# Eccentric Connectivity Index oft-Polyacenic Nanotubes

Authors: Jia-Bao Liu; Hani Shaker; Imran Nadeem; Mohammad Reza Farahani
Journal: Advances in Materials Science and Engineering
 (2019)

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## Abstract

The eccentric connectivity index ECI is a chemical structure descriptor that is currently being used for the modeling of biological activities of a chemical compound. This index has been proved to provide a high degree of predictability as compared to some other well-known indices in case of anticonvulsant, anti-inflammatory, and diuretic activities. The ECI of an infinite class of 1-polyacenic (phenylenic) nanotubes has been recently studied. In this article, we extend this study to generalized polyacenic nanotubes and find ECI oft-polyacenic nanotubes for t>1.


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## Body

## 1. Introduction

A basic concept of chemistry is that the properties/activities of a molecule depend upon its structural characteristics. Molecular graphs can be used to model the chemical structures of molecules and molecular compounds, by considering atoms as vertices and the chemical bonds between the atoms as edges. In the study of quantitative structure-property and structure-activity relationships (QSPR/QSAR), the topological indices are very helpful in detecting the biological activities of a chemical compound [1–4].A topological index is a numerical graph invariant that is used to correlate the chemical structure of a molecule with its physicochemical properties and biological activities. Generally, topological indices are classified into five generations: first-generation topological indices are integer numbers obtained by simple operations from local vertex invariants involving only one vertex at a time. Some of the famous topological indices of this class are Wiener index, Hosoya index, and Centric indices of Balaban [5]. Second-generation topological indices are real numbers based on integer graph properties. These indices were obtained via structural operations from integer local vertex invariants, involving more than one vertex at a time. Some examples of this class include molecular connectivity indices, Balaban J index, bond connectivity indices, and kappa shape indices [5]. Third-generation topological indices are real numbers which are based on local properties of the molecular graph. These indices are of recent introduction and have very low degeneracy. These are based on information theory applied to the terms of distance sums or on newly introduced nonsymmetrical matrices. Some examples include information indices [6], the hyper-Wiener index [5], the Kirchhoff index [7], and electrotopological state indices [2]. Recently, fourth- and fifth-generation topological indices are placed as new generations topological indices. Fourth-generation topological indices are of highly discriminating power, i.e., ≥100. The examples of fourth-generation topological indices include eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and superaugmented eccentric connectivity topochemical indices [10]. Detour matrix-based adjacent path eccentric distance sum indices [11] belong to the fifth-generation topological indices.LetG be a connected molecular graph with vertex set VG and edge EG. Let IG be the set of those edges of G that are incident to a vertex k∈VG, and then the degree of k is denoted by dk and is defined as the cardinality of IG. The distance from a vertex k∈VG to a vertex l∈VG is denoted by dk,l and is defined as the minimum number of edges lying between them. The eccentricity εk of a given vertex k∈VG is defined as the largest distance between k and any vertex l of G.Sharma et al. in [8] have presented a distance-based chemical structure descriptor, called the eccentric connectivity index (ECI), which is presented as(1)ξcG=∑k∈VGdk⋅εk.It is recorded in [12–16] that ECI provides good correlations with regard to physicochemical properties and biological activities. This index is reported as a highly discriminating descriptor for QSPR/QSAR studies [8, 9, 17]. The degree of prediction of ECI is better than the Wiener index in case of diuretic activity [18] and anti-inflammatory activity in [19]. Also, this index has been proved to provide a high degree of predictability with regard to anticonvulsant activity [20] in comparison to Zagreb indices. Recently, the eccentric connectivity index has been studied for certain nanotubes [21–26] and for several molecular graphs [27–29].Polyacenes relate to a family of polycyclic aromatic hydrocarbon (PAH) compounds which are formed by the linearly fused benzene rings. Numerous molecules of this class have interesting optical, thermodynamic, electronic, ferromagnetic, and photoconductive properties [30–33]. In the first organic solid-state injection laser, the lasing was discovered by using the single crystals of tetracene [34, 35]. They have application in rechargeable Li-ion batteries [36] and also have presence in various celestial objects like planetary nebulae [37]. In this sense, the polyacenes have received much attention. The PI index of linear polyacenes has been studied in [38]. The molecular graphs of certain linear polyacene molecules are given in Figure 1.Figure 1
The polyacene molecules. (a) L1: Benzene; (b) L2: Naphthalene; (c) L3: Anthracene; (d) L4: Tetracene; (e) L5: Pentacene; (f) L6: Hexacene; (g) L7: Heptacene.
(a)
(b)
(c)
(d)
(e)
(f)
(g)Recently, the Zagreb indices of 3-polyacenic (anthracenic), 4-polyacenic (tetracenic), and 5-polyacenic (pentacenic) nanotubes have been studied in [39–41], respectively. The ECI of 1-polyacenic (phenylenic) nanotubes has been presented in [25]. In this paper, we generalize these results to t-polyacenic nanotubes for t>1 and present the ECI for these nanotubes.


