# Topological Indices of Certain Transformed Chemical Structures

Authors: Xuewu Zuo; Jia-Bao Liu; Hifza Iqbal; Kashif Ali; Syed Tahir Raza Rizvi
Journal: Journal of Chemistry
 (2020)

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## Abstract

Topological indices like generalized Randić index, augmented Zagreb index, geometric arithmetic index, harmonic index, product connectivity index, general sum-connectivity index, and atom-bond connectivity index are employed to calculate the bioactivity of chemicals. In this paper, we define these indices for the line graph ofk-subdivided linear [n] Tetracene, fullerene networks, tetracenic nanotori, and carbon nanotube networks.


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## Body

## 1. Introduction

In chemical graph theory, we apply the concepts of graph theory to describe the mathematical model of a variety of chemical structures. The atoms of the molecules correspond to the vertices, and the chemical bond is reflected by edges. Topological indices are numerical parameters of chemical graphs associated with quantitative structure property relationship (QSPR) and quantitative structure activity relationship (QSAR). The major topological indices are distance based, degree based, and eccentricity based. Among these classes, degree-based topological indices are of great importance and are helpful tools for chemists. The concept of topological index came from the work done by Wiener, when he was working on boiling point of paraffin [1]. The Wiener index is the first and most studied topological index. The degree-based topological indices for line graph of some subdivided graphs were studied in [2]. In [3], the bounds of topological indices for some graph operations are discussed. Baĉa et al. studied some indices for families of fullerene graph in [4]. Baig et al. found the topological indices for poly oxide, poly silicate, DOX, and DSL networks in [5]. Liu et al. found the different topological indices for Eulerian graphs, fractal graphs, and generalized Sierpinski networks in [6–8]. The number of spanning trees and normalized Laplacian of linear octagonal quadrilateral networks were studied in [9]. Recently, Liu et al. in [10] calculated the generalized adjacency, Laplacian, and signless Laplacian spectra of the weighted edge corona networks. In [11], Gao et al. found the forgotten topological index on chemical structure in drugs. Imran et al. calculated the degree-based topological indices for different networks in [12–15]. In 2018, Mufti et al. [16] found the topological indices for para-line graphs of pentacene. Nadeem et al. calculated the degree-based topological indices for para-line graphs of V-Phenylenic nanostructures in [17].Randić in 1975 introduced the Randić index [18]. Bollobas and Erdos generalized the Randić index for any real number α and named it as generalized Randić index:(1)RαG=∑uv∈EGdudvα.Recall that the augmented Zagreb index is [19](2)AG=∑uv∈EGdudvdu+dv−23.Furtula et al. defined the geometric arithmetic index as [19](3)GAG=∑uv∈EG2dudvdu+dv.Moreover, the harmonic index is defined as follows [20]:(4)HG=∑uv∈EG2du+dv.The first degree-based connectivity index for graphs evolved by using vertex degree is product connectivity index (Randić index), proposed by the chemist Randić, as [18](5)RG=∑uv∈EG1dudv.The general sum-connectivity index has been introduced in 2010, as [21](6)χαG=∑uv∈EGdu+dvα.One of the well-known degree-based topological indices is the atom-bond connectivity (ABC) index of a graph, proposed by Estrada et al. and defined as [22](7)ABCG=∑uv∈EGdu+dv−2dudv.The remaining article is characterized as follows. In Section2, the topological indices for the line graph of subdivided graph of different nanostructures have been discussed. The conclusion has been drawn in Section 3.


## 2. Main Results

LetG be a finite, simple, and connected graph with order p and size q. For k≥1, a k-subdivided graph Gk of G is obtained by replacing each edge of graph G by a path Pk+2. A line graph LG of graph G is a transformed graph having q vertices and two vertices have common neighbourhood in LG if and only if their corresponding edges are adjacent in G.


