Topological Indices of Certain Transformed Chemical Structures

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 of k-subdivided linear [n] Tetracene, fullerene networks, tetracenic nanotori, and carbon nanotube networks.


Introduction
In chemical graph theory, we apply the concepts of graph theory to describe the mathematical model of a variety of chemical structures. e 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). e 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. e concept of topological index came from the work done by Wiener, when he was working on boiling point of paraffin [1]. e Wiener index is the first and most studied topological index. e 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][7][8].
e 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][13][14][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 paraline 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: Recall that the augmented Zagreb index is [19] A(G) � Furtula et al. defined the geometric arithmetic index as [19] Moreover, the harmonic index is defined as follows [20]: e 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] e general sum-connectivity index has been introduced in 2010, as [21] 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] e remaining article is characterized as follows. In Section 2, the topological indices for the line graph of subdivided graph of different nanostructures have been discussed. e conclusion has been drawn in Section 3.

Main Results
Let G be a finite, simple, and connected graph with order p and size q. For k ≥ 1, a k-subdivided graph G(k) of G is obtained by replacing each edge of graph G by a path P k+2 . A line graph L(G) of graph G is a transformed graph having q vertices and two vertices have common neighbourhood in L(G) if and only if their corresponding edges are adjacent in G.
e generalized Randić index of G(n, k) is computed as follows: e augmented Zagreb index is computed as Next, the geometric arithmetic index is computed as 2 Journal of Chemistry Moreover, the harmonic index is defined as follows: which completes the proof of the theorem. □ Theorem 2. e product connectivity index, general sumconnectivity index, and atom-bond connectivity index of G(n, k) are Proof.
e product connectivity index of G(n, k) is computed as e general sum-connectivity index is computed as e atom-bond connectivity (ABC) index is computed as which completes the proof of the theorem.

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

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 of k-subdivided graph, having any kind of edge degree sequence and type of edges?
Data Availability e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.