On the Computation of Some Topological Descriptors to Find Closed Formulas for Certain Chemical Graphs

In this research paper, we will compute the topological indices (degree based) such as the ordinary generalized geometricarithmetic (OGA) index, first and second Gourava indices, first and second hyper-Gourava indices, general Randic ́ index Rc(G), for c � ±1, ±1/2 { }, harmonic index, general version of the harmonic index, atom-bond connectivity (ABC) index, SK, SK1, and SK2 indices, sum-connectivity index, general sum-connectivity index, and first general Zagreb and forgotten topological indices for various types of chemical networks such as the subdivided polythiophene network, subdivided hexagonal network, subdivided backbone DNA network, and subdivided honeycomb network. &e discussion on the aforementioned networks will give us very remarkable results by using the aforementioned topological indices.


Introduction
e branch of mathematics that is related to the study of implementation of chemistry and graph theory together is called chemical graph theory. is theory is used to model the molecules of a chemical compound mathematically. is theory helps us to understand the physical properties of that chemical/molecular compound. In this theory, we construct the structure of a chemical compound in the form of a graph. In chemical graph theory, atoms are used as nodes, and bonds between the atoms are utilized as edges. A topological index is a numerical parameter of a graph that explains its topology. e topological index is also called a molecular descriptor and a connectivity index. It is obtained by transforming the chemical information into a numerical quantity. Topological indices are used as molecular descriptors in the construction of quantitative structure-activity relationships and quantitative structure-property relationships as well.
e theoretical models such as quantitative structure-activity relationships (QSARs) relate the quantitative measure of a chemical structure to a biological property or a physical property, and quantitative structure-property relationships (QSPRs) relate mathematically physical/chemical properties to the structure of a molecule. Topological indices such as ordinary generalized geometric-arithmetic (OGA) index, first and second Gourava indices, first and second hyper-Gourava indices, general Randic index R c (G), for c � ±1, ±1/2 { }, harmonic index, general version of harmonic index [1,2], atom-bond connectivity (ABC) index [3,4], SK, SK 1 , and SK 2 indices, sumconnectivity index, general sum-connectivity index, and first general Zagreb [5] and forgotten topological indices have very significant roles in QSAR and QSPR studies and are used to discuss the bioactivity of molecular structures.
In 2009, D. Vukičević and B. Furtula established the first GA index in [6][7][8][9][10][11]. e first geometric-arithmetic (GA) index of a graph ξ was calculated by (2) In 2017, V. R. Kulli proposed the first and second Gourava and hyper-Gourava indices in [13,14]. e first and second Gourava and hyper-Gourava indices of a graph ξ were formulated by (3) In 1975, Randic´index [15][16][17] was introduced by Milan Randic´. It is often used in chemoinformatics to investigate the compounds of chemicals. e Randic´index is also called "the connectivity index of the graph" and formulated by where d u and d v are the degrees of the nodes. Later, Bollobás and Erdos furnished its generalized version for c, where c ∈ R, known as the general Randicí ndex [18][19][20][21] defined as In 2012, L. Zhong described the harmonic index in [22,23], and it is given by In 2015, L. Yan introduced the general version of the harmonic index in [24] and defined by In 2008, Ernesto¨Estrada et al. [25,26] introduced a new topological index, named atom-bond connectivity (ABC) index, calculated by e ABC index is an excellent valuable index in the formation of heat in alkanes [25,26]. Definition 1. For a graph ξ, the SK index [27] can be computed by Let d g and d h be the degrees of nodes g and h in ξ, respectively.
Definition 2. For a graph ξ, the SK 1 index can be computed by .
Let d g and d h be the degrees of nodes g and h in ξ, respectively.
Definition 3. For a graph ξ, the SK 2 index can be computed by Let d g and d h be the degrees of nodes g and h in ξ, respectively.
In 2009, B. Lučić proposed the sum-connectivity index (χ) in [28] calculated by In 2010, B. Zhou and Trinajstić furnished an index named general sum-connectivity index in [24,29] and formulated as follows: In 2005, X. Li and J. Zheng produced the generalized form of the first Zagreb index by calling it the "first general Zagreb index." e first general Zagreb index [30][31][32][33][34][35] of a graph ξ was computed by k M 1 ; k belongs to R, and k ≠ 0 and k ≠ 1.