### 2.1. Linear [n] Tetracene

We will start the debate from linear [n] Tetracene, by defining its topology. It has the appearance of a pale orange powder. Tetracene is a four ringed member of the series of acnes. The original graph has order 18n and size 23n−2. The line graph of subdivided graph has 2,2, 2,3, and 3,3 types of edges. For subdivision k=1, the number of edges will be 8n+10, 16n−4, and 37n−16, respectively. Similarly, for subdivision k=2, the number of edges will be 24n+12, 30n−12, and 30n−12, respectively. Further, for subdivision k≥3, the number of edges will be 24n+12+k−223n−2, 30n−12, and 30n−12, respectively. Let Gn,k be the line graph of k-subdivided linear[n] Tetracene with k≥3 vertices. Its topological indices are calculated in the next theorems.Theorem 1.
LetGn,k be the line graph of k-subdividing linear [n] Tetracene with k≥3. The generalized Randić index, augmented Zagreb index, geometric arithmetic index, and harmonic index of Gn,k are(1)
RαGn,k=22αk−223n−2+3.22α+22n+1+2α+1.3α+1+2.32α+15n−2(2)
AGn,k=8k−223n−2+24759n/32−2187/16(3)
GAGn,k=k−223n−2+626+9n−246/5(4)
HGn,k=k−2/223n−2+34n−14/5Proof.
The generalized Randić index ofGn,k is computed as follows:(8)RαGn,k=∑uv∈EGn,kdudvα=∑uv∈E2,2dudvα+∑uv∈E2,3dudvα+∑uv∈E3,3dudvα=k−223n−2+24n+124α+30n−126α+30n−129α=22αk−223n−2+3.22α+22n+1+2α+1⋅3α+1+2.32α+15n−2.
The augmented Zagreb index is computed as(9)AGn,k=∑uv∈EGn,kdudvdu+dv−23=∑uv∈E2,2dudvdu+dv−23+∑uv∈E2,3dudvdu+dv−23+∑uv∈E3,3dudvdu+dv−23=k−223n−2+24n+1242+2−23+30n−1262+3−23+30n−1293+3−23=8k−223n−2+24759n32−218716.
Next, the geometric arithmetic index is computed as(10)GAGn,k=∑uv∈EGn,k2dudvdu+dv=∑uv∈E2,22dudvdu+dv+∑uv∈E2,32dudvdu+dv+∑uv∈E3,32dudvdu+dv=k−223n−2+24n+12242+2+30n−12262+3+30n−12293+3=k−223n−2+626+9n−2465.
Moreover, the harmonic index is defined as follows:(11)HGn,k=∑uv∈EGn,k2du+dv=∑uv∈E2,22du+dv+∑uv∈E2,32du+dv+∑uv∈E3,32du+dv=k−223n−2+24n+1222+2+30n−1222+3+30n−1223+3=k−2223n−2+34n−145,which completes the proof of the theorem.Theorem 2.
The product connectivity index, general sum-connectivity index, and atom-bond connectivity index ofGn,k are(1)
RGn,k=k−2/223n−2+22+103/2n−26+1(2)
χαGn,k=22αk−223n−2+6.5α+2α+1⋅3α+15n−2(3)
ABCGn,k=k−2/223n−2+272+20n−8Proof.
The product connectivity index ofGn,k is computed as(12)RGn,k=∑uv∈EGn,k1dudv=∑uv∈E2,21dudv+∑uv∈E2,31dudv+∑uv∈E3,31dudv=k−223n−2+24n+1214+30n−1216+30n−1219=k−2223n−2+22+1032n−26+1.
The general sum-connectivity index is computed as(13)χαGn,k=∑uv∈EGn,kdu+dvα=∑uv∈E2,2du+dvα+∑uv∈E2,3du+dvα+∑uv∈E3,3du+dvα=k−223n−2+24n+122+2α+30n−122+3α+30n−123+3α=22αk−223n−2+6.5α+2α+1⋅3α+15n−2.
The atom-bond connectivity (ABC) index is computed as(14)ABCGn,k=∑uv∈EGn,kdu+dv−2dudv=∑uv∈E2,2du+dv−2dudv+∑uv∈E2,3du+dv−2dudv+∑uv∈E2,3du+dv−2dudv=k−223n−2+24n+122+2−24+30n−122+3−26+30n−123+3−29=k−2223n−2+272+20n−8,which completes the proof of the theorem.


