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

Authors: Muhammad Haroon Aftab; Muhammad Rafaqat; M. Hussain; Tariq Zia
Journal: Journal of Chemistry
 (2021)

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## Abstract

In this research paper, we will compute the topological indices (degree based) such as the ordinary generalized geometric-arithmetic (OGA) index, first and second Gourava indices, first and second hyper-Gourava indices, general Randic´ indexRγG,forγ=±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. The discussion on the aforementioned networks will give us very remarkable results by using the aforementioned topological indices.


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## Body

## 1. Introduction

The branch of mathematics that is related to the study of implementation of chemistry and graph theory together is called chemical graph theory. This theory is used to model the molecules of a chemical compound mathematically. This 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. The 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. The 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 indexRγG,forγ=±1,±1/2, harmonic index, general version of harmonic index [1, 2], atom-bond connectivity (ABC) index [3, 4], SK, SK1, and SK2 indices, sum-connectivity 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 [6–11]. The first geometric-arithmetic (GA) index of a graph ξ was calculated by(1)GAξ=∑gh∈Eξ2dgdhdg+dh.An ordinary geometric-arithmetic index ofξ was produced in 2011 in [12] and formulated by, for each positive real number k,(2)OGAkξ=∑gh∈Eξ4dgdhdg+dhk.In 2017, V. R. Kulli proposed the first and second Gourava and hyper-Gourava indices in [13, 14]. The first and second Gourava and hyper-Gourava indices of a graph ξ were formulated by(3)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2.In 1975, Randic´ index [15–17] was introduced by Milan Randic´. It is often used in chemoinformatics to investigate the compounds of chemicals. The Randic´ index is also called “the connectivity index of the graph” and formulated by(4)R1/2ξ=∑uv∈Eξ1du.dv,where du and dv are the degrees of the nodes.Later, Bollobás and Erdos furnished its generalized version forγ, where γ∈ R, known as the general Randic´ index [18–21] defined as(5)RγG=∑gh∈EGdg·dhγ,forγ=−1,1,−12,12.In 2012, L. Zhong described the harmonic index in [22, 23], and it is given by(6)HIG=∑gh∈EG2dg+dh.In 2015, L. Yan introduced the general version of the harmonic index in [24] and defined by(7)HkIG=∑gh∈EG2dg+dhk.In 2008, Ernesto¨ Estrada et al. [25, 26] introduced a new topological index, named atom-bond connectivity (ABC) index, calculated by(8)ABCG=∑gh∈EGdg+dh−2dgdh.The ABC index is an excellent valuable index in the formation of heat in alkanes [25, 26].Definition 1.
For a graphξ, the SK index [27] can be computed by(9)SKξ=∑gh∈Eξdg+dh2.
Letdganddh be the degrees of nodes g and h in ξ, respectively.Definition 2.
For a graphξ, the SK1 index can be computed by(10)SK1ξ=∑gh∈Eξdgdh2.
Letdganddh be the degrees of nodes g and h in ξ, respectively.Definition 3.
For a graphξ, the SK2 index can be computed by(11)SK2ξ=∑gh∈Eξdg+dh22.
Letdganddh be the degrees of nodes g and h in ξ, respectively.
In 2009, B. Lučić proposed the sum-connectivity indexχ in [28] calculated by(12)χ−1/2G=∑gh∈EGdg+dh−1/2.
In 2010, B. Zhou and Trinajstić furnished an index named general sum-connectivity index in [24, 29] and formulated as follows:(13)χkξ=∑gh∈Eξdg+dhk.
In 2005, X. Li and J. Zheng produced the generalized form of the ﬁrst Zagreb index by calling it the “ﬁrst general Zagreb index.” The first general Zagreb index [30–35] of a graph ξ was computed by Mk1ξ=∑gh∈Eξdgk−1+dhk−1; k belongs to R, and k ≠ 0 and k ≠ 1.
In 2015, Boris Furtula and Ivan Gutman discovered an index named as “forgotten topological index” [36–38] and computed as(14)FG=∑gh∈EGdg2+dh2.


## 2. 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 indexRγξ,forγ=±1,±1/2, harmonic index, general version of harmonic index, atom-bond connectivity (ABC) index, SK, SK1, and SK2 indices, sum-connectivity 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.


