# On Distance-Based Topological Descriptors of Chemical Interconnection Networks

Authors: Min Hu; Haidar Ali; Muhammad Ahsan Binyamin; Bilal Ali; Jia-Bao Liu; Chengmei Fan
Journal: Journal of Mathematics
 (2021)

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## Abstract

Structure-based topological descriptors of chemical networks enable us the prediction of physico-chemical properties and the bioactivities of compounds through QSAR/QSPR methods. Topological indices are the numerical values to represent a graph which characterises the graph. One of the latest distance-based topological index is the Mostar index. In this paper, we study the Mostar index, Szeged index, PI index,ABCGG index, and NGG index, for chain oxide network COXn, chain silicate network CSn, ortho chain Sn, and para chain Qn, for the first time. Moreover, analytically closed formulae for these structures are determined.


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## Body

## 1. Introduction and Preliminary Results

All the graphsG in this paper are considered to be finite, undirected, and loopless. Graph G is the set made up of vertices (also called the nodes) which are connected with the edges (also called links). It consists on two sets V and E, where V is called the vertex set and E is called the edge set. In order to understand the properties and information contained in the connectivity pattern of graphs, there are many numbers of numerical quantities, known as structure invariants, topological indices, or topological descriptors, which have been derived and studied over the past few decades. The topological indices have vast number of applications in the chemical graph theory which is the special branch of mathematical chemistry.Graph theory has a wide range of applications in engineering due to its diagrammatic nature. It is used in computer science to study the algorithms and flow of information. In engineering, it is used to model the graphics and designs of different networks by converting them in the form of graph.The topological indices are very much used for characterizing the chemical graphs on the basis of their numerical values. They establish the relationship between the structure and properties of molecule. Topological indices are widely used in QSAR and QSPR research studies [1]. Till now, many topological indices have been derived. For any two graphs G and H which are isomorphic to each other, then TopG=TopH [2]. Due to the success of simple topological indices, such as Wiener Index [3], Zagreb index [4], and Szeged index [5], motivated others, hundreds of topological indices are introduced. Wiener index is one of the first index which was introduced by Harold Wiener in 1947 [6], when he was working on the boiling point of paraffins. The Wiener index [7] of a graph G is defined as the sum of all the distances between pairs of vertices of G:(1)WG=∑u,v⊆VGdu,v,where du,v denotes the shortest-path distance in G.The Szeged index is defined as(2)SzG=∑e=uv∈EGnunv,where nu denotes the number of vertices of G closer to u than to v and nv is defined as the number of vertices of G closer to v than to u. This was first studied by Gutman. Later, it is known as the Szeged index [8].The PI index [9], of a graph G, is defined as(3)PIvG=∑e=uv∈EGnu+nv.The Graovac–Ghorbani index is defined as(4)ABCGGG=∑e=uv∈EGnu+nv−2nunv,and this index is introduced by Graovac and Ghorbani [10], and Furtula [11] used the name Graovac–Ghorbani index.The normalizedABCGG index is NGG index, first studied by Dimitrov et al. [12], and is defined as(5)NGGG=∑e=uv∈EG1nunv.A chemical graph is a simple graph in which atoms correspond to the vertices and edge denotes the bond between two atoms. A topological index, specially, the Mostar index is one of the latest topological index, derived in 2018 [13]. Previously, Arockiaraj [14] found the Mostar indices of carbon nanostructures, and Hayata and Zhou [15] calculated the large Mostar index on cacti. The Mostar index for a graph G is defined as the sum of all the absolutes values of the difference between nu and nv, where u and v are the adjacent vertices of an edge:(6)MoG=∑e=uv∈EGnu−nv.


## 2. Main Results

The main goal of this article is to compute the Mostar index of ortho chain and para chain using the edge cut method; also, we find the Mostar index, Szeged index, PI index,ABCGG index, and NGG index of oxide chains, chain silicates, ortho chain, and para chain by using the technique of edge partition. The notations used in this paper are standard and taken from the book of west [16]. For the concepts and terms not defined here, we refer the reader to concern with the book of Harary [17] and also concern with [18–25].