## 2. Main Results

The generalized molecular graph of thet-polyacenic nanotube Ptp,q is shown in Figure 2. In this graphical representation, q counts the number of polyacene units in a row and p counts the number of alternative polyacene units in a column of the t-polyacenic nanotube, where a polyacene unit consists of t hexagons. The molecular graph of the t-polyacenic nanotube Ptp,q has 4p rows and q columns. For 1≤t≤6, the t-polyacenic nanotube Ptp,q is known as phenylenic, naphthalenic, anthacenic, tetracenic, pentacenic, and hexacenic nanotubes, respectively. The molecular graphs of these nanotubes are presented in Figure 3. Let G be a molecular graph of the Ptp,q nanotube and then we can observe that 2≤dk≤3 for each k∈VG. So, we have the vertex partitions of G as follows:(2)V2=k∈VG∣dk=2,V3=k∈VG∣dk=3.Figure 2
The generalized molecular graph of thet-polyacenic nanotube Ptp,q.Figure 3
The molecular graphs oft-polyacenic nanotubes Ptp,q for 1≤t≤6. (a) P14,3: phenylenic nanotube; (b) P24,3: naphthalenic nanotube; (c) P34,3: anthracenic nanotube; (d) P44,3: tetracenic nanotube; (e) P54,3: pentacenic nanotube; (f) P64,3: hexacenic nanotube.
(a)
(b)
(c)
(d)
(e)
(f)The vertex partitions ofG along with their cardinalities corresponding to each row are presented in Table 1. In the following theorems, we formulate the eccentric connectivity index for Ptp,q nanotubes for t>1.Table 1
The vertex partitions of thePtp,q nanotube along with their cardinalities corresponding to each row.
Vertex partition
Rows
Cardinality for each row
V
1
1,4
p
t
q
V
2
2,3,6,7
,
…
,
4
p
−
2,4
p
−
1
t
+
1
q
V
3
4,5,8,9
,
…
,
4
p
−
4,4
p
−
3
t
qTheorem 1.
LetPtp,q be the graph of the t-polyacenic nanotube, and then for q even, we have(3)ξcPtp,q=10t2+11t+3q2+25t+3q,ifp=1andt≥1,92t+1p2q+34t2+4t+1pq2−10t+3pq−t2t+1q2+2tq,if2≤p≤tq2andt≥1,3t22t+1q3+212t+1p2q−34t2−1pq2−14t+3pq−tq2+2tq,iftq+22≤p≤tq−1andt≥1,182t+1p2q+32t+1pq2−14t+3pq−tq2+2tq,ifp≥tq,pis  even  andt≥1,182t+1p2q+32t+1pq2−14t+3pq−tq2+2t−3q,ifp≥tq,pis  odd  andt≥1.Proof.
ConsiderG=Ptp,q. Let vi represents the vertices in the ith row. With respect to εvi or dvi, we have the following cases.Case 1 (whenp=1 and t≥1).
In this case, the eccentricity of each vertex in each row is2t+1q+2/2. Hence, from Table 1 and (1), we have(4)ξcG=2tq22t+1q+22+2t+1q32t+1q+22=10t2+11t+3q2+25t+3q.Case 2 (when2≤p≤tq/2 and t≥1).
In this case,(5)εvi=εvi+1=εv4p+1−i=εv4p−i=4p+2t+1q2−i,where i=1,2,…,p. Hence, from Table 1 and (1), we have(6)ξcG=2tq2+tq+q34p+2t+1q2−1+2tq3+tq+q3∑i=2p4p+2t+1q2−i=92t+1p2q+34t2+4t+1pq2−10t+3pq−t2t+1q2+2tq.Case 3 (whentq+2/2≤p≤tq−1 and t≥1).
In this case,(7)εvi=εv4p+1−i=8p+q2−i,wherei=1,2,…,4p−tq2.