Topological Indices on Certain Chemical Graphs
In this part of the research paper, we will compute the topological indices (degree based) such as ordinary generalized geometric-arithmetic (OGA) index, first and second Gourava indices, first and second hyper-Gourava indices, general Randic index R c (ξ), for c � ± 1, ± 1/2 { }, harmonic index, general version of harmonic index, atom-bond connectivity (ABC) index, SK, SK 1 , and SK 2 indices, sumconnectivity index, general sum-connectivity index, first general Zagreb index, and forgotten topological indices for various types of chemical networks such as subdivided polythiophene network, subdivided hexagonal network, subdivided backbone DNA network, and subdivided honeycomb network.

Results for the Subdivided Polythiophene Network.
Polythiophenes are rings with five elements having one heteroatom together with their benzo and other carbocylic. Polythiophene is used in electronic devices such as water purification devices, biosensors, and light-emitting diodes and in hydrogen storage [39]. In a subdivided polythiophene network, shown in Figure 1, we insert another vertex (degree 2) in every edge of ξ. In this way, we get a subdivided polythiophene network. In this section, we compute the subdivided polythiophene network using the above-defined topological indices. In the subdivided polythiophene network SPLY n , we have the number of nodes 11n − 1 and edges 12n − 2. A subdivided polythiophene network for n � 5 is shown in Figure 1. We get two kinds of edges (degree based) that are (2,2) and (2,3). Table 1 gives us two types of edges. A subdivided polythiophene network SPLY 5 is displayed in Figure 1.

Theorem 1.
For the subdivided polythiophene network, SPLY n , the ordinary generalized geometric-arithmetic index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Proof. By letting ξ as a subdivided polythiophene network SPLY n of n dimensions, we have the number of nodes and  Table 1: Division of edges of a graph ξ found on the degree of terminating nodes of each of the edges.

Theorem 5. For the subdivided polythiophene network, SPLY n , the harmonic index is calculated by
Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Theorem 6. For the subdivided polythiophene network, SPLY n , the general version of the harmonic index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Theorem 7. For the subdivided polythiophene network, SPLY n , the atom-bond connectivity index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get and by doing some calculations, we get Theorem 9. For the subdivided polythiophene network, SPLY n , the sum-connectivity index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Theorem 10. For the subdivided polythiophene network, SPLY n , the general sum-connectivity index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Theorem 11. For the subdivided polythiophene network, SPLY n , the first general Zagreb index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get Theorem 12. For the subdivided polythiophene network, SPLY n , the forgotten index is calculated by Proof. By letting ξ as a subdivided polythiophene network SPLY n , from Table 1, we know and by doing some calculations, we get

Results for the Subdivided Hexagonal Network.
We construct a subdivided hexagonal network shown in Figure 2 by adding a new vertex in each edge. For this process, a triangular tiling is used. In this way, an n-dimensional subdivided hexagonal network is obtained and denoted by SHX n . A subdivided hexagonal network for n � 6 is shown in Figure 2, whereas n shows the number of nodes. e order of SHX n is 12n 2 − 18n + 7 for n > 1, and the size is 18n 2 − 30n + 12 for n > 1. After the subdivision of this network, we have three types of edges that are (2, 3), (2,4), and (2,6). e division of edges is shown in Table 2. A subdivided hexagonal network SHX 6 is displayed in Figure 2.
Theorem 13. For the subdivided hexagonal network, SHX n , the ordinary generalized geometric-arithmetic index is calculated by Journal of Chemistry 5 Proof. By letting 1 as a subdivided hexagonal network SHX n , from Table 2, we know Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know and by doing some calculations, we get Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know Proof. By letting ξ as a subdivided hexagonal network SHX n of n dimensions, we have the number of nodes and edges in SHX n as |V(SHX n )| � 12n 2 − 18N + 7 for n > 1 and |E(SHX n )| � 18n 2 − 30n + 12 for n > 1, respectively. We know that using (63). From Table 2, we know By doing some calculations, we get R − 1 (ξ) � (3n 2 − 3n + 1/2). Case 2: if c � − (1/2), the application of Randic´index R c (ξ) using (63), By doing some calculations, we get Case 3: if c � (1/2), the application of Randic´index R c (ξ) using (63), √ .