### 2.1. Results for the Chain Oxide NetworkCOXn

In this section, we discussCOXn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. If we remove the silicon atom from the silicate network, then the resulting network is an oxide network [26], which consists of three oxygen atoms. Oxide network has the triangular structure. If an oxide network shares its oxygen with other oxide network linearly, then the oxide chain is formed, as shown in Figure 1.Figure 1
Oxide network.Theorem 1.
LetG1 be the oxide network of n order, then its Szeged index is 2n3+6n2+n/3.Proof.
LetG1≅OXn, where n≥2; also, n is an integer.(7)SzG1=∑e∈EG1nunv,SzG1=∑e∈EG1n2n2+∑e∈EG1n2n4+∑e∈EG1n4n4.
By using Table1, we have(8)SzG1=2+2n2+4n−4+2n3−11n+63,SzG1=2n3+6n2+n3,which is required.Table 1
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2222, 42n2n2+4n−44, 4n−22n3−11n+6/3Theorem 2.
LetG1 be the oxide network of n order; then, its PI index is 4n2+2n.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(9)PIvG1=∑e∈EG1nu+nv,PIvG1=∑e∈EG1n2+n2+∑e∈EG1n2+n4+∑e∈EG1n4+n4.
By using Table2, we have(10)PIvG1=4+2n2+6n−4+2n2−4n,PIvG1=4n2+2n,which is required.Table 2
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2242, 42n2n2+6n−44, 4n−22n2−4nTheorem 3.
LetG1 be the oxide network of n order; then, its ABCGG index is 1+n2+3n−3/n2+2n−2+6n2−12n−6/2n3−11n+6.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(11)ABCGGG1=∑e∈EG1nu+nv−2nunv,ABCGGG1=∑e∈EG1n2+n2−2n2n2+∑e∈EG1n2+n4−2n2n4+∑e∈EG1n4+n4−2n4n4.
By using Table3, we have(12)ABCGGG1=1+n2+3n−3n2+2n−2+6n2−12n−62n3−11n+6,which is required.Table 3
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2,2212,42nn2+3n−3/n2+2n−24,4n−26n2−12n−6/2n3−11n+6Theorem 4.
LetG1 be the oxide network of n order; then, its NGGG1 index is 1/2+1/2n2+4n−4+3/2n3−11n+6.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(13)NGGG1=∑e∈EG11nunv,NGGG1=∑e∈EG11n2n2+∑e∈EG11n2n4+∑e∈EG11n4n4.
By using Table4, we have(14)NGGG1=12+12n2+4n−4+32n3−11n+6,which is required.Table 4
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 221/22, 42n1/2n2+4n−44, 4n−23/2n3−11n+6Theorem 5.
LetG1 be the oxide network of even order; then, its Mostar index is 3n2−2n.Proof.
LetG1≅OXn, where n≥2; also, n is even:(15)MoG1=∑uv∈EG1nu−nv,MoG1=∑uv∈EG1n2−n2+∑uv∈EG1n2−n4+∑uv∈EG1n4−n4.
By using Table5, we have(16)MoG1=0+2n2+2n−4+n2−4n+4,MoG1=3n2−2n,which is required.Table 5
Edge partition of oxide network of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2202, 42n2n2+2n−44, 4n−2n2−4n+4Theorem 6.
LetG1 be the oxide network of odd order; then, its Mostar index is 3n2−2n−1.Proof.
LetG1≅OXn, where n≥1; also, n is odd:(17)MoG1=∑uv∈EG1nu−nv,MoG1=∑uv∈EG1n2−n2+∑uv∈EG1n2−n4+∑uv∈EG1n4−n4.
By using Table6, we have(18)MoG1=0+2n2+2n−4+n2−4n+3,MoG1=3n2−2n−1,which is required.Table 6
Edge partition of oxide network of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2202, 42n2n2+2n−44, 4n−2n2−4n+3