Also,(8)εv4p−tq/2+i=εv4p−tq/2+1+i=εv2tq+1−i=εv2tq−i=4t+1q2−i,where i=1,2,…,tq−p. Hence, from Table 1 and (1), we have(9)ξcG=2tq28p+q2−1+2tq+q3+2p−tq28p+q2−2+6+⋯+4p−tq2−2+3+7+⋯+4p−tq2−1+2tq32p−tq2−18p+q2−4+8+⋯+4p−tq2+2tq35+9+⋯+4p−tq2−3+2tq3+tq+q3∑i=1tq−p4t+1q2−i,=3t22t+1q3+212t+1p2q−34t2−1pq2−14t+3pq−tq2+2tq.Case 4 (whenp≥tq, p is even and t≥1).
In this case,(10)εvi=εv4p+1−i=8p+q2−i,wherei=1,2,…,2p.
Hence, from Table1 and (1), we have(11)ξcG=2tq28p+q2−1+2tq+q3p8p+q2−2+6+⋯+2p−2+3+7+⋯+2p−1+2tq3p−18p+q2−4+8+⋯+2p+5+9+⋯+2p−3=182t+1p2q+32t+1pq2−14t+3pq−tq2+2tq.Case 5 (whenp≥tq, p is odd and t≥1).
In this case, we use the eccentricities of vertices as given in case 4. From Table1 and (1), we have(12)ξcG=2tq28p+q2−1+2tq+q3p8p+q2−2+6+⋯+2p+3+7+⋯+2p−3+2tq3p−18p+q2−4+8+⋯+2p−2+5+9+⋯+2p−1=182t+1p2q+32t+1pq2−14t+3pq−tq2+2t−3q.Theorem 2.
LetPtp,q be the graph of the t-polyacenic nanotube, and then for q odd, we have(13)ξcPtp,q=25t2+8t+3,ifp=1,q=1andt≥2,10t2+11t+3q2+5t+3q,ifp=1,q≥3andt≥1,92t+1p2q+34t2+4t+1pq2−28t+3pq−t2t+1q2+3tq,if2≤p≤tq2andtq≥4,3t22t+1q3+212t+1p2q−34t2−1pq2−210t+3pq−tq2+3tq,iftq+22≤p≤tq−1andteven,3t6t2+5t+1q3−32t+1p2q−34t2+4t+1pq2+62t+1pq+t+18t+3q2+11t+9q,ifp=tq+12andtodd,3t3q3+310t+3p2q+34t+1pq2+8t+3pq−t9t+5q2−63t−2q,iftq+32≤p≤tq−2andtodd,67t+3p2q+32t+1pq2−234t+15pq+t48t+23q2−39t+4q,ifp=tq−1andtodd,182t+1p2q+32t+1pq2−210t+3pq−tq2+3tq,ifp≥tq,pis  even  andt≥1,182t+1p2q+32t+1pq2−210t+3pq−tq2+3t−1q,ifp≥tq,pis  odd  andt≥1.Proof.
ConsiderG=Ptp,q. Let vi represent the vertices in the ith row of G. With respect to εvi or dvi, we have the following cases.Case 1 (whenp=1, q=1 and t≥2).
In this case, the eccentricity of each vertex in each row ist+1. Hence, from Table 1 and (1), we have(14)ξcG=2t2t+1+2t+13t+1=25t2+8t+3.Case 2 (whenp=1, q≥3 and t≥1).
In this case, the eccentricity of each vertex in each row is2t+1q+1/2. Hence, from Table 1 and (1), we have(15)ξcG=2tq22t+1q+12+2tq+q32t+1q+12=10t2+11t+3q2+5t+3q.Case 3 (when2≤p≤tq/2 and tq≥4).
In this case,(16)εvi=εvi+1=εv4p+1−i=εv4p−i=4p+2t+1q−12−i,where i=1,2,…,p. Hence, from Table 1 and (1), we have(17)ξcG=2tq2+t+1q34p+2t+1q−12−1+2tq3+t+1q3∑i=2p4p+2t+1q−12−i=92t+1p2q+34t2+4t+1pq2−28t+3pq−t2t+1q2+3tq.Case 4 (whentq+2/2≤p≤tq−1 and t is even).
In this case,(18)εvi=εv4p+1−i=8p+q−12−i,wherei=1,2,…,4p−tq2.
Also,(19)εv4p−tq/2+i=εv4p−tq/2+1+i=εv2tq+1−i=εv2tq−i=4t+1q−12−i,where i=1,2,…,tq−p. Hence, from Table 1 and (1), we have(20)ξcG=2tq28p+q−12−1+2tq+q32p−tq28p+q−12−2+6+⋯+4p−tq2−2+3+7+⋯+4p−tq2−1+2tq32p−tq2−18p+q−12−4+8+⋯+4p−tq2+2tq35+9+⋯+4p−tq2−3+2tq3+t+1q3∑i=1tq−p4t+1q−12−i=3t22t+1q3+212t+1p2q−34t2−1pq2−210t+3pq−tq2+3tq.Case 5 (whenp=tq+1/2 and t is odd).
In this case,(21)εv1=εv2=εv4p=εv4p−1=4tq+q+12,εvi+2=εv4p−1−i=εv4p=εv4p−1=4t+1q+12−i,wherei=1,2.