### 2.2. Tetracenic Nanotori

Next, we have nanostructuresF, G, K, and L. The original graph of each structure has same order 18pq, and size of F is 27pq−4p−2q, G is 27pq−4p, K is 27pq−2q, and L is 27pq. Let r and s be the number of vertices with degree 2 and 3, respectively. The types of edges of line graph of subdivided graph for each nanostructure will be 2,2, 2,3, and 3,3. For subdivision k=1, the number of edges for F will be 2q+r+4, 16p+4q−8, and 27pq−20p−8q+3s+4, respectively, the number of edges for G will be r, 16p, and 27pq−20p+3s, respectively, the number of edges for K will be 2q+r, 4q, and 27pq−8q+3s, respectively, and the number of edges for L will be r, 0, and 27pq+3s, respectively. Similarly, for subdivision k=2, the number of edges for F will be 16p+8q+r, 54pq−24p−12q, and 3s, respectively, the number of edges for G will be 16p+r, 54pq−24p, and 3s, respectively, the number of edges for K will be 8q+r, 54pq−12q, and 3s, respectively, and the number of edges for L will be r, 54pq, and 3s, respectively. Further, for subdivision k≥3, the number of edges for F will be 16p+8q+r+27pq−4p−2qk−2, 54pq−24p−12q, and 3s, respectively, the number of edges for G will be 16p+r+27pq−4pk−2, 54pq−24p, and 3s, respectively, the number of edges for K will be 8q+r+27pq−2qk−2, 54pq−12q, and 3s, respectively, and the number of edges for L will be r+27pqk−2, 54pq, and 3s, respectively.Theorem 3.
LetGn,k be the line graph of subdivided nanostructures F, G, K, and L, by k≥3 vertices. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index are as follows.
ForF:(1)
RαGn,k=22αk−227pq−4p−2q+16p+8q+r+2α+1⋅3α+19pq−4p−2q+32α+1s(2)
AGn,k=8k−227pq−4p−2q+54pq−8p−4q+r+2187s/64(3)
GAGn,k=k−227pq−4p−2q+16p+8q+r+3s+126/59pq−4p−2q(4)
HGn,k=k−2/227pq−4p−2q+108/5pq−8/5p−4/5q+r/2+s
ForG:(1)
RαGn,k=22αk−227pq−4p+16p+r+2α+1⋅3α+19pq−4p+32α+1s(2)
AGn,k=8k−227pq−4p+54pq−8p+r+2187s/64(3)
GAGn,k=k−227pq−4p+16p+r+3s+126/59pq−4p(4)
HGn,k=k−2/227pq−4p+108/5pq−8/5p+r/2+s
ForK:(1)
RαGn,k=22αk−227pq−2q+8q+r+2α+1⋅3α+19pq−2q+32α+1s(2)
AGn,k=8k−227pq−2q+54pq−4q+r+2187s/64(3)
GAGn,k=k−227pq−2q+8q+r+3s+126/59pq−2q(4)
HGn,k=k−2/227pq−2q+108/5pq−4/5q+r/2+s
ForL:(1)
RαGn,k=22αk−227pq+r+2α+1⋅3α+3pq+32α+1s(2)
AGn,k=8k−227pq+54pq+r+2187s/64(3)
GAGn,k=k−227pq+r+3s+1086/5pq(4)
HGn,k=k−2/227pq+108/5pq+r/2+sProof.
By using equations (1)–(4), we get the required results.Theorem 4.
LetGn,k be the line graph obtained after subdividing nanostructures F, G, K, and L by k≥3 vertices. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, are as follows.
ForF:(1)
RGn,k=k−2/227pq−4p−2q+96pq+42−6p+22−6q+r/2+s(2)
χαGn,k=22αk−227pq−4p−2q+16p+8q+r+6.5α9pq−4p−2q+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−4p−2q+54pq−8p−4q+r+2s
ForG:(1)
RGn,k=k−2/227pq−4p+96pq+42−6p+r/2+s(2)
χαGn,k=22αk−227pq−4p+16p+r+6.5α9pq−4p+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−4p+54pq−8p+r+2s
ForK:(1)
RGn,k=k−2/227pq−2q+96pq+22−6q+r/2+s(2)
χαGn,k=22αk−227pq−2q+8q+r+6.5α9pq−2q+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−2q+54pq−4q+r+2s
ForL:(1)
RGn,k=k−2/227pq+96pq+r/2+s(2)
χαGn,k=22αk−227pq+r+54.5αpq+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq+54pq+r+2sProof.
By using equations (5), (6), and (8), we get the required results.


