversity of chemical databases. There are two major classes of topological indices such as distance based topological indices and degree based topological indices. Among these classes, distance based topological indices are of great importance and play a vital role in chemical graph theory and particularly in chemistry.
A graph G with vertex set V(G) and edge set E(G) is connected if there exists a path between any pair of vertices in G. The degree of a vertex u e V is the number of edges incident to u and denoted by deg(u). For two vertices u, v of a graph G their distance d (u, v) is defined as the length of any shortest path connecting u and v in G. For a given vertex u of G its eccentricity e(u) is the largest distance between u and any vertex v of G.
Sharma et al.9 introduced a distance based topological index, the eccentric connectivity index (ECI) of G, defined as

(1)
It is reported in4-8 that ECI provides excellent correlations with regard to both physical and biological properties. The eccentric connectivity index is successfully used for mathematical models of biological activities of diverse nature. The simplicity amalgamated with high correlating ability of this index can easily be exploited in QSPR/QSAR studies.9-11 The prediction accuracy rate of ECI is better than the Wiener index with regard to diuretic activity12 and anti-inflammatory activity.13 Compare to Zagreb indices, the ECI has been shown to give a high degree of predictability in case of anticonvulsant activity.14 Recently, the eccentric connectivity index was studied for
Imran Nadeem and Hani Shaker: On Eccentric Connectivity Index of TiO2 Nanotubes
364
Acta Chim. Slov. 2016, 63, 363-368
certain nanotubes15-19 and for various classes of
graphs.20-22
The titanium nanotubular materials, called titania by a generic name, are of high interest metal oxide substances due to their widespread applications in production of catalytic, gas-sensing and corrosionresistance materials.23 As a well-known semiconductor with numerous technological applications, Titania (TiO2) nanotubes are comprehensively studied in materials science.24 The TiO2 nanotubes were systematically synthesized using different met-hods25 and carefully studied as prospective technological materials. Theoretical studies on the stability and electronic characteristics of titania nanostructures have extensively been studied.26-28 The numerous studies on the use of titania in technological applications also required theoretical studies on stability and other properties of such struc-
tures
29-31
Recently, M. A. Malik et al.32 studied the Zagreb indices of an infinite class of TiO2 nanotubes. In this paper, we study eccentric connectivity index of these nanotubes.
Figure 2: The labeled vertices of TiO2[m,n] nanotube.
In the molecular graph, G, of TiO2 nanotubes, we can see that 2<deg(v)< 5. So, we have the vertex partitions as follows.

(2)
2. Main Results
The cardinalities of all vertex partitions are presented in Table 1.
The molecular graph of titania nanotubes TiO2[m,n] is presented in Figure 1, where m denotes the number of octagons in a row and n denotes the number of octagons in a column of the titania nanotube.
Figure 1: The molecular graph of TiO2[m,n] nanotube.
The molecular graph of TiO2[m,n] nanotube has 2n+2 rows and m columns. For each Ith row and j"1 column, we label the vertices of TiO2[m,n] nanotube by ulj,
vij, x.. and yij as shown in Figure 2.
ij ij ij
Table 1: The vertex partitions of the TiO2 nanotubes along with their cardinalities.
Vertex partition	Cardinality
V,	2mn + 4m
V2	2mn
V3	2m
V4	2mn
In the following, we compute the exact formulas for eccentric connectivity index of TiO2[m,n] nanotubes.
Theorem 2.1 Let TiO2[m,n] be the graph of titania nano-
m— 2
tube, then for n<
we have
f(Ti02[m,«]) = 40 trrn + 32 m~
Proof. Consider G = Ti02[m,n]. When n< the eccentricity of every vertex in every row is 2m. From Table 1, we have
(3)
f lC) = 2 degMfM
.. I ; 1.1
= (2nw + 2m)(2)(2m) + (2m)(2)(2m) +
+ (2mnX3X2íM) + {2fl¡K4X2"'H{2m/i)(5)(2m) = 40í«:M + 32m'!.
(15)
Imran Nadeem and Hani Shaker: On Eccentric Connectivity Index of TiO2 Nanotubes
Acta Chim. Slov. 2016, 63, 363-368
365
Theorem 2.2 Let TiO2[m,n] be the graph of titania nano-tube, where m = 2p then for p even we have
?(TiO.\m,n\) =
1 60/th + 4&pn+l04p* + 2Ap
if p = 2 tr,
l&p1 + 4^p~n + \ 12pir + ]2S pn + 64 p1 + 24 p if ~^-<n<p- \ waAp*2n\
\20pJn+68pn2+U2pn+96p2+52p 120p~n + 60 prr + 12 pn +104/r + 24 p
if ii> p- \ and a is odd; ¡f n > p~ I and n is even.
(5)
Proof. Consider G = TiO2[m,n]. With respect to the eccentricity of vertices, we have the following cases.