### 2.2. Results for the Chain Silicate NetworkCSn

In this section, we discussCSn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. Silicates are the compounds which consist of silicon and oxygen, having the tetrahedron structure with bond angle of 109.5°. SiO4 is found in almost all of the silicates. A single tetrahedron has a shape like a pyramid with triangular base. It has four oxygen atoms at its corners, and silicon atom is bounded equally with oxygen atoms with bond length of 162pm. A single tetrahedron is shown in Figure 2(a). If a single tetrahedron shares its oxygen with other tetrahedrons; then, a linear silicate chain [27] is formed, as shown in Figure 2(b).Figure 2
(a) Single silicate and (b) chain silicate.
(a)(b)Theorem 7.
LetG2 be the chain silicate network of n order; then, its Szeged index is 3n3+9n2/2.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(19)SzG2=∑e∈EG2nunv,SzG2=∑e∈EG2n3n3+∑e∈EG2n3n6+∑e∈EG2n6n6.
By using Table7, we have(20)SzG2=n+4+6n2+4n−8+3n3−3n2−10n+82,SzG2=3n3+9n22,which is required.Table 7
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesSzeged index3, 3n+4n+43, 622n−16n2+4n−86, 6n−23n3−3n2−10n+8/2Theorem 8.
LetG2 be the chain silicate network of n order; then, its PI index is 9n2+3n.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(21)PIvG2=∑e∈EG2nu+nv,PIvG2=∑e∈EG2n3+n3+∑e∈EG2n3+n6+∑e∈EG2n6+n6.
By using Table8, we have(22)PIvG2=2n+8+6n2+8n−10+3n2−7n+2,PIvG2=9n2+3n,which is required.Table 8
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesPI index3, 3n+42n+83, 622n−16n2+8n−106, 6n−23n2−7n+2Theorem 9.
LetG2 be the chain silicate network of n order then its ABCGG index is 2n+6/n+4+3n2+4n−6/3n2+2n−4+6n2−14n/3n3−3n2−10n+8.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(23)ABCGGG2=∑e∈EG2nu+nv−2nunv,ABCGGG2=∑e∈EG2n3+n3−2n3n3+∑e∈EG2n3+n6−2n3n6+∑e∈EG2n6+n6−2n6n6.
By using Table9, we have(24)ABCGGG2=2n+6n+4+3n2+4n−63n2+2n−4+6n2−14n3n3−3n2−10n+8,which is required.Table 9
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index3, 3n+42n+6/n+43, 622n−13n2+4n−6/3n2+2n−46, 6n−26n2−14n/3n3−3n2−10n+8Theorem 10.
LetG2 be the chain silicate network of n order; then, its NGG index is 1/n+4+1/6n2+4n−8+2/3n3−3n2−10n+8.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(25)NGGG2=∑e∈EG21nunv,NGGG2=∑e∈EG21n3n3+∑e∈EG21n3n6+∑e∈EG21n6n6.
By using Table10, we have(26)NGGG2=1n+4+16n2+4n−8+23n3−3n2−10n+8,which is required.Table 10
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index3, 3n+41/n+43, 622n−11/6n2+4n−86, 6n−22/3n3−3n2−10n+8Theorem 11.
LetG2 be the chain silicate network of even order; then, its Mostar index is 15n2−12n/2.Proof.
LetG2≅CSn, where n≥2; also, n is even.(27)MoG2=∑uv∈EG2nu−nv,MoG2=∑uv∈EG2n3−n3+∑uv∈EG2n3−n6+∑uv∈EG2n6−n6.
By using Table11, we have(28)MoG2=0+6n2−6+3n2−12n+122,MoG2=15n2−12n2,which is required.Table 11
Edge partition of chain silicate network of even order.
Edge partitiondu,dvNumber of edgesMostar index3, 3n+403, 622n−16n2−66, 6n−23n2−12n+12/2Theorem 12.
LetG2 be the chain silicate network of odd order; then, its Mostar index is 15n2−12n−3/2.Proof.
LetG2≅CSn, where n≥1; also, n is odd:(29)MoG2=∑uv∈EG2nu−nv,MoG2=∑uv∈EG2n3−n3+∑uv∈EG2n3−n6+∑uv∈EG2n6−n6.
By using Table12, we have(30)MoG2=0+6n2−6+3n2−12n+92,MoG2=15n2−12n−32,which is required.Table 12
Edge partition of chain silicate network of odd order.
Edge partitiondu,dvNumber of edgesMostar index3, 3n+403, 622n−16n2−66, 6n−23n2−12n+9/2