### 2.1. 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 SPLYn, 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 SPLY5 is displayed in Figure 1.Figure 1
SPLY5.Table 1
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)6n+4(2, 3)6n−1Theorem 1.
For the subdivided polythiophene network, SPLYn, the ordinary generalized geometric-arithmetic index is calculated by(15)OGAkξ=6n1+245k+4−6245k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(16)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=6n+4162+2k+6n−6245k,and by doing some calculations, we get(17)OGAkξ=6n1+245k+4−6245k.Theorem 2.
For the subdivided polythiophene network, SPLYn, the first and second Gourava indices are calculated by GO1ξ=114n−34 and GO2ξ=276n−116.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(18)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(19)GO1ξ=6n+44+4+6n−65+6=114n−34,GO2ξ=6n+416+6n−630=276n−116.Theorem 3.
For the subdivided polythiophene network, SPLYn, the first and second hyper-Gourava indices are calculated by HGO1ξ=1110n−470 and HGO2ξ=6936n−4376.Proof.
By lettingξ as a subdivided polythiophene network, SPLYn, from Table 1, we know(20)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get HGO1ξ=6n+464+6n−6121=1110n−470, and(21)HGO2ξ=6n+4256+6n−6900=6936n−4376.Theorem 4.
For the subdivided polythiophene network, SPLYn, the general Randic´ index is calculated by(22)RγSPLYn=52n,forγ=−1,n10920+85,forγ=−12,n26710−6710,forγ=12,60n−20,forγ=1.Proof.
By lettingξ as a subdivided polythiophene network SPLYn of n dimensions, we have the number of nodes and edges in SPLYn as VSPLYn=11n−1 and ESPLYn=12n−2, respectively.
We know that(23)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(24)R−1ξ=∑gh∈Eξ1dgdh,using (23). From Table 1, we know R−1ξ=6n+44−1+6n−66−1. By doing some calculations, we get R−1ξ=5/2n.Case 2: ifγ = − 1/2, the application of Randic´ index Rγξ(25)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (23),(26)R−1/2ξ=6n+414+6n−616,and by doing some calculations, we getR−1/2ξ=n109/20+8/5.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(27)R1/2ξ=∑gh∈Eξdg.dh,using (23),(28)R1/2ξ=6n+44+6n−66,and by doing some calculations, we getR1/2ξ=n267/10−67/10.Case 4: ifγ = 1, the application of Randic´ index Rγξ(29)R1ξ=∑gh∈Eξdg.dh1,  using (23),(30)R1ξ=6n+44+6n−66,  and by doing some calculations, we get R1ξ=60n−20.Theorem 5.
For the subdivided polythiophene network, SPLYn, the harmonic index is calculated by(31)HIξ=2n−25.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(32)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(33)HIξ=6n+412+6n−625=25n−1.Theorem 6.
For the subdivided polythiophene network, SPLYn, the general version of the harmonic index is calculated by(34)HkIξ=6n5k+4k+45k−64k10k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(35)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(36)HkIξ=6n+422+2k+6n−622+3k,=6n5k+4k+45k−64k10k.Theorem 7.
For the subdivided polythiophene network, SPLYn, the atom-bond connectivity index is calculated by(37)ABCξ=62n−2.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(38)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(39)ABCξ=6n+42+2−24+6n−62+3−26=26n−1.Theorem 8.
For the subdivided polythiophene network, SPLYn, SK, SK1, and SK2 indices are calculated by SKξ=27n−7,SK1ξ=30n−10, and SK2ξ=1/2123n−43, respectively.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(40)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(41)SKξ=6n+42+6n−652=27n−7,SK1ξ=6n+42+6n−63=30n−10,SK2ξ=6n+44+6n−6254=12123n−43.Theorem 9.
For the subdivided polythiophene network, SPLYn, the sum-connectivity index is calculated by(42)χ−1/2ξ=n3+65+2−65.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(43)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=6n+412+6n−615,and by doing some calculations, we get(44)χ−1/2ξ=n3+65+2−65.Theorem 10.
For the subdivided polythiophene network, SPLYn, the general sum-connectivity index is calculated by(45)χkξ=6n5k+4k+44k−65k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(46)χkξ=∑gh∈Eξdg+dhk,χkξ=6n+44k+6n−65k,and by doing some calculations, we get(47)χkξ=6n5k+4k+44k−65k.Theorem 11.
For the subdivided polythiophene network, SPLYn, the first general Zagreb index is calculated by(48)Mk1ξ=9n2k+2n3k+22k2−3k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(49)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=6n+42k+6n−62k−1+3k−1,and by doing some calculations, we get(50)Mk1ξ=9n2k+2n3k+22k2−3k.Theorem 12.
For the subdivided polythiophene network, SPLYn, the forgotten index is calculated by(51)Fξ=263n−23.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(52)Fξ=∑gh∈Eξdg2+dh2,Fξ=6n+48+6n−613,and by doing some calculations, we get(53)Fξ=126n−46.


### 2.2. Results for the Subdivided Hexagonal Network

We construct a subdivided hexagonal network shown in Figure2 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 SHXn. A subdivided hexagonal network for n = 6 is shown in Figure 2, whereas n shows the number of nodes. The order of SHXn is 12n2 − 18n + 7 for n > 1, and the size is 18n2 − 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). The division of edges is shown in Table 2. A subdivided hexagonal network SHX6 is displayed in Figure 2.Figure 2
SHX6.Table 2
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ2,3182,424n−22,663n2−9n+7Theorem 13.
For the subdivided hexagonal network, SHXn, the ordinary generalized geometric-arithmetic index is calculated by(54)OGAkξ=18n2488k+n24326k−54488k+42488k−48326k.Proof.
By letting1 as a subdivided hexagonal network SHXn, from Table 2, we know(55)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=18245k+24n−2326k+63n2−9n+7488k,and by doing some calculations, we get(56)OGAkξ=18n2488k+n24326k−54488k+42488k−48326k.Theorem 14.
For the subdivided hexagonal network, SHXn, the first and second Gourava indices are calculated by(57)GO1ξ=360n2−744n+366,GO2ξ=1728n2−4032n+2268.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(58)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(59)GO1ξ=185+6+24n−26+8+63n2−9n+78+12=360n2−744n+366,GO2ξ=1830+24n−248+63n2−9n+796,=1728n2−4032n+2268.Theorem 15.
For the subdivided hexagonal network, SHXn, the first and second hyper-Gourava indices are calculated by HGO1ξ=7200n2−16896n+9570 and HGO2ξ=165888n2−442368n+292680.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(60)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(61)HGO1ξ=18121+24n−2196+63n2−9n+7400=7200n2−16896n+9570,HGO2ξ=18900+24n−22304+63n2−9n+79216,=165888n2−442368n+292680.Theorem 16.
For the subdivided hexagonal network, SHXn, n > 1, the general Randic´ index is calculated by(62)RγSHXn=3n2−3n+12,forγ=−1,3n23+n72−93+2.501,forγ=−12,62.35n2−119.17n+53.82,forγ=12,216n2−456n+228,forγ=1.Proof.
By lettingξ as a subdivided hexagonal network SHXn of n dimensions, we have the number of nodes and edges in SHXn as VSHXn=12n2−18N+7 for n > 1 and ESHXn=18n2−30n+12 for n > 1, respectively. We know that(63)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(64)R−1ξ=∑gh∈Eξ1dgdh,using (63). From Table 2, we know(65)R−1ξ=186−1+24n−28−1+63n2−9n+712−1.By doing some calculations, we getR−1ξ=3n2−3n+1/2.Case 2: ifγ = − 1/2, the application of Randic´ index Rγξ(66)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (63),(67)R−1/2ξ=1816+24n−218+63n2−9n+7112.By doing some calculations, we get(68)R−1/2ξ=3n23+n72−93+2.501.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(69)R1/2ξ=∑gh∈Eξdg·dh,using (63),(70)R1/2ξ=186+24n−28+63n2−9n+712.By doing some calculations, we get(71)R1/2ξ=62.35n2−119.17n+53.82.Case 4: ifγ = 1, the application of Randic´ index Rγξ(72)R1ξ=∑gh∈Eξdg.dh1,
using (63),(73)R1ξ=186+24n−28+63n2−9n+712.
By doing some calculations, we get(74)R1ξ=216n2−456n+228.Theorem 17.
For the subdivided hexagonal network, SHXn, the harmonic index is calculated by(75)HIξ=4.5n2−5.5n+1.7.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(76)HIξ=∑gh∈Eξ2dg+dh.
By doing some calculations, we get(77)HIξ=1825+24n−226+63n2−9n+728,=11045n2−55n+17=4.5n2−5.5n+1.7.Theorem 18.
For the subdivided hexagonal network, SHXn, the general version of the harmonic index is calculated by(78)HkIξ=18n24−k+2n123−k−274−k+182k5−k−483−k+424−k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(79)HkIξ=∑gh∈Eξ2dg+dhk.
By doing some calculations, we get(80)HkIξ=1822+3k+24n−222+4k+63n2−9n+722+6k=n2184k+n243k−544k+182k5k−483k+424k=18n24−k+2n123−k−274−k+182k5−k−483−k+424−k.Theorem 19.
For the subdivided hexagonal network, SHXn, the atom-bond connectivity index is calculated by(81)ABCξ=92n2−152n+62.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(82)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(83)ABCξ=182+3−26+24n−22+4−28+63n2−9n+72+6−212=92n2−152n+62.Theorem 20.
For the subdivided hexagonal network, SHXn, SK, SK1, and SK2 indices are calculated by SKξ=72n2−144n+69,SK1ξ=108n2−228n+498,andSK2ξ=288n2−648n+7052, respectively.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(84)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(85)SKξ=1852+24n−262+63n2−9n+782=72n2−144n+69.SK1ξ=183+24n−24+363n2−9n+7=108n2−228n+498.SK2ξ=18254+24n−29+63n2−9n+716=288n2−648n+7052.Theorem 21.
For the subdivided hexagonal network, SHXn, the sum-connectivity index is calculated by(86)χ−1/2ξ=18n218+n246−548+185−486+428.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(87)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=1816+24n−218+63n2−9n+7112,and by doing some calculations, we get(88)χ−1/2ξ=n2188+n246−548+185−486+428=18n218+n246−548+185−486+428.Theorem 22.
For the subdivided hexagonal network, SHXn, the general sum-connectivity index is calculated by(89)χkξ=18n28k+n246k−548k+185k−486k+428k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(90)χkξ=∑gh∈Eξdg+dhk,χkξ=185k+24n−26k+63n2−9n+78k,and by doing some calculations, we get(91)χkξ==18n28k+n246k−548k+185k−486k+428k.Theorem 23.
For the subdivided hexagonal network, SHXn, the first general Zagreb index is calculated by(92)Mk1ξ=n236k+92k+n64k−152k−96k+62k+63k−124k+76k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(93)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=182k2+3k3+24n−482k2+4k4+63n2−9n+72k−1+6k−1,and by doing some calculations, we get(94)Mk1ξ=n236k+92k+n64k−152k−96k+62k+63k−124k+76k.Theorem 24.
For the subdivided hexagonal network, SHXn, the forgotten index is calculated by(95)Fξ=2360n2−840n+477.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(96)Fξ=∑gh∈Eξdg2+dh2,Fξ=1813+24n−220+63n2−9n+740,and by doing some calculations, we get(97)Fξ=2360n2−840n+477.