Case 1. When p = 2n
In this case the eccentricity of the vertices uij, is 3p + 2n + 1 where i = l,2n + 2. The eccentricity of each vertex in the remaining 2n rows is 4p. Hence
Case 3. When n > p -1 and n is odd
In this case the eccentricity of vertices ui;, v„ is same as the eccentricity of vertices u(2n+3 i)j, v(2n+3 i)j where i = 1,2, •••, n + 1. The eccentricity of these vertices in ith row is given by
e(uij) = s(vii) = 3p+2n+2-i where i-1,2,■■■,« +1
f'(G) = £ deg(v)£(v)
ref(G)
- 2(2/7}(2)(3/? + 2n + ]) + 2(2p)(4)0p + 2r7 + \) + (2p)(2n)(3)(4p) + (2p)(2n)(5)(,4p) + 2p(2n + 2){2)(4p) = 160 p-n + 48 pn + 104 p2 + 24 p.
(10) (6)
Case 2. when ^ < » < p -1 and p ^ 2n
In this case the eccentricity of the vertices , v„ is
v j
same as the eccentricity of vertices u(2n+3 i)j, v(2n+3 ijj. where i = 1,2, •••, 2n - p + 1. The eccentricity of these vertices in ith row is given by
Also, the eccentricity of the vertices xijt yijt x(i+1)J, y^j is same as the eccentricity of the vertices x(2n+3i)j,
y(+n+3-iff X(2n+2-i)j , y(2n+2-i)j where i = 1,2 (n + 1)/2. The
eccentricity of these vertices in ith row is given by

e(xu) = £(yi/)=3p+2n+2-2i where / = 1,2,■ -(re +1)/2
(11)
(7)
The eccentricity of vertices uij, vij in remaining 2p -
ij ij
2n rows is 4p.
Also, the eccentricity of the vertices xij, yij, X(i+1)j, y(i+j j is same as the eccentricity of the vertices x(2n+3i)j,
y (2n+3-ij X(2n+2-i)j, y(2n+2-i)j where i = 1,2, "', (2n - p)/2 The
eccentricity of these vertices in ith row is given by
The shortest paths having maximal length in Ti-O2[8,7] nanotube are shown in Figure 3. Hence
= 2(2 pX2)(3 p +2n+1) + 2(2 p){4\3 p + 2n + \) +
s(x:j ) = e{ys ) = 3p + 2n + 2 - 2/ where f = 1j2,--,(2b- p)/2
(8)
The eccentricity of the vertices xij, yij in the remai-
ij ij
ning (2p - 2n + 2) rows is 4p. Hence ie(G)= X deg(vMv)
veK(G)
= 2(2p)(2)(3p + 2n + l) + 2(2p)m3p + 2n + l) + 2(2p)(3) £ (3p + 2n + 2-i)
(-2
+ 2(2;;)(5) £ (3p + 2n + 2-i) + (2p)(2p~2„)(3)(4p) + {2p)(2p-2n)(5)(4p)
¡-i
{2a-pV2
+ 4(2p)(2) X (3p + 2n + 2-2i) + (2p)(2p-2n + 2)(2)(4p)
¡=i
= 28/>1+48paflf+l \2pn2 + )28pn + 64p2 +24p
= 120 p2n + 68 pn2 +U2pn+96p2 + 52 p
(12)
(9)
Imran Nadeem and Hani Shaker: On Eccentric Connectivity Index of TiO2 Nanotubes
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Acta Chim. Slov. 2016, 63, 363-368
Figure 3: The shortest paths having maximal length in TiO2[8,7] nanotube.
Case 4. When n > p - 1 and n is even
In this case the eccentricity of vertices uij, v. is same
v v
as we discussed in case 3. Also, the eccentricity of the vertices xtj, yVj, x(i+1)j, y(i+1)j is same as the eccentricity of the
vertices ^-j ydn+l-iff X(2n+2-i)j , y(2n+2-i)j where V = 1,2
•••, n/2. The eccentricity of these vertices in ith row is given by
e(x,) = e{y,) = 3p + 2n + 2-2t where / = 1,2,-■ -,nj2
(13)
The eccentricity of the vertices xij, yij in the remaining 2 rows is 4p. Hence
= 2(2p)(2)Op + 2n + ]) + 2(2p)(4)(lp + 2n + \) +
J^(3p+2n + 2-2i)+2(2p)(2)(4p)
Theorem 2.3 Let TiO2[m,n] be the graph of titania nanotube, where m = 2p then for p odd we have
In this case the eccentricity of the vertices uij, vij is
ij ij
"(2n+3-ij V(2n+3-i)j. where i = 1,2. The eccentricity of these vertices in ith row is gi-
Case 1. When p = 2n - 1
In this case the ecc same as the eccentricity of vertices u re i = 1 ven by
The eccentricity of vertices u., vij in remaining 2n rows is 4p. Also, the eccentricity of the vertices x1;, y1;. is same as the eccentricity of vertices x(2n+2)j X(2n+2j. The eccentricity of the vertices x., y1j is given by
£(*i;) = «fry) = 3P + 2n + 1
(17)
The eccentricity of the vertices x ., y■■ in the remai-
ij ij
ning 2n rows is 4p. The shortest paths having maximal length in TiO2[14,4] nanotube are shown in Figure 4.