### 2.3. Results for the Ortho ChainSn

In this section, we discussSn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. The single molecule of para and ortho chain has the same structure. Basically, it is a cycle graph having 4 sides denoted as C4 and represented as a four-sided regular polygon. The ortho chain has a zig-zag structure where each corner of C4 is attached linearly, as shown in Figure 3. The para chain has a structure in which each C4 is attached at corner to corner with other C4 but not linearly, as shown in Figure 4 [28].Figure 3
Ortho chain ofn vertices.Figure 4
Para chain ofn order.Theorem 13.
LetG3 be the ortho chain of n order; then, its Szeged index is 3n3+15n2−2n.Proof.
LetG3≅Sn, where n≥2; also, n is an integer.(31)SzG3=∑e∈EG3nunv,SzG3=∑e∈EG3n2n2+∑e∈EG3n2n4+∑e∈EG3n4n4.
By using Table13, we have(32)SzG3=3n3+3n2+26n−82+12n2−4n+3n3+3n2−22n+82,SzG3=3n3+15n2−2n,which is required.Table 13
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2n+23n3+3n2+26n−8/22, 42n12n2−4n4, 4n−23n3+3n2−22n+8/2Theorem 14.
LetG3 be the ortho chain of n order; then, its PI index is 12n2+4n.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(33)PIvG3=∑e∈EG3nu+nv,PIvG3=∑e∈EG3n2+n2+∑e∈EG3n2+n4+∑e∈EG3n4+n4.
By using Table14, we have(34)PIvG3=3n2+7n+2+6n2+2n+3n2−5n−2,PIvG3=12n2+4n,which is required.Table 14
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2n+23n2+7n+22, 42n6n2+2n4, 4n−23n2−5n−2Theorem 15.
LetG3 be the ortho chain of n order; then, its ABCGG index is 6n2+14n/3n3+3n2+26n−8+3n2+n−1/6n2−2n+6n2−10n−8/3n3+3n2−22n+8.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(35)ABCGGG3=∑e∈EG3nu+nv−2nunv,ABCGGG3=∑e∈EG3n2+n2−2n2n2+∑e∈EG3n2+n4−2n2n4+∑e∈EG3n4+n4−2n4n4.
By using Table15, we have(36)ABCGGG3=6n2+14n3n3+3n2+26n−8+3n2+n−16n2−2n+6n2−10n−83n3+3n2−22n+8,which is required.Table 15
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2, 2n+26n2+14n/3n3+3n2+26n−82, 42n3n2+n−1/6n2−2n4, 4n−26n2−10n−8/3n3+3n2−22n+8Theorem 16.
LetG3 be the ortho chain of n order; then, its NGGG3 index is 2/3n3+3n2+26n−8+1/23n2−n+2/3n3+3n2−22n+8.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(37)NGGG3=∑e∈EG31nunv,NGGG3=∑e∈EG31n2n2+∑e∈EG31n2n4+∑e∈EG31n4n4.
By using Table16, we have(38)NGGG3=23n3+3n2+26n−8+123n2−n+23n3+3n2−22n+8,which is required.Table 16
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 2n+22/3n3+3n2+26n−82, 42n1/23n2−n4, 4n−22/3n3+3n2−22n+8Theorem 17.
LetG3 be the ortho chain of even order; then, its Mostar index is 9n2−6n.Proof.
LetG3≅Sn, where n≥2; also, n is even:(39)MoG3=∑uv∈EG3nu−nv,MoG3=∑uv∈EG3n2−n2+∑uv∈EG3n2−n4+∑uv∈EG3n4−n4.
By using Table17, we have(40)MoG3=3n2+12n−122+6n2−6n+3n2−12n+122,MoG3=9n2−6n,which is required.Table 17
Edge partition of ortho chain of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2n+23n2+12n−12/22, 42n6n2−6n4, 4n−23n2−12n+12/2Theorem 18.
LetG3 be the ortho chain of odd order; then, its Mostar index is 9n2−6n−3.Proof.
LetG3≅Sn, where n≥1; also, n is odd:(41)MoG3=∑uv∈EG3nu−nv,MoG3=∑uv∈EG3n2−n2+∑uv∈EG3n2−n4+∑uv∈EG3n4−n4.
By using Table18, we have(42)MoG3=3n2+12n−152+6n2−6n+3n2−12n+92,MoG3=9n2−6n−3,which is required.Table 18
Edge partition of ortho chain of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2n+23n2+12n−15/22, 42n6n2−6n4, 4n−23n2−12n+9/2