### 2.3. Fullerene Networks

The next model we are going to add is of fullerene. It is a regular graph of degree 3. LetKBmn be the Klein-bottle fullerene and Hmn be the toroidal fullerene for n≥2even and m≥2 having order 2mn and size 3mn and KBm+1/2n be the Klein-bottle fullerene for n≥2even and m≥1 with order 2nm+1/2 and size 3nm+1/2. The line graph of subdivided graph for k=1 each structure is a 3-regular graph, with KBmn and Hmn having same 9mn edge count, whereas KBm+1/2n has 9nm+1/2 edges. Similarly, for subdivision k=2, the types of edges are 2,3 and 3,3. Both KBmn and Hmn have same count 6mn in each type of edges, and for KBm+1/2n, each type has again same count 6nm+1/2. For subdivision k≥3, the fullerenes KBmn and Hmn have (2,3), (3,3) and (2,2) types of edges with counts 6mn, 6mn and 3mn(k−2), respectively. The fullerene KBm+1/2n also has (2,3), (3,3) and (2,2) types of edges, and their count is 3n(2m+1), 3n(2m+1), and 3n(m+1/2)(k−2), respectively.Theorem 5.
LetGn,k be the line graph of k-subdivided Klein-bottle fullerenes, KBmn and, Hmn for n≥2even and m≥2, further Klein-bottle fullerene, KBm+1/2n for n≥2even and m≥1, with k≥3. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index are
ForKBmn:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1mn(2)
AGn,k=38k−2+1241/32mn(3)
GAGn,k=3k−2+46/5+2mn(4)
HGn,k=3k−2/2+22/5mn
ForKBm+1/2n:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1nm+1/2(2)
AGn,k=38k−2+1241/32nm+1/2(3)
GAGn,k=3k−2+46/5+2nm+1/2(4)
HGn,k=3k−2/2+22/5nm+1/2
ForHmn:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1mn(2)
AGn,k=38k−2+1241/32mn(3)
GAGn,k=3k−2+46/5+2mn(4)
HGn,k=3k−2/2+22/5mnProof.
By using equations (1)–(4), we get the required results.Theorem 6.
LetGn,k be the line graph of k-subdivided Klein-bottle fullerene, KBmn and toroidal fullerene, Hmn for n≥2even and m≥2, further Klein-bottle fullerene, KBm+1/2n for n≥2even and m≥1, by k≥3. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, are
ForKBmn:(1)
RGn,k=3k−2/2+6+2mn(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=3k−2/2+2+4mn
ForKBm+1/2n:(1)
RGn,k=3k−2/2+6+2nm+1/2(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1nm+1/2(3)
ABCGn,k=3k−2/2+2+4nm+1/2
ForHmn:(1)
RGn,k=3k−2/2+6+2mn(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=3k−2/2+2+4mnProof.
By using equations (5)–(7), we get the required results.