### 2.3. Results for the Subdivided Backbone DNA Network

The 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. These sugar and phosphates make up the backbone, while the nitrogen bases are found in the centre and hold the two strands together. There 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 SBBDNA(n). The order and size of SBBDNA(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 SBBDNA(4) is shown in Figure 3.Figure 3
SBBDNA(4).Table 3
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)10n(2, 3)6n−1Theorem 25.
For the subdivided backbone DNA network, SBBDNA(n), the ordinary generalized geometric-arithmetic index is calculated by(98)OGAkξ=2n5+3245k−3245k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(99)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=10n162+2k+6n−6245k,and by doing some calculations, we get(100)OGAkξ=2n5+3245k−3245k.Theorem 26.
For the subdivided backbone DNA network, SBBDNA(n), the first and second Gourava indices are calculated by GO1ξ=146n−66 and GO2ξ=340n−180.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(101)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(102)GO1ξ=10n4+4+6n−65+6=146n−66,GO2ξ=10n16+6n−630=340n−180.Theorem 27.
For the subdivided backbone DNA network, SBBDNA(n), the first and second hyper-Gourava indices are calculated by(103)HGO1ξ=1366n−726,HGO2ξ=7960n−5400.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(104)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(105)HGO1ξ=10n64+6n−6121=1366n−726,HGO2ξ=10n256+6n−6900=7960n−5400.Theorem 28.
For the subdivided backbone DNA network, SBBDNA(n), the general Randic´ index is calculated by(106)RγSBBDNAn=72n−1,forγ=−1,n14920−4920,forγ=−12,n34710−14710,forγ=12,76n−36,forγ=1.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n) of n dimensions, we have the order and size of ξ in SBBDNA(n) as VSBBDNAn=15n−5 and ESBBDNAn=16n−6, respectively.
We know that(107)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(108)R−1ξ=∑gh∈Eξ1dgdh,using (107). From Table 3, we know R−1ξ=10n4−1+6n−66−1. By doing some calculations, we get R−1ξ=7/2n−1.Case 2: ifγ = −1/2, the application of Randic´ index Rγξ(109)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (107),(110)R−1/2ξ=10n14+6n−616,and by doing some calculations, we getR−1/2ξ=n149/20−49/20.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(111)R1/2ξ=∑gh∈Eξdg.dh,using (107),(112)R1/2ξ=10n4+6n−66,and by doing some calculations, we getR1/2ξ=n347/10−147/10.Case 4: ifγ = 1, the application of Randic´ index Rγξ(113)R1ξ=∑gh∈Eξdg·dh1,  using (107),(114)R1ξ=10n4+6n−66,  and by doing some calculations, we get R1ξ=76n−36.Theorem 29.
For the subdivided backbone DNA network, SBBDNA(n), the harmonic index is calculated by(115)HIξ=37n−125.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(116)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(117)HIξ=10n12+6n−625=37n−125.Theorem 30.
For the subdivided backbone DNA network, SBBDNA(n), the general version of the harmonic index is calculated by(118)HkIξ=2n52k+35k2k−65k2k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(119)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(120)HkIξ=10n22+2k+6n−622+3k=2n52k+35k2k−65k2k.Theorem 31.
For the subdivided backbone DNA network, SBBDNA(n), the atom-bond connectivity index is calculated by(121)ABCξ=28n−32.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(122)ABCξ=∑gh∈Eξdg+dh−2dgdh.
By doing some calculations, we get(123)ABCξ=10n2+2−24+6n−62+3−26=28n−32.Theorem 32.
For the subdivided backbone DNA network, SBBDNA(n), SK, SK1, and SK2 indices are calculated by SKξ=35n−15,SK1ξ=38n−18, and SK2ξ=1/2155n−75, respectively.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(124)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22.
By doing some calculations, we get(125)SKξ=10n2+6n−652=35n−15,SK1ξ=10n2+6n−63=38n−18,SK2ξ=10n4+6n−6254=12155n−75.Theorem 33.
For the subdivided backbone DNA network, SBBDNA(n), the sum-connectivity index is calculated by(126)χ−1/2ξ=n5+65−65.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(127)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=10n12+6n−615.
By doing some calculations, we get(128)χ−1/2ξ=n5+65−65.Theorem 34.
For the subdivided backbone DNA network, SBBDNA(n), the general sum-connectivity index is calculated by(129)χkξ=n104k+65k−65k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(130)χkξ=∑gh∈Eξdg+dhk,χkξ=10n4k+6n−65k.
By doing some calculations, we get(131)χkξ=n104k+65k−65k.Theorem 35.
For the subdivided backbone DNA network, SBBDNA(n), the first general Zagreb index is calculated by(132)Mk1ξ=n102k+32k+23k−32k+23k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(133)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=10n2k+6n−62k−1+3k−1,and by doing some calculations, we get(134)Mk1ξ=n102k+32k+23k−32k+23k.Theorem 36.
For the subdivided backbone DNA network, SBBDNA(n), the forgotten index is calculated by(135)Fξ=279n−39.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(136)Fξ=∑gh∈Eξdg2+dh2,Fξ=10n8+6n−613,and by doing some calculations, we get(137)Fξ=279n−39.