### 2.4. Results for the Para ChainQn

In this section, we discussQn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index.Theorem 19.
LetG4 be the para chain of n order; then, its Szeged index is 6n3+6n2+4n.Proof.
LetG4≅Qn, where n≥2; also, n is an integer.(43)SzG4=∑e∈EG4nunv,SzG4=∑e∈EG4n2n2+∑e∈EG4n2n4.
By using Table19, we have(44)SzG4=24n−8+6n3+6n2−20n+8,SzG4=6n3+6n2+4n,which is required.Table 19
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2424n−82, 44n−46n3+6n2−20n+8Theorem 20.
LetG4 be the para chain of n order; then, its PI index is 12n2+4n.Proof.
LetG4≅Qn, where n≥2; also, n is an integer:(45)PIvG4=∑e∈EG4nu+nv,PIvG4=∑e∈EG4n2+n2+∑e∈EG4n2+n4.
By using Table20, we have(46)PIvG4=12n+4+12n2−8n−4,PIvG4=12n2+4n,which is required.Table 20
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2412n+42, 44n−412n2−8n−4Theorem 21.
LetG4 be the para chain of n order; then, its ABCGG index is 6n+1/12n−4+6n2−4n−3/3n3+3n2−10n+4.Proof.
LetG4≅OXn, where n≥2; also, n is an integer:(47)ABCGGG4=∑e∈EG4nu+nv−2nunv,ABCGGG4=∑e∈EG4n2+n2−2n2n2+∑e∈EG4n2+n4−2n2n4.
By using Table21, we have(48)ABCGGG4=6n+112n−4+6n2−4n−33n3+3n2−10n+4,which is required.Table 21
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2, 246n+1/12n−42, 44n−46n2−4n−3/3n3+3n2−10n+4Theorem 22.
LetG4 be the para chain of n order; then, its NGGG4 index is 1/26n−2+1/6n3+6n2−20n+8.Proof.
LetG4≅Qn, where n≥2 also n is an integer:(49)NGGG4=∑e∈EG41nunv,NGGG4=∑e∈EG41n2n2+∑e∈EG41n2n4.
By using Table22, we have(50)NGGG4=126n−2+16n3+6n2−20n+8,which is required.Table 22
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 241/26n−22, 44n−41/6n3+6n2−20n+8Theorem 23.
LetG4 be the para chain of even order; then, its Mostar index is 6n2.Proof.
LetG4≅Qn, where n≥2; also, n is even:(51)MoG4=∑uv∈EG4nu−nv,MoG4=∑uv∈EG4n2−n2+∑uv∈EG4n2−n4.
By using Table23, we have(52)MoG4=12n−12+6n2−12n+12,MoG4=6n2,which is required.Table 23
Edge partition of para chain of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2412n−122, 44n−46n2−12n+12Theorem 24.
LetG4 be the para chain of odd order; then, its Mostar index is 6n2−6.Proof.
LetG4≅Qn, where n≥1; also, n is odd:(53)MoG4∑uv∈EG4nu−nv,MoG4=∑uv∈EG4n2−n2+∑uv∈EG4n2−n4.
By using Table24, we have(54)MoG4=12n−12+6n2+12n+6,MoG4=6n2−6,which is required.For the comparison of Szeged, PI,ABCGG, and NGG index of COXn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 25 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of CSn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 26 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of Sn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 27 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of Qn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 28 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.Table 24
Edge partition of para chain of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2412n−122, 44n−46n2+12n+6Table 25
Comparison table for chain oxide network.
OXnSzGPIABCNGG136——21420——33742——476722.751.0451351102.710.9562181562.660.9073292102.620.8684722722.590.8496513422.560.83108704202.530.81Table 26
Comparison table for chain silicate.
CSnSzGPIABCNGG1612——23042——38190——41681562.910.5853002402.890.4964863422.880.4577354622.860.41810566002.830.38914587562.810.351019509302.800.33Table 27
Comparison table for ortho chain.
SnSzGPIABCNGG11616——28056——3210120——44242081.980.2657403201.880.19611764561.800.15717506161.740.12824808001.680.109338410061.630.0910448012401.580.08Table 28
Comparison table for para chain.
QnSzGPIABCNGG11616——280561.610.3232281201.460.2044962081.370.1659203201.300.13615364561.260.11723806161.220.10834888001.190.099489610081.160.0810664012401.140.07