### 2.4. Carbon Nanotube Networks

LetNAmn be the nanotube, for m,n≥2 with order 2mn+1 and size m3n+2. Let NBmn be the nanotube, for n≥2even and m≥1 with order 2nm+1 and size n3m+2. In the line graph of subdivided graph, both structures have 2,2, 2,3, and 3,3 types of edges. For k=1, NAmn has 2m, 4m, and 9mn−2m edges, respectively, whereas NBmn has 3n, 2n, and 9mn−n edges, respectively. Similarly, for subdivision k=2, NAmn has 6m and last two types have same 6mn count of edges, whereas NBmn has 6n and again last two types have same 6mn count of edges. Further, for subdivision k≥3, NAmn has 6m+k−23mn+2m and last two types have same 6mn count of edges, whereas NBmn has 6n+k−23mn+2n and last two types have same 6mn count of edges.Theorem 7.
LetGn,k be the line graph of k-subdivided NAmn nanotube, for m,n≥2, and NBmn nanotube, for n≥2even and m≥1 with k≥3. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index areForNAmn(1)
RαGn,k=22αk−23mn+2m+3.22α+1m+3α+1⋅2α+1+2.32α+1mn(2)
AGn,k=8k−23mn+2m+48m+2187/32mn(3)
GAGn,k=k−23mn+2m+6m+626/5+1mn(4)
HGn,k=k−2/23mn+2m+3m+17/5mnForNBmn(1)
RαGn,k=22αk−23mn+2n+3.22α+1n+3α+1⋅2α+1+2.32α+1mn(2)
AGn,k=8k−23mn+2n+48n+2187/32mn(3)
GAGn,k=k−23mn+2n+6n+626/5+1mn(4)
HGn,k=k−2/23mn+2n+3n+17/5mnProof.
By using equations (1)–(4), we get the required results.Theorem 8.
LetGn,k be the line graph of k-subdivided NAmn nanotube, for m,n≥2, and NBmn nanotube, for n≥2even and m≥1 with k≥3. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, areForNAmn:(1)
RGn,k=k−2/23mn+2m+3m+6+2mn(2)
χαGn,k=22αk−23mn+2m+3.22α+1m+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=k−2/23mn+2m+32m+32+4mnForNBmn:(1)
RGn,k=k−2/23mn+2n+3n+6+2mn(2)
χαGn,k=22αk−23mn+2n+3.22α+1n+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=k−2/23mn+2n+32n+32+4mnProof.
By using equations (5)–(7), we get the required results.


## 2.1. Linear [n] Tetracene

We will start the debate from linear [n] Tetracene, by defining its topology. It has the appearance of a pale orange powder. Tetracene is a four ringed member of the series of acnes. The original graph has order 18n and size 23n−2. The line graph of subdivided graph has 2,2, 2,3, and 3,3 types of edges. For subdivision k=1, the number of edges will be 8n+10, 16n−4, and 37n−16, respectively. Similarly, for subdivision k=2, the number of edges will be 24n+12, 30n−12, and 30n−12, respectively. Further, for subdivision k≥3, the number of edges will be 24n+12+k−223n−2, 30n−12, and 30n−12, respectively. Let Gn,k be the line graph of k-subdivided linear[n] Tetracene with k≥3 vertices. Its topological indices are calculated in the next theorems.Theorem 1.
LetGn,k be the line graph of k-subdividing linear [n] Tetracene with k≥3. The generalized Randić index, augmented Zagreb index, geometric arithmetic index, and harmonic index of Gn,k are(1)
RαGn,k=22αk−223n−2+3.22α+22n+1+2α+1.3α+1+2.32α+15n−2(2)
AGn,k=8k−223n−2+24759n/32−2187/16(3)
GAGn,k=k−223n−2+626+9n−246/5(4)
HGn,k=k−2/223n−2+34n−14/5Proof.
The generalized Randić index ofGn,k is computed as follows:(8)RαGn,k=∑uv∈EGn,kdudvα=∑uv∈E2,2dudvα+∑uv∈E2,3dudvα+∑uv∈E3,3dudvα=k−223n−2+24n+124α+30n−126α+30n−129α=22αk−223n−2+3.22α+22n+1+2α+1⋅3α+1+2.32α+15n−2.
The augmented Zagreb index is computed as(9)AGn,k=∑uv∈EGn,kdudvdu+dv−23=∑uv∈E2,2dudvdu+dv−23+∑uv∈E2,3dudvdu+dv−23+∑uv∈E3,3dudvdu+dv−23=k−223n−2+24n+1242+2−23+30n−1262+3−23+30n−1293+3−23=8k−223n−2+24759n32−218716.
Next, the geometric arithmetic index is computed as(10)GAGn,k=∑uv∈EGn,k2dudvdu+dv=∑uv∈E2,22dudvdu+dv+∑uv∈E2,32dudvdu+dv+∑uv∈E3,32dudvdu+dv=k−223n−2+24n+12242+2+30n−12262+3+30n−12293+3=k−223n−2+626+9n−2465.
Moreover, the harmonic index is defined as follows:(11)HGn,k=∑uv∈EGn,k2du+dv=∑uv∈E2,22du+dv+∑uv∈E2,32du+dv+∑uv∈E3,32du+dv=k−223n−2+24n+1222+2+30n−1222+3+30n−1223+3=k−2223n−2+34n−145,which completes the proof of the theorem.Theorem 2.
The product connectivity index, general sum-connectivity index, and atom-bond connectivity index ofGn,k are(1)
RGn,k=k−2/223n−2+22+103/2n−26+1(2)
χαGn,k=22αk−223n−2+6.5α+2α+1⋅3α+15n−2(3)
ABCGn,k=k−2/223n−2+272+20n−8Proof.
The product connectivity index ofGn,k is computed as(12)RGn,k=∑uv∈EGn,k1dudv=∑uv∈E2,21dudv+∑uv∈E2,31dudv+∑uv∈E3,31dudv=k−223n−2+24n+1214+30n−1216+30n−1219=k−2223n−2+22+1032n−26+1.
The general sum-connectivity index is computed as(13)χαGn,k=∑uv∈EGn,kdu+dvα=∑uv∈E2,2du+dvα+∑uv∈E2,3du+dvα+∑uv∈E3,3du+dvα=k−223n−2+24n+122+2α+30n−122+3α+30n−123+3α=22αk−223n−2+6.5α+2α+1⋅3α+15n−2.
The atom-bond connectivity (ABC) index is computed as(14)ABCGn,k=∑uv∈EGn,kdu+dv−2dudv=∑uv∈E2,2du+dv−2dudv+∑uv∈E2,3du+dv−2dudv+∑uv∈E2,3du+dv−2dudv=k−223n−2+24n+122+2−24+30n−122+3−26+30n−123+3−29=k−2223n−2+272+20n−8,which completes the proof of the theorem.