(70)
By doing some calculations, we get Case 4: if c � 1, the application of Randic´index R c (ξ) using (63), By doing some calculations, we get

Theorem 17. For the subdivided hexagonal network, SHX n , the harmonic index is calculated by
Proof. By letting ξ as a subdivided hexagonal network SHX n , from (77)

Theorem 18. For the subdivided hexagonal network, SHX n , the general version of the harmonic index is calculated by
Proof. By letting ξ as a subdivided hexagonal network SHX n , from (80)

Theorem 19. For the subdivided hexagonal network, SHX n , the atom-bond connectivity index is calculated by
Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know and by doing some calculations, we get (86) Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know and by doing some calculations, we get (89) Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know   8 Journal of Chemistry and by doing some calculations, we get χ k (ξ) � � 18n 2 8 k + n 24 6 k − 54 8 k + 18 5 k − 48 6 k + 42 8 k .
(91) Theorem 23. For the subdivided hexagonal network, SHX n , the first general Zagreb index is calculated by k M 1 (ξ) � n 2 3 6 k + 9 2 k + n 6 4 k − 15 2 k − 9 6 k + 6 2 k + 6 3 k − 12 4 k + 7 6 k . (92) Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know and by doing some calculations, we get Theorem 24. For the subdivided hexagonal network, SHX n , the forgotten index is calculated by Proof. By letting ξ as a subdivided hexagonal network SHX n , from Table 2, we know and by doing some calculations, we get 2.3. Results for the Subdivided Backbone DNA Network. e structure of DNA is called a double helix as it is made of two strands that wind around each other that looks like a staircase [40]. Each strand has a backbone made of deoxyribose, sugar, and a phosphate group. ese sugar and phosphates make up the backbone, while the nitrogen bases are found in the centre and hold the two strands together.
ere are 4 bases attached to each sugar which are adenine, cytosine, guanine, and thymine. Both ends of DNA have a number, i.e., one end is´5 and the other is´3. In a subdivided backbone DNA network, shown in Figure 3, we insert another node (degree 2) in each edge of ξ. In this way, we get a subdivided backbone DNA network of n dimensions. A subdivided backbone DNA network for n � 4 is displayed in Figure 3. A subdivided backbone DNA network is symbolized as SBB DNA (n). e order and size of SBB DNA (n) are 15n − 5 and 16n − 6, respectively. We obtain two types of edges (degree based) that are (2, 2) and (2, 3). Table 3 gives us two kinds of edges. A subdivided backbone DNA network SBB DNA (4) is shown in Figure 3.

Theorem 25. For the subdivided backbone DNA network, SBB DNA (n), the ordinary generalized geometric-arithmetic index is calculated by
Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know and by doing some calculations, we get Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know and by doing some calculations, we get Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n) of n dimensions, we have the order and size of ξ in SBB DNA (n) as |V(SBB DNA (n))| � 15n − 5 and |E(SBB DNA (n))| � 16n − 6, respectively.

Theorem 29.
For the subdivided backbone DNA network, SBB DNA (n), the harmonic index is calculated by Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know Table 3: Division of edges of a graph ξ found on the degree of terminating nodes of each of the edges.
6(n − 1) Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know and by doing some calculations, we get Theorem 31. For the subdivided backbone DNA network, SBB DNA (n), the atom-bond connectivity index is calculated by Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know By doing some calculations, we get Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know By doing some calculations, we get Theorem 34. For the subdivided backbone DNA network, SBB DNA (n), the general sum-connectivity index is calculated by Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know By doing some calculations, we get χ k (ξ) � n 10 4 k + 6 5 k − 6 5 k .
Theorem 35. For the subdivided backbone DNA network, SBB DNA (n), the first general Zagreb index is calculated by k M 1 (ξ) � n 10 2 k + 3 2 k + 2 3 k − 3 2 k + 2 3 k . (132) Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from Table 3, we know and by doing some calculations, we get k M 1 (ξ) � n 10 2 k + 3 2 k + 2 3 k − 3 2 k + 2 3 k . (134) Theorem 36. For the subdivided backbone DNA network, SBB DNA (n), the forgotten index is calculated by Proof. By letting ξ as a subdivided backbone DNA network SBB DNA (n), from e first honeycomb network is symbolized by HC (1) . e next honeycomb network is produced by attaching more hexagons to each of its edges.
is newly formed honeycomb network is symbolized by HC (2) ; similarly, the next honeycomb network is produced by attaching more hexagons to each of its edges. In this way, the newly formed honeycomb network is denoted by HC (3) . By repeating this process, we finally obtain a honeycomb network of n dimensions and denote by HC (n) . e honeycomb network is being used in computer graphics, image processing, and cellular phone base stations; moreover, it is used in chemistry for the representation of benzenoid hydrocarbons. To get the subdivided honeycomb network shown in Figure 4, we insert a new node on each of its edges. e n-dimensional subdivided honeycomb network is symbolized by SHC n . A subdivided honeycomb network for n � 4 is displayed in Figure 4. e number of nodes and edges in the subdivided honeycomb networks are 15n 2 − 3n and 18n 2 − 6n, respectively. We have obtained two different types of edges in SHC 4 shown in Table 4, whereas Figure 4 shows SHC 4 .
Proof. By letting ξ as a subdivided honeycomb network SHC n , from