### 2.4. Results for the Subdivided Honeycomb Network

The honeycomb network is a hexagon. It can be made in different methods. The first honeycomb network is symbolized by HC(1). The next honeycomb network is produced by attaching more hexagons to each of its edges. This 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). The 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. The n-dimensional subdivided honeycomb network is symbolized by SHCn. A subdivided honeycomb network for n = 4 is displayed in Figure 4. The number of nodes and edges in the subdivided honeycomb networks are 15n2 − 3n and 18n2 − 6n, respectively. We have obtained two different types of edges in SHC4 shown in Table 4, whereas Figure 4 shows SHC4.Figure 4
4-dimensional SHC4.Table 4
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)12n(2, 3)18nn−1Theorem 37.
For the subdivided honeycomb network, SHCn, the ordinary generalized geometric-arithmetic index is calculated by(138)OGAkξ=63n2245k+n2−3245k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(139)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=12n162+2k+18nn−1245k,and by doing some calculations, we get(140)OGAkξ=63n2245k+n2−3245k.Theorem 38.
For the subdivided honeycomb network, SHCn, the first and second Gourava indices are calculated by GO1ξ=198n2−102n and GO2ξ=540n2−492n.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(141)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(142)GO1ξ=12n4+4+18nn−15+6=198n2−102n,GO2ξ=12n16+18nn−130=540n2−492n.Theorem 39.
For the subdivided honeycomb network, SHCn, the first and second hyper-Gourava indices are calculated by HGO1ξ=2178n2−1410n and HGO2ξ=16200n2−13128n.Proof.
By lettingξ as the subdivided honeycomb network SHCn, from Table 4, we know(143)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(144)HGO1ξ=12n64+18nn−1121=2178n2−1410n,HGO2ξ=12n256+18nn−1900=16200n2−13128n.Theorem 40.
For the subdivided honeycomb network, SHCn, the general Randic´ index is calculated by(145)RγSHCn=3n2,forγ=−1,3n26+n2−6,forγ=−12,63n26+n4−36,forγ=12,108n2−60n,forγ=1.Proof.
By lettingξ as a subdivided honeycomb network SHCn of n dimensions, we have the order and size of ξ in SHCn as VSHCn=15n2−3n and ESHCn=18n2−6n, respectively. We know that(146)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(147)R−1ξ=∑gh∈Eξ1dgdh,using (146). From Table 4, we obtain R−1ξ=12n4−1+18nn−16−1.By doing some calculations, we obtainR−1ξ=3n2.Case 2: ifγ = −1/2, the application of Randic´ index Rγξ(148)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (146). From Table 4, we obtain R−1/2ξ=12n1/4+18nn−11/6.By doing some calculations, we obtainR−1/2ξ=3n26+n2−6.Case 3: ifγ = 1/2, the application of Randic index Rγξ(149)R1/2ξ=∑gh∈Eξdg.dh,using (146). From Table 4, we obtain R1/2ξ=12n4+18nn−16.By doing some calculations, we obtainR1/2ξ=63n26+n4−36.Case 4: ifγ = 1, the application of Randic´ index Rγξ(150)R1ξ=∑gh∈Eξdg.dh1,using (146). From Table 4, we obtain R1ξ=12n4+18nn−16.By doing some calculations, we obtainR1ξ=108n2−60n.Theorem 41.
For the subdivided honeycomb network, SHCn, the harmonic index is calculated by(151)HIξ=36n2−6n5.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(152)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(153)HIξ=12n12+18nn−125=36n2−6n5.Theorem 42.
For the subdivided honeycomb network, SHCn, the general version of the harmonic index is calculated by(154)HkIξ=63n225k+n22k−325k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(155)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(156)HkIξ=12n22+2k+18nn−122+3k,=63n225k+n22k−325k.Theorem 43.
For the subdivided honeycomb network, SHCn, the atom-bond connectivity index is calculated by(157)ABCξ=n29n−3.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(158)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(159)ABCξ=12n2+2−24+18nn−12+3−26,=n29n−3.Theorem 44.
For the subdivided honeycomb network, SHCn, SK, SK1, and SK2 indices are calculated by SKξ=45n2−21n,SK1ξ=54n2−30n, and SK2ξ=1/2225n2−129n, respectively.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(160)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(161)SKξ=12n2+18nn−152=45n2−21n,SK1ξ=12n2+18nn−13=54n2−30n,SK2ξ=12n4+18nn−1254=12225n2−129n.Theorem 45.
For the subdivided honeycomb network, SHCn, the sum-connectivity index is calculated by(162)χ−1/2ξ=1518n2+n65−18.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(163)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=12n12+18nn−115,and by doing some calculations, we get(164)χ−1/2ξ=1518n2+n65−18.Theorem 46.
For the subdivided honeycomb network, SHCn, the general sum-connectivity index is calculated by(165)χkξ=185kn2+6n24k−35k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(166)χkξ=∑gh∈Eξdg+dhk,χkξ=12n4k+18nn−15k,and by doing some calculations, we get(167)χkξ=185kn2+6n24k−35k.Theorem 47.
For the subdivided honeycomb network, SHCn, the first general Zagreb index is calculated by(168)Mk1ξ=18n22k−1+3k−1+6n22k−3n2k−1+3k−1.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(169)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=12n2k+18nn−12k−1+3k−1,and by doing some calculations, we get(170)Mk1ξ=18n22k−1+3k−1+6n22k−3n2k−1+3k−1.Theorem 48.
For the subdivided honeycomb network, SHCn, the forgotten index is calculated by(171)Fξ=2n117n−69.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(172)Fξ=∑gh∈Eξdg2+dh2,Fξ=12n8+18nn−113,and by doing some calculations, we get Fξ=2n117n−69.