Figure 4: The shortest paths having maximal length in TiO2[14,4] nanotube.
Hence ?{G) = X deg(vMv)
+ 2(2p)Q){3p + 2 n) + 2(2p)(5)(3p + 2 n)
(18)
+ 2(2p)(2)(3p + 2n + \) + (2p){2n){2)(4p)
Case 2. when ^ < n < p - 1 and p^2n-l
In this case the eccentricity of the vertices uijt v;. is same as we discussed in Case 2 of Theorem 2.2. The eccentricity of the vertices x., y1j, x(2n+2)j x(2n+2)j is same as we discussed in Case 1.
(15)
Proof. Consider G = TiO2[m,n]. With respect to the eccentricity of vertices, we have the following cases.
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Also, the eccentricity of the vertices x(i+1);., y(i+1j, x(i+2)j y(i+2j is same as the eccentricity of the vertices
x(2n.i)j> y(2n+2-ijj' X(2n+l-i)j, y^nyl-ij Where i = 1,2, '", 2 -P - 1)/2. The eccentricity of these vertices in (i y 1)th row is
given by
e(x{Mii ) = «(JV+Dj) = 3p + 2« +1 - 2i where / = 1,2,---,{2« - p—1)/ 2
(19)
The eccentricity of the vertices x., y. in the remaining (2p - 2n + 2) rows is 4p. Hence
<T(G)= T deg(v)ff(v)
œi'(fï)
= 2(2p)(2)Op + 2» +1) + 2(2p)(4)(ip +2«+l) +2(2p)(3) J + 2h + 2 - i)
+2(2^X5) ï'c3p+2»+2-0+<2p)(2p-mWP)+(2p){2p-2nXWp)
ja2
+2(2pX2)(3p+2n+l)+4(2pK2) X (3p + 2n +1 -2/) + 2/j(2p -2n + 2)<2)(4p)
The eccentricity of the remaining vertices is same as we discussed in case 3. Hence
+2(2^X5)^(3/) + 2n+ 2-i) + 2(2p)(2)	(24)
(20)
= 20 p* + 80 p'n + 80 pn2 + 96pn + 80p: + 28 p
Case 3. When n > p - 1 and n is odd
In this case the eccentricity of the vertices ui., v..,x,.,
ij ij 1 j
y1j., x(2ny2)j x(2ny2)j is same as we discussed in Case 2. Also, the eccyitritity of the vertices x(i+1)j, y(i+Dj, x(i+2)j, y(i+2)jis same as the eccentricity of the vertices x(2ny2_ijj, y(2ny2-i)j,
x(2n+1-i)j, y(2n+1-i)j where i = 1,2 (n - 1)/2. The eccentricity of these vertices in (i y 1)th row is given by
Hence
?{G) = £ deg(v)£(v)
(21)
3. Conclusion
The eccentric connectivity index provides excellent prediction accuracy rate compare to other indices in certain biological activities of diverse nature such as diuretic activity, anticonvulsant activity and anti-inflammatory activity. In this sense, this index is very useful in QSPR/QSAR studies. In this paper, we study eccentric connectivity index of an infinite class of TiO2 nanotubes. By using this index, we can find mathematical models of certain biological activities for this material. With the help of these models, we can predict about certain biological activities for this material.
= 2{2p){2)Op + 2n + \) + 2{2p){A)Qp + 2n + \) + 2{2p)Q)Yj<^P+'in + 2-i)
(U-IK2
(22)
+2(2p){5)Ypp + 2n + 2~i) + 2{2p)(2)Qp + 2n + \) + 4(2p)(2) £ (3p +2« +1-2i)
M	M
= \20 p2n +60 pn~ + 12pn + 12p2 + 28 p
Case 4. When n > n - 1 and n is even.
In this case the eccentricity of the vertices x^., y(i+1)j, x(iy2)j y(iy2j is same as the eccentricity of the vertices
x(2n+2-i)r y^-j x(2n+1-i)j, y^n+l-ij where i = 1,2, "', n/2.
The eccentricity of these vertices in (i y 1)th row is given by
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Povzetek
Med molekulske strukturne deskriptorje spada tudi »eccentric connectivity« indeks (ECI), ki je bil pred kratkim uporabljen za matematično modeliranje raznovrstnih bioloških aktivnosti. V primerjavi z Wienerjevim indeksom, daje ECI visoko stopnjo predvidljivosti v primeru diuretične in protivnetne aktivnosti. Stopnja natančnosti napovedi indeksa ECI je boljša od zagebškega indeksa v primeru antikonvulzivne aktivnosti. Med kovinskimi oksidi predstavljajo nanocevke Ti-O2 material, ki ima veliko tehnološko uporabnost. Številne študije tega materiala zahtevajo tudi teoretične študije njegovih lastnosti. Nedavno je bil za nanocevke TiO2 določen zagrebški indeks, v tem prispevku pa preučujemo indeks ECI.
Imran Nadeem and Hani Shaker: On Eccentric Connectivity Index of TiO2 Nanotubes