## 2.2. Tetracenic Nanotori

Next, we have nanostructuresF, G, K, and L. The original graph of each structure has same order 18pq, and size of F is 27pq−4p−2q, G is 27pq−4p, K is 27pq−2q, and L is 27pq. Let r and s be the number of vertices with degree 2 and 3, respectively. The types of edges of line graph of subdivided graph for each nanostructure will be 2,2, 2,3, and 3,3. For subdivision k=1, the number of edges for F will be 2q+r+4, 16p+4q−8, and 27pq−20p−8q+3s+4, respectively, the number of edges for G will be r, 16p, and 27pq−20p+3s, respectively, the number of edges for K will be 2q+r, 4q, and 27pq−8q+3s, respectively, and the number of edges for L will be r, 0, and 27pq+3s, respectively. Similarly, for subdivision k=2, the number of edges for F will be 16p+8q+r, 54pq−24p−12q, and 3s, respectively, the number of edges for G will be 16p+r, 54pq−24p, and 3s, respectively, the number of edges for K will be 8q+r, 54pq−12q, and 3s, respectively, and the number of edges for L will be r, 54pq, and 3s, respectively. Further, for subdivision k≥3, the number of edges for F will be 16p+8q+r+27pq−4p−2qk−2, 54pq−24p−12q, and 3s, respectively, the number of edges for G will be 16p+r+27pq−4pk−2, 54pq−24p, and 3s, respectively, the number of edges for K will be 8q+r+27pq−2qk−2, 54pq−12q, and 3s, respectively, and the number of edges for L will be r+27pqk−2, 54pq, and 3s, respectively.Theorem 3.
LetGn,k be the line graph of subdivided nanostructures F, G, K, and L, by k≥3 vertices. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index are as follows.
ForF:(1)
RαGn,k=22αk−227pq−4p−2q+16p+8q+r+2α+1⋅3α+19pq−4p−2q+32α+1s(2)
AGn,k=8k−227pq−4p−2q+54pq−8p−4q+r+2187s/64(3)
GAGn,k=k−227pq−4p−2q+16p+8q+r+3s+126/59pq−4p−2q(4)
HGn,k=k−2/227pq−4p−2q+108/5pq−8/5p−4/5q+r/2+s
ForG:(1)
RαGn,k=22αk−227pq−4p+16p+r+2α+1⋅3α+19pq−4p+32α+1s(2)
AGn,k=8k−227pq−4p+54pq−8p+r+2187s/64(3)
GAGn,k=k−227pq−4p+16p+r+3s+126/59pq−4p(4)
HGn,k=k−2/227pq−4p+108/5pq−8/5p+r/2+s
ForK:(1)
RαGn,k=22αk−227pq−2q+8q+r+2α+1⋅3α+19pq−2q+32α+1s(2)
AGn,k=8k−227pq−2q+54pq−4q+r+2187s/64(3)
GAGn,k=k−227pq−2q+8q+r+3s+126/59pq−2q(4)
HGn,k=k−2/227pq−2q+108/5pq−4/5q+r/2+s
ForL:(1)
RαGn,k=22αk−227pq+r+2α+1⋅3α+3pq+32α+1s(2)
AGn,k=8k−227pq+54pq+r+2187s/64(3)
GAGn,k=k−227pq+r+3s+1086/5pq(4)
HGn,k=k−2/227pq+108/5pq+r/2+sProof.
By using equations (1)–(4), we get the required results.Theorem 4.
LetGn,k be the line graph obtained after subdividing nanostructures F, G, K, and L by k≥3 vertices. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, are as follows.
ForF:(1)
RGn,k=k−2/227pq−4p−2q+96pq+42−6p+22−6q+r/2+s(2)
χαGn,k=22αk−227pq−4p−2q+16p+8q+r+6.5α9pq−4p−2q+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−4p−2q+54pq−8p−4q+r+2s
ForG:(1)
RGn,k=k−2/227pq−4p+96pq+42−6p+r/2+s(2)
χαGn,k=22αk−227pq−4p+16p+r+6.5α9pq−4p+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−4p+54pq−8p+r+2s
ForK:(1)
RGn,k=k−2/227pq−2q+96pq+22−6q+r/2+s(2)
χαGn,k=22αk−227pq−2q+8q+r+6.5α9pq−2q+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq−2q+54pq−4q+r+2s
ForL:(1)
RGn,k=k−2/227pq+96pq+r/2+s(2)
χαGn,k=22αk−227pq+r+54.5αpq+2α⋅3α+1s(3)
ABCGn,k=1/2k−227pq+54pq+r+2sProof.
By using equations (5), (6), and (8), we get the required results.