## 2.1. 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 SPLYn, 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 SPLY5 is displayed in Figure 1.Figure 1
SPLY5.Table 1
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)6n+4(2, 3)6n−1Theorem 1.
For the subdivided polythiophene network, SPLYn, the ordinary generalized geometric-arithmetic index is calculated by(15)OGAkξ=6n1+245k+4−6245k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(16)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=6n+4162+2k+6n−6245k,and by doing some calculations, we get(17)OGAkξ=6n1+245k+4−6245k.Theorem 2.
For the subdivided polythiophene network, SPLYn, the first and second Gourava indices are calculated by GO1ξ=114n−34 and GO2ξ=276n−116.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(18)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(19)GO1ξ=6n+44+4+6n−65+6=114n−34,GO2ξ=6n+416+6n−630=276n−116.Theorem 3.
For the subdivided polythiophene network, SPLYn, the first and second hyper-Gourava indices are calculated by HGO1ξ=1110n−470 and HGO2ξ=6936n−4376.Proof.
By lettingξ as a subdivided polythiophene network, SPLYn, from Table 1, we know(20)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get HGO1ξ=6n+464+6n−6121=1110n−470, and(21)HGO2ξ=6n+4256+6n−6900=6936n−4376.Theorem 4.
For the subdivided polythiophene network, SPLYn, the general Randic´ index is calculated by(22)RγSPLYn=52n,forγ=−1,n10920+85,forγ=−12,n26710−6710,forγ=12,60n−20,forγ=1.Proof.
By lettingξ as a subdivided polythiophene network SPLYn of n dimensions, we have the number of nodes and edges in SPLYn as VSPLYn=11n−1 and ESPLYn=12n−2, respectively.
We know that(23)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(24)R−1ξ=∑gh∈Eξ1dgdh,using (23). From Table 1, we know R−1ξ=6n+44−1+6n−66−1. By doing some calculations, we get R−1ξ=5/2n.Case 2: ifγ = − 1/2, the application of Randic´ index Rγξ(25)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (23),(26)R−1/2ξ=6n+414+6n−616,and by doing some calculations, we getR−1/2ξ=n109/20+8/5.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(27)R1/2ξ=∑gh∈Eξdg.dh,using (23),(28)R1/2ξ=6n+44+6n−66,and by doing some calculations, we getR1/2ξ=n267/10−67/10.Case 4: ifγ = 1, the application of Randic´ index Rγξ(29)R1ξ=∑gh∈Eξdg.dh1,  using (23),(30)R1ξ=6n+44+6n−66,  and by doing some calculations, we get R1ξ=60n−20.Theorem 5.
For the subdivided polythiophene network, SPLYn, the harmonic index is calculated by(31)HIξ=2n−25.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(32)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(33)HIξ=6n+412+6n−625=25n−1.Theorem 6.
For the subdivided polythiophene network, SPLYn, the general version of the harmonic index is calculated by(34)HkIξ=6n5k+4k+45k−64k10k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(35)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(36)HkIξ=6n+422+2k+6n−622+3k,=6n5k+4k+45k−64k10k.Theorem 7.
For the subdivided polythiophene network, SPLYn, the atom-bond connectivity index is calculated by(37)ABCξ=62n−2.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(38)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(39)ABCξ=6n+42+2−24+6n−62+3−26=26n−1.Theorem 8.
For the subdivided polythiophene network, SPLYn, SK, SK1, and SK2 indices are calculated by SKξ=27n−7,SK1ξ=30n−10, and SK2ξ=1/2123n−43, respectively.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(40)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(41)SKξ=6n+42+6n−652=27n−7,SK1ξ=6n+42+6n−63=30n−10,SK2ξ=6n+44+6n−6254=12123n−43.Theorem 9.
For the subdivided polythiophene network, SPLYn, the sum-connectivity index is calculated by(42)χ−1/2ξ=n3+65+2−65.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(43)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=6n+412+6n−615,and by doing some calculations, we get(44)χ−1/2ξ=n3+65+2−65.Theorem 10.
For the subdivided polythiophene network, SPLYn, the general sum-connectivity index is calculated by(45)χkξ=6n5k+4k+44k−65k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(46)χkξ=∑gh∈Eξdg+dhk,χkξ=6n+44k+6n−65k,and by doing some calculations, we get(47)χkξ=6n5k+4k+44k−65k.Theorem 11.
For the subdivided polythiophene network, SPLYn, the first general Zagreb index is calculated by(48)Mk1ξ=9n2k+2n3k+22k2−3k.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(49)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=6n+42k+6n−62k−1+3k−1,and by doing some calculations, we get(50)Mk1ξ=9n2k+2n3k+22k2−3k.Theorem 12.
For the subdivided polythiophene network, SPLYn, the forgotten index is calculated by(51)Fξ=263n−23.Proof.
By lettingξ as a subdivided polythiophene network SPLYn, from Table 1, we know(52)Fξ=∑gh∈Eξdg2+dh2,Fξ=6n+48+6n−613,and by doing some calculations, we get(53)Fξ=126n−46.