## 2.1. Results for the Chain Oxide NetworkCOXn

In this section, we discussCOXn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. If we remove the silicon atom from the silicate network, then the resulting network is an oxide network [26], which consists of three oxygen atoms. Oxide network has the triangular structure. If an oxide network shares its oxygen with other oxide network linearly, then the oxide chain is formed, as shown in Figure 1.Figure 1
Oxide network.Theorem 1.
LetG1 be the oxide network of n order, then its Szeged index is 2n3+6n2+n/3.Proof.
LetG1≅OXn, where n≥2; also, n is an integer.(7)SzG1=∑e∈EG1nunv,SzG1=∑e∈EG1n2n2+∑e∈EG1n2n4+∑e∈EG1n4n4.
By using Table1, we have(8)SzG1=2+2n2+4n−4+2n3−11n+63,SzG1=2n3+6n2+n3,which is required.Table 1
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2222, 42n2n2+4n−44, 4n−22n3−11n+6/3Theorem 2.
LetG1 be the oxide network of n order; then, its PI index is 4n2+2n.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(9)PIvG1=∑e∈EG1nu+nv,PIvG1=∑e∈EG1n2+n2+∑e∈EG1n2+n4+∑e∈EG1n4+n4.
By using Table2, we have(10)PIvG1=4+2n2+6n−4+2n2−4n,PIvG1=4n2+2n,which is required.Table 2
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2242, 42n2n2+6n−44, 4n−22n2−4nTheorem 3.
LetG1 be the oxide network of n order; then, its ABCGG index is 1+n2+3n−3/n2+2n−2+6n2−12n−6/2n3−11n+6.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(11)ABCGGG1=∑e∈EG1nu+nv−2nunv,ABCGGG1=∑e∈EG1n2+n2−2n2n2+∑e∈EG1n2+n4−2n2n4+∑e∈EG1n4+n4−2n4n4.
By using Table3, we have(12)ABCGGG1=1+n2+3n−3n2+2n−2+6n2−12n−62n3−11n+6,which is required.Table 3
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2,2212,42nn2+3n−3/n2+2n−24,4n−26n2−12n−6/2n3−11n+6Theorem 4.
LetG1 be the oxide network of n order; then, its NGGG1 index is 1/2+1/2n2+4n−4+3/2n3−11n+6.Proof.
LetG1≅OXn, where n≥2; also, n is an integer:(13)NGGG1=∑e∈EG11nunv,NGGG1=∑e∈EG11n2n2+∑e∈EG11n2n4+∑e∈EG11n4n4.
By using Table4, we have(14)NGGG1=12+12n2+4n−4+32n3−11n+6,which is required.Table 4
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 221/22, 42n1/2n2+4n−44, 4n−23/2n3−11n+6Theorem 5.
LetG1 be the oxide network of even order; then, its Mostar index is 3n2−2n.Proof.
LetG1≅OXn, where n≥2; also, n is even:(15)MoG1=∑uv∈EG1nu−nv,MoG1=∑uv∈EG1n2−n2+∑uv∈EG1n2−n4+∑uv∈EG1n4−n4.
By using Table5, we have(16)MoG1=0+2n2+2n−4+n2−4n+4,MoG1=3n2−2n,which is required.Table 5
Edge partition of oxide network of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2202, 42n2n2+2n−44, 4n−2n2−4n+4Theorem 6.
LetG1 be the oxide network of odd order; then, its Mostar index is 3n2−2n−1.Proof.
LetG1≅OXn, where n≥1; also, n is odd:(17)MoG1=∑uv∈EG1nu−nv,MoG1=∑uv∈EG1n2−n2+∑uv∈EG1n2−n4+∑uv∈EG1n4−n4.
By using Table6, we have(18)MoG1=0+2n2+2n−4+n2−4n+3,MoG1=3n2−2n−1,which is required.Table 6
Edge partition of oxide network of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2202, 42n2n2+2n−44, 4n−2n2−4n+3