## 2.3. Fullerene Networks

The next model we are going to add is of fullerene. It is a regular graph of degree 3. LetKBmn be the Klein-bottle fullerene and Hmn be the toroidal fullerene for n≥2even and m≥2 having order 2mn and size 3mn and KBm+1/2n be the Klein-bottle fullerene for n≥2even and m≥1 with order 2nm+1/2 and size 3nm+1/2. The line graph of subdivided graph for k=1 each structure is a 3-regular graph, with KBmn and Hmn having same 9mn edge count, whereas KBm+1/2n has 9nm+1/2 edges. Similarly, for subdivision k=2, the types of edges are 2,3 and 3,3. Both KBmn and Hmn have same count 6mn in each type of edges, and for KBm+1/2n, each type has again same count 6nm+1/2. For subdivision k≥3, the fullerenes KBmn and Hmn have (2,3), (3,3) and (2,2) types of edges with counts 6mn, 6mn and 3mn(k−2), respectively. The fullerene KBm+1/2n also has (2,3), (3,3) and (2,2) types of edges, and their count is 3n(2m+1), 3n(2m+1), and 3n(m+1/2)(k−2), respectively.Theorem 5.
LetGn,k be the line graph of k-subdivided Klein-bottle fullerenes, KBmn and, Hmn for n≥2even and m≥2, further Klein-bottle fullerene, KBm+1/2n for n≥2even and m≥1, with k≥3. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index are
ForKBmn:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1mn(2)
AGn,k=38k−2+1241/32mn(3)
GAGn,k=3k−2+46/5+2mn(4)
HGn,k=3k−2/2+22/5mn
ForKBm+1/2n:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1nm+1/2(2)
AGn,k=38k−2+1241/32nm+1/2(3)
GAGn,k=3k−2+46/5+2nm+1/2(4)
HGn,k=3k−2/2+22/5nm+1/2
ForHmn:(1)
RαGn,k=3.22αk−2+2α+1⋅3α+1+2⋅32α+1mn(2)
AGn,k=38k−2+1241/32mn(3)
GAGn,k=3k−2+46/5+2mn(4)
HGn,k=3k−2/2+22/5mnProof.
By using equations (1)–(4), we get the required results.Theorem 6.
LetGn,k be the line graph of k-subdivided Klein-bottle fullerene, KBmn and toroidal fullerene, Hmn for n≥2even and m≥2, further Klein-bottle fullerene, KBm+1/2n for n≥2even and m≥1, by k≥3. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, are
ForKBmn:(1)
RGn,k=3k−2/2+6+2mn(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=3k−2/2+2+4mn
ForKBm+1/2n:(1)
RGn,k=3k−2/2+6+2nm+1/2(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1nm+1/2(3)
ABCGn,k=3k−2/2+2+4nm+1/2
ForHmn:(1)
RGn,k=3k−2/2+6+2mn(2)
χαGn,k=3.22αk−2+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=3k−2/2+2+4mnProof.
By using equations (5)–(7), we get the required results.