## 2.2. Results for the Subdivided Hexagonal Network

We construct a subdivided hexagonal network shown in Figure2 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 SHXn. A subdivided hexagonal network for n = 6 is shown in Figure 2, whereas n shows the number of nodes. The order of SHXn is 12n2 − 18n + 7 for n > 1, and the size is 18n2 − 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). The division of edges is shown in Table 2. A subdivided hexagonal network SHX6 is displayed in Figure 2.Figure 2
SHX6.Table 2
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ2,3182,424n−22,663n2−9n+7Theorem 13.
For the subdivided hexagonal network, SHXn, the ordinary generalized geometric-arithmetic index is calculated by(54)OGAkξ=18n2488k+n24326k−54488k+42488k−48326k.Proof.
By letting1 as a subdivided hexagonal network SHXn, from Table 2, we know(55)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=18245k+24n−2326k+63n2−9n+7488k,and by doing some calculations, we get(56)OGAkξ=18n2488k+n24326k−54488k+42488k−48326k.Theorem 14.
For the subdivided hexagonal network, SHXn, the first and second Gourava indices are calculated by(57)GO1ξ=360n2−744n+366,GO2ξ=1728n2−4032n+2268.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(58)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(59)GO1ξ=185+6+24n−26+8+63n2−9n+78+12=360n2−744n+366,GO2ξ=1830+24n−248+63n2−9n+796,=1728n2−4032n+2268.Theorem 15.
For the subdivided hexagonal network, SHXn, the first and second hyper-Gourava indices are calculated by HGO1ξ=7200n2−16896n+9570 and HGO2ξ=165888n2−442368n+292680.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(60)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(61)HGO1ξ=18121+24n−2196+63n2−9n+7400=7200n2−16896n+9570,HGO2ξ=18900+24n−22304+63n2−9n+79216,=165888n2−442368n+292680.Theorem 16.
For the subdivided hexagonal network, SHXn, n > 1, the general Randic´ index is calculated by(62)RγSHXn=3n2−3n+12,forγ=−1,3n23+n72−93+2.501,forγ=−12,62.35n2−119.17n+53.82,forγ=12,216n2−456n+228,forγ=1.Proof.
By lettingξ as a subdivided hexagonal network SHXn of n dimensions, we have the number of nodes and edges in SHXn as VSHXn=12n2−18N+7 for n > 1 and ESHXn=18n2−30n+12 for n > 1, respectively. We know that(63)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(64)R−1ξ=∑gh∈Eξ1dgdh,using (63). From Table 2, we know(65)R−1ξ=186−1+24n−28−1+63n2−9n+712−1.By doing some calculations, we getR−1ξ=3n2−3n+1/2.Case 2: ifγ = − 1/2, the application of Randic´ index Rγξ(66)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (63),(67)R−1/2ξ=1816+24n−218+63n2−9n+7112.By doing some calculations, we get(68)R−1/2ξ=3n23+n72−93+2.501.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(69)R1/2ξ=∑gh∈Eξdg·dh,using (63),(70)R1/2ξ=186+24n−28+63n2−9n+712.By doing some calculations, we get(71)R1/2ξ=62.35n2−119.17n+53.82.Case 4: ifγ = 1, the application of Randic´ index Rγξ(72)R1ξ=∑gh∈Eξdg.dh1,
using (63),(73)R1ξ=186+24n−28+63n2−9n+712.
By doing some calculations, we get(74)R1ξ=216n2−456n+228.Theorem 17.
For the subdivided hexagonal network, SHXn, the harmonic index is calculated by(75)HIξ=4.5n2−5.5n+1.7.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(76)HIξ=∑gh∈Eξ2dg+dh.
By doing some calculations, we get(77)HIξ=1825+24n−226+63n2−9n+728,=11045n2−55n+17=4.5n2−5.5n+1.7.Theorem 18.
For the subdivided hexagonal network, SHXn, the general version of the harmonic index is calculated by(78)HkIξ=18n24−k+2n123−k−274−k+182k5−k−483−k+424−k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(79)HkIξ=∑gh∈Eξ2dg+dhk.
By doing some calculations, we get(80)HkIξ=1822+3k+24n−222+4k+63n2−9n+722+6k=n2184k+n243k−544k+182k5k−483k+424k=18n24−k+2n123−k−274−k+182k5−k−483−k+424−k.Theorem 19.
For the subdivided hexagonal network, SHXn, the atom-bond connectivity index is calculated by(81)ABCξ=92n2−152n+62.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(82)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(83)ABCξ=182+3−26+24n−22+4−28+63n2−9n+72+6−212=92n2−152n+62.Theorem 20.
For the subdivided hexagonal network, SHXn, SK, SK1, and SK2 indices are calculated by SKξ=72n2−144n+69,SK1ξ=108n2−228n+498,andSK2ξ=288n2−648n+7052, respectively.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(84)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(85)SKξ=1852+24n−262+63n2−9n+782=72n2−144n+69.SK1ξ=183+24n−24+363n2−9n+7=108n2−228n+498.SK2ξ=18254+24n−29+63n2−9n+716=288n2−648n+7052.Theorem 21.
For the subdivided hexagonal network, SHXn, the sum-connectivity index is calculated by(86)χ−1/2ξ=18n218+n246−548+185−486+428.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(87)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=1816+24n−218+63n2−9n+7112,and by doing some calculations, we get(88)χ−1/2ξ=n2188+n246−548+185−486+428=18n218+n246−548+185−486+428.Theorem 22.
For the subdivided hexagonal network, SHXn, the general sum-connectivity index is calculated by(89)χkξ=18n28k+n246k−548k+185k−486k+428k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(90)χkξ=∑gh∈Eξdg+dhk,χkξ=185k+24n−26k+63n2−9n+78k,and by doing some calculations, we get(91)χkξ==18n28k+n246k−548k+185k−486k+428k.Theorem 23.
For the subdivided hexagonal network, SHXn, the first general Zagreb index is calculated by(92)Mk1ξ=n236k+92k+n64k−152k−96k+62k+63k−124k+76k.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(93)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=182k2+3k3+24n−482k2+4k4+63n2−9n+72k−1+6k−1,and by doing some calculations, we get(94)Mk1ξ=n236k+92k+n64k−152k−96k+62k+63k−124k+76k.Theorem 24.
For the subdivided hexagonal network, SHXn, the forgotten index is calculated by(95)Fξ=2360n2−840n+477.Proof.
By lettingξ as a subdivided hexagonal network SHXn, from Table 2, we know(96)Fξ=∑gh∈Eξdg2+dh2,Fξ=1813+24n−220+63n2−9n+740,and by doing some calculations, we get(97)Fξ=2360n2−840n+477.