Also,(22)εvi+4=εvi+5=εv2tq−1−i=εv2tq−2−i=4t+1q−32−i,where i=1,2,…,tq−p−1. Hence, from Table 1 and (1), we have(23)ξcG=2tq2+tq+q34t+1q+12+2tq+q34t+1q+12−1+2tq34t+1q+12−2+2tq3+tq+q3∑i=1tq−p−14t+1q−32−i=3t6t2+5t+1q3−32t+1p2q−34t2+4t+1pq2+62t+1pq+t+18t+3q2+11t+9q.Case 6 (whentq+3/2≤p≤tq−2 and t is odd).
In this case,(24)εvi=εv4p+1−i=8p+q−12−i,wherei=1,2,…,4p−tq+12,εv4p−tq+1/2+1=εv4p−tq+1/2+2=εv2tq+2=εv2tq+1=4t+1q+12,εv4p−tq+1/2+2+i=εv2tq+1−i=4t+1q+12−i,wherei=1,2.
Also,(25)εv4p−tq+1/2+4+i=εv4p−tq+1/2+5+i=εv2tq−1−i=εv2tq−2−i=4t+1q−32−i,where i=1,2,…,tq−p−1. Hence, from Table 1 and (1), we have(26)ξcG=2tq28p+q−12−1+2tq+q32p−tq+128p+q−12−2+6+⋯+4p−tq+12−2+2tq+q33+7+⋯+4p−tq+12−1+2tq32p−tq+12−18p+q−12−4+8+⋯+4p−tq+12+2tq35+9+⋯+4p−tq+12−3+2tq3+tq+q34t+1q+12+2tq+q34t+1q+12−1+2tq34t+1q+12−2+2tq3+tq+q3∑i=1tq−p−14t+1q−32−i=3t3q3+310t+3p2q+34t+1pq2+8t+3pq−t9t+5q2−63t−2q.Case 7 (whenp=tq−1 and t is odd).
In this case,(27)εvi=εv4p+1−i=8p+q−12−i,wherei=1,2,…,2p−2,εv2p−3=εv2p−2=εv2p+3=εv2p+4=4t+1q+12.
Also,(28)εv2p−2+i=εv2p+3−i=εv2p+3=εv2p+4=4t+1q+12−i,wherei=1,2.
Hence, from Table1 and (1), we have(29)ξcG=2tq28p+q−12−1+2tq+q3p−28p+q−12−2+6+⋯+2p−2+3+7+⋯+2p−2−1+2tq3p−38p+q−12−4+8+⋯+2p−2+5+9+⋯+2p−2−3+2tq3+tq+q34t+1q+12+2tq+q34t+1q+12−1+2tq34t+1q+12−2=67t+3p2q+32t+1pq2−234t+15pq+t48t+23q2−39t+4q.Case 8 (whenp≥tq, p is even and t≥1).
In this case,(30)εvi=εv4p+1−i=8p+q−12−i,wherei=1,2,…,2p.
Hence, from Table1 and (1), we have(31)ξcG=2tq28p+q−12−1+2tq+q3p8p+q−12−2+6+⋯+2p−2+3+7+⋯+2p−1+2tq3p−18p+q−12−4+8+⋯+2p+5+9+⋯+2p−3=182t+1p2q+32t+1pq2−210t+3pq−tq2+3tq.Case 9 (whenp≥tq, p is odd and t≥1).
In this case, we use the eccentricities of vertices as given in case 8. From Table1 and (1), we have(32)ξcG=2tq28p+q−12−1+2tq+q3p8p+q−12−2+6+⋯+2p+3+7+⋯+2p−3+2tq3p−18p+q−12−4+8+⋯+2p−2+5+9+⋯+2p−1=182t+1p2q+32t+1pq2−210t+3pq−tq2+3t−1q.Remark 1.
The results presented by Rao and Lakshmi in [25] become special cases of the results given in Theorems 1 and 2 for t=1.


## 3. Conclusion

In this paper, we present generalized formulae of ECI fort-polyacenic nanotubes. The comparability about biological activities of chemical compounds is of immense interest in QSAR/QSPR studies. The eccentric connectivity index ECI provides the best prediction accuracy rate compared to other indices in various biological activities of diverse nature such as anti-inflammatory activity, anticonvulsant activity, and diuretic activity. In this sense, this index can be very helpful in QSAR/QSPR studies, and by using the given results, we can present mathematical models of several biological activities of all chemical compounds, which correspond to t-polyacenic nanotubes such as phenylenic nanotubes, naphthalenic nanotubes, anthracenic nanotubes, tetracenic nanotubes, pentacenic nanotubes, and hexacenic nanotubes.



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