## 2.4. Carbon Nanotube Networks

LetNAmn be the nanotube, for m,n≥2 with order 2mn+1 and size m3n+2. Let NBmn be the nanotube, for n≥2even and m≥1 with order 2nm+1 and size n3m+2. In the line graph of subdivided graph, both structures have 2,2, 2,3, and 3,3 types of edges. For k=1, NAmn has 2m, 4m, and 9mn−2m edges, respectively, whereas NBmn has 3n, 2n, and 9mn−n edges, respectively. Similarly, for subdivision k=2, NAmn has 6m and last two types have same 6mn count of edges, whereas NBmn has 6n and again last two types have same 6mn count of edges. Further, for subdivision k≥3, NAmn has 6m+k−23mn+2m and last two types have same 6mn count of edges, whereas NBmn has 6n+k−23mn+2n and last two types have same 6mn count of edges.Theorem 7.
LetGn,k be the line graph of k-subdivided NAmn nanotube, for m,n≥2, and NBmn nanotube, for n≥2even and m≥1 with k≥3. Their generalized Randić index, general Zagreb index, augmented Zagreb index, geometric arithmetic index, and harmonic index areForNAmn(1)
RαGn,k=22αk−23mn+2m+3.22α+1m+3α+1⋅2α+1+2.32α+1mn(2)
AGn,k=8k−23mn+2m+48m+2187/32mn(3)
GAGn,k=k−23mn+2m+6m+626/5+1mn(4)
HGn,k=k−2/23mn+2m+3m+17/5mnForNBmn(1)
RαGn,k=22αk−23mn+2n+3.22α+1n+3α+1⋅2α+1+2.32α+1mn(2)
AGn,k=8k−23mn+2n+48n+2187/32mn(3)
GAGn,k=k−23mn+2n+6n+626/5+1mn(4)
HGn,k=k−2/23mn+2n+3n+17/5mnProof.
By using equations (1)–(4), we get the required results.Theorem 8.
LetGn,k be the line graph of k-subdivided NAmn nanotube, for m,n≥2, and NBmn nanotube, for n≥2even and m≥1 with k≥3. Their connectivity indices, i.e., product connectivity index, general sum-connectivity index, and atom-bond connectivity index, areForNAmn:(1)
RGn,k=k−2/23mn+2m+3m+6+2mn(2)
χαGn,k=22αk−23mn+2m+3.22α+1m+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=k−2/23mn+2m+32m+32+4mnForNBmn:(1)
RGn,k=k−2/23mn+2n+3n+6+2mn(2)
χαGn,k=22αk−23mn+2n+3.22α+1n+6.5α+2α+1⋅3α+1mn(3)
ABCGn,k=k−2/23mn+2n+32n+32+4mnProof.
By using equations (5)–(7), we get the required results.


## 3. Conclusion

All graphs are simple in this article. We have found different degree-based topological indices for the line graph of subdividedk≥3 graph of linear [n] Tetracene, Klein bottle fullerene, V-tetracenic nanotube, H-tetracenic nanotube, tetracenic nanotori, toroidal fullerene, and carbon nanotubes.


## 4. Future Work

In future, degree-based topological indices for some additional structures can be studied. Moreover, can we study degree-based topological indices for line graph ofk-subdivided graph, having any kind of edge degree sequence and type of edges?



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*Source: 3045646-2020-04-15.xml*