## 2.3. Results for the Subdivided Backbone DNA Network

The 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. These sugar and phosphates make up the backbone, while the nitrogen bases are found in the centre and hold the two strands together. There 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 SBBDNA(n). The order and size of SBBDNA(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 SBBDNA(4) is shown in Figure 3.Figure 3
SBBDNA(4).Table 3
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)10n(2, 3)6n−1Theorem 25.
For the subdivided backbone DNA network, SBBDNA(n), the ordinary generalized geometric-arithmetic index is calculated by(98)OGAkξ=2n5+3245k−3245k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(99)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=10n162+2k+6n−6245k,and by doing some calculations, we get(100)OGAkξ=2n5+3245k−3245k.Theorem 26.
For the subdivided backbone DNA network, SBBDNA(n), the first and second Gourava indices are calculated by GO1ξ=146n−66 and GO2ξ=340n−180.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(101)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(102)GO1ξ=10n4+4+6n−65+6=146n−66,GO2ξ=10n16+6n−630=340n−180.Theorem 27.
For the subdivided backbone DNA network, SBBDNA(n), the first and second hyper-Gourava indices are calculated by(103)HGO1ξ=1366n−726,HGO2ξ=7960n−5400.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(104)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(105)HGO1ξ=10n64+6n−6121=1366n−726,HGO2ξ=10n256+6n−6900=7960n−5400.Theorem 28.
For the subdivided backbone DNA network, SBBDNA(n), the general Randic´ index is calculated by(106)RγSBBDNAn=72n−1,forγ=−1,n14920−4920,forγ=−12,n34710−14710,forγ=12,76n−36,forγ=1.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n) of n dimensions, we have the order and size of ξ in SBBDNA(n) as VSBBDNAn=15n−5 and ESBBDNAn=16n−6, respectively.
We know that(107)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(108)R−1ξ=∑gh∈Eξ1dgdh,using (107). From Table 3, we know R−1ξ=10n4−1+6n−66−1. By doing some calculations, we get R−1ξ=7/2n−1.Case 2: ifγ = −1/2, the application of Randic´ index Rγξ(109)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (107),(110)R−1/2ξ=10n14+6n−616,and by doing some calculations, we getR−1/2ξ=n149/20−49/20.Case 3: ifγ =  1/2, the application of Randic´ index Rγξ(111)R1/2ξ=∑gh∈Eξdg.dh,using (107),(112)R1/2ξ=10n4+6n−66,and by doing some calculations, we getR1/2ξ=n347/10−147/10.Case 4: ifγ = 1, the application of Randic´ index Rγξ(113)R1ξ=∑gh∈Eξdg·dh1,  using (107),(114)R1ξ=10n4+6n−66,  and by doing some calculations, we get R1ξ=76n−36.Theorem 29.
For the subdivided backbone DNA network, SBBDNA(n), the harmonic index is calculated by(115)HIξ=37n−125.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(116)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(117)HIξ=10n12+6n−625=37n−125.Theorem 30.
For the subdivided backbone DNA network, SBBDNA(n), the general version of the harmonic index is calculated by(118)HkIξ=2n52k+35k2k−65k2k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(119)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(120)HkIξ=10n22+2k+6n−622+3k=2n52k+35k2k−65k2k.Theorem 31.
For the subdivided backbone DNA network, SBBDNA(n), the atom-bond connectivity index is calculated by(121)ABCξ=28n−32.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(122)ABCξ=∑gh∈Eξdg+dh−2dgdh.
By doing some calculations, we get(123)ABCξ=10n2+2−24+6n−62+3−26=28n−32.Theorem 32.
For the subdivided backbone DNA network, SBBDNA(n), SK, SK1, and SK2 indices are calculated by SKξ=35n−15,SK1ξ=38n−18, and SK2ξ=1/2155n−75, respectively.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(124)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22.
By doing some calculations, we get(125)SKξ=10n2+6n−652=35n−15,SK1ξ=10n2+6n−63=38n−18,SK2ξ=10n4+6n−6254=12155n−75.Theorem 33.
For the subdivided backbone DNA network, SBBDNA(n), the sum-connectivity index is calculated by(126)χ−1/2ξ=n5+65−65.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(127)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=10n12+6n−615.
By doing some calculations, we get(128)χ−1/2ξ=n5+65−65.Theorem 34.
For the subdivided backbone DNA network, SBBDNA(n), the general sum-connectivity index is calculated by(129)χkξ=n104k+65k−65k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(130)χkξ=∑gh∈Eξdg+dhk,χkξ=10n4k+6n−65k.
By doing some calculations, we get(131)χkξ=n104k+65k−65k.Theorem 35.
For the subdivided backbone DNA network, SBBDNA(n), the first general Zagreb index is calculated by(132)Mk1ξ=n102k+32k+23k−32k+23k.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(133)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=10n2k+6n−62k−1+3k−1,and by doing some calculations, we get(134)Mk1ξ=n102k+32k+23k−32k+23k.Theorem 36.
For the subdivided backbone DNA network, SBBDNA(n), the forgotten index is calculated by(135)Fξ=279n−39.Proof.
By lettingξ as a subdivided backbone DNA network SBBDNA(n), from Table 3, we know(136)Fξ=∑gh∈Eξdg2+dh2,Fξ=10n8+6n−613,and by doing some calculations, we get(137)Fξ=279n−39.


## 2.4. Results for the Subdivided Honeycomb Network

The honeycomb network is a hexagon. It can be made in different methods. The first honeycomb network is symbolized by HC(1). The next honeycomb network is produced by attaching more hexagons to each of its edges. This 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). The 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. The n-dimensional subdivided honeycomb network is symbolized by SHCn. A subdivided honeycomb network for n = 4 is displayed in Figure 4. The number of nodes and edges in the subdivided honeycomb networks are 15n2 − 3n and 18n2 − 6n, respectively. We have obtained two different types of edges in SHC4 shown in Table 4, whereas Figure 4 shows SHC4.Figure 4
4-dimensional SHC4.Table 4
Division of edges of a graphξ found on the degree of terminating nodes of each of the edges.
dg,dh for gh∈EξNumber ofEξ(2, 2)12n(2, 3)18nn−1Theorem 37.
For the subdivided honeycomb network, SHCn, the ordinary generalized geometric-arithmetic index is calculated by(138)OGAkξ=63n2245k+n2−3245k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(139)OGAkξ=∑gh∈Eξ4dgdhdg+dhk,OGAkξ=12n162+2k+18nn−1245k,and by doing some calculations, we get(140)OGAkξ=63n2245k+n2−3245k.Theorem 38.
For the subdivided honeycomb network, SHCn, the first and second Gourava indices are calculated by GO1ξ=198n2−102n and GO2ξ=540n2−492n.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(141)GO1ξ=∑gh∈Eξdg+dh+dgdh,GO2ξ=∑gh∈Eξdg+dh+dgdh,and by doing some calculations, we get(142)GO1ξ=12n4+4+18nn−15+6=198n2−102n,GO2ξ=12n16+18nn−130=540n2−492n.Theorem 39.
For the subdivided honeycomb network, SHCn, the first and second hyper-Gourava indices are calculated by HGO1ξ=2178n2−1410n and HGO2ξ=16200n2−13128n.Proof.
By lettingξ as the subdivided honeycomb network SHCn, from Table 4, we know(143)HGO1ξ=∑gh∈Eξdg+dh+dgdh2,HGO2ξ=∑gh∈Eξdg+dh+dgdh2,and by doing some calculations, we get(144)HGO1ξ=12n64+18nn−1121=2178n2−1410n,HGO2ξ=12n256+18nn−1900=16200n2−13128n.Theorem 40.
For the subdivided honeycomb network, SHCn, the general Randic´ index is calculated by(145)RγSHCn=3n2,forγ=−1,3n26+n2−6,forγ=−12,63n26+n4−36,forγ=12,108n2−60n,forγ=1.Proof.
By lettingξ as a subdivided honeycomb network SHCn of n dimensions, we have the order and size of ξ in SHCn as VSHCn=15n2−3n and ESHCn=18n2−6n, respectively. We know that(146)Rγξ=∑gh∈Eξdg⋅dhγ,for γ=−1,1,−1/2,1/2.Case 1: ifγ = −1, the application of Randic´ index Rγξ(147)R−1ξ=∑gh∈Eξ1dgdh,using (146). From Table 4, we obtain R−1ξ=12n4−1+18nn−16−1.By doing some calculations, we obtainR−1ξ=3n2.Case 2: ifγ = −1/2, the application of Randic´ index Rγξ(148)R−1/2ξ=∑gh∈Eξ1dg⋅dh,using (146). From Table 4, we obtain R−1/2ξ=12n1/4+18nn−11/6.By doing some calculations, we obtainR−1/2ξ=3n26+n2−6.Case 3: ifγ = 1/2, the application of Randic index Rγξ(149)R1/2ξ=∑gh∈Eξdg.dh,using (146). From Table 4, we obtain R1/2ξ=12n4+18nn−16.By doing some calculations, we obtainR1/2ξ=63n26+n4−36.Case 4: ifγ = 1, the application of Randic´ index Rγξ(150)R1ξ=∑gh∈Eξdg.dh1,using (146). From Table 4, we obtain R1ξ=12n4+18nn−16.By doing some calculations, we obtainR1ξ=108n2−60n.Theorem 41.
For the subdivided honeycomb network, SHCn, the harmonic index is calculated by(151)HIξ=36n2−6n5.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(152)HIξ=∑gh∈Eξ2dg+dh,and by doing some calculations, we get(153)HIξ=12n12+18nn−125=36n2−6n5.Theorem 42.
For the subdivided honeycomb network, SHCn, the general version of the harmonic index is calculated by(154)HkIξ=63n225k+n22k−325k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(155)HkIξ=∑gh∈Eξ2dg+dhk,and by doing some calculations, we get(156)HkIξ=12n22+2k+18nn−122+3k,=63n225k+n22k−325k.Theorem 43.
For the subdivided honeycomb network, SHCn, the atom-bond connectivity index is calculated by(157)ABCξ=n29n−3.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(158)ABCξ=∑gh∈Eξdg+dh−2dgdh,and by doing some calculations, we get(159)ABCξ=12n2+2−24+18nn−12+3−26,=n29n−3.Theorem 44.
For the subdivided honeycomb network, SHCn, SK, SK1, and SK2 indices are calculated by SKξ=45n2−21n,SK1ξ=54n2−30n, and SK2ξ=1/2225n2−129n, respectively.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(160)SKξ=∑gh∈Eξdg+dh2,SK1ξ=∑gh∈Eξdgdh2,SK2ξ=∑gh∈Eξdg+dh22,and by doing some calculations, we get(161)SKξ=12n2+18nn−152=45n2−21n,SK1ξ=12n2+18nn−13=54n2−30n,SK2ξ=12n4+18nn−1254=12225n2−129n.Theorem 45.
For the subdivided honeycomb network, SHCn, the sum-connectivity index is calculated by(162)χ−1/2ξ=1518n2+n65−18.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(163)χ−1/2ξ=∑gh∈Eξdg+dh−1/2,χ−1/2ξ=12n12+18nn−115,and by doing some calculations, we get(164)χ−1/2ξ=1518n2+n65−18.Theorem 46.
For the subdivided honeycomb network, SHCn, the general sum-connectivity index is calculated by(165)χkξ=185kn2+6n24k−35k.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(166)χkξ=∑gh∈Eξdg+dhk,χkξ=12n4k+18nn−15k,and by doing some calculations, we get(167)χkξ=185kn2+6n24k−35k.Theorem 47.
For the subdivided honeycomb network, SHCn, the first general Zagreb index is calculated by(168)Mk1ξ=18n22k−1+3k−1+6n22k−3n2k−1+3k−1.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(169)Mk1ξ=∑gh∈Eξdgk−1+dhk−1,k>1,Mk1ξ=12n2k+18nn−12k−1+3k−1,and by doing some calculations, we get(170)Mk1ξ=18n22k−1+3k−1+6n22k−3n2k−1+3k−1.Theorem 48.
For the subdivided honeycomb network, SHCn, the forgotten index is calculated by(171)Fξ=2n117n−69.Proof.
By lettingξ as a subdivided honeycomb network SHCn, from Table 4, we know(172)Fξ=∑gh∈Eξdg2+dh2,Fξ=12n8+18nn−113,and by doing some calculations, we get Fξ=2n117n−69.


## 3. Conclusions

In this paper, we have computed 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´ indexRγξ,forγ=±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 different kinds of chemical networks such as the subdivided polythiophene network, subdivided hexagonal network, subdivided backbone DNA network, and subdivided honeycomb network. The above computed topological indices are used as molecular descriptors in the construction of “quantitative structure-activity relationships and quantitative structure-property relationships.” These indices give us results that can be correlated with the molecular structures to understand their chemical and physical properties.For the next research papers, our goal is to compute more topological indices for some new graphs to know their topologies.



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*Source: 5533619-2021-05-21.xml*