## 2.2. Results for the Chain Silicate NetworkCSn

In this section, we discussCSn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. Silicates are the compounds which consist of silicon and oxygen, having the tetrahedron structure with bond angle of 109.5°. SiO4 is found in almost all of the silicates. A single tetrahedron has a shape like a pyramid with triangular base. It has four oxygen atoms at its corners, and silicon atom is bounded equally with oxygen atoms with bond length of 162pm. A single tetrahedron is shown in Figure 2(a). If a single tetrahedron shares its oxygen with other tetrahedrons; then, a linear silicate chain [27] is formed, as shown in Figure 2(b).Figure 2
(a) Single silicate and (b) chain silicate.
(a)(b)Theorem 7.
LetG2 be the chain silicate network of n order; then, its Szeged index is 3n3+9n2/2.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(19)SzG2=∑e∈EG2nunv,SzG2=∑e∈EG2n3n3+∑e∈EG2n3n6+∑e∈EG2n6n6.
By using Table7, we have(20)SzG2=n+4+6n2+4n−8+3n3−3n2−10n+82,SzG2=3n3+9n22,which is required.Table 7
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesSzeged index3, 3n+4n+43, 622n−16n2+4n−86, 6n−23n3−3n2−10n+8/2Theorem 8.
LetG2 be the chain silicate network of n order; then, its PI index is 9n2+3n.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(21)PIvG2=∑e∈EG2nu+nv,PIvG2=∑e∈EG2n3+n3+∑e∈EG2n3+n6+∑e∈EG2n6+n6.
By using Table8, we have(22)PIvG2=2n+8+6n2+8n−10+3n2−7n+2,PIvG2=9n2+3n,which is required.Table 8
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesPI index3, 3n+42n+83, 622n−16n2+8n−106, 6n−23n2−7n+2Theorem 9.
LetG2 be the chain silicate network of n order then its ABCGG index is 2n+6/n+4+3n2+4n−6/3n2+2n−4+6n2−14n/3n3−3n2−10n+8.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(23)ABCGGG2=∑e∈EG2nu+nv−2nunv,ABCGGG2=∑e∈EG2n3+n3−2n3n3+∑e∈EG2n3+n6−2n3n6+∑e∈EG2n6+n6−2n6n6.
By using Table9, we have(24)ABCGGG2=2n+6n+4+3n2+4n−63n2+2n−4+6n2−14n3n3−3n2−10n+8,which is required.Table 9
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index3, 3n+42n+6/n+43, 622n−13n2+4n−6/3n2+2n−46, 6n−26n2−14n/3n3−3n2−10n+8Theorem 10.
LetG2 be the chain silicate network of n order; then, its NGG index is 1/n+4+1/6n2+4n−8+2/3n3−3n2−10n+8.Proof.
LetG2≅CSn, where n≥2; also, n is an integer:(25)NGGG2=∑e∈EG21nunv,NGGG2=∑e∈EG21n3n3+∑e∈EG21n3n6+∑e∈EG21n6n6.
By using Table10, we have(26)NGGG2=1n+4+16n2+4n−8+23n3−3n2−10n+8,which is required.Table 10
Edge partition of chain silicate network ofn order.
Edge partitiondu,dvNumber of edgesABCGG index3, 3n+41/n+43, 622n−11/6n2+4n−86, 6n−22/3n3−3n2−10n+8Theorem 11.
LetG2 be the chain silicate network of even order; then, its Mostar index is 15n2−12n/2.Proof.
LetG2≅CSn, where n≥2; also, n is even.(27)MoG2=∑uv∈EG2nu−nv,MoG2=∑uv∈EG2n3−n3+∑uv∈EG2n3−n6+∑uv∈EG2n6−n6.
By using Table11, we have(28)MoG2=0+6n2−6+3n2−12n+122,MoG2=15n2−12n2,which is required.Table 11
Edge partition of chain silicate network of even order.
Edge partitiondu,dvNumber of edgesMostar index3, 3n+403, 622n−16n2−66, 6n−23n2−12n+12/2Theorem 12.
LetG2 be the chain silicate network of odd order; then, its Mostar index is 15n2−12n−3/2.Proof.
LetG2≅CSn, where n≥1; also, n is odd:(29)MoG2=∑uv∈EG2nu−nv,MoG2=∑uv∈EG2n3−n3+∑uv∈EG2n3−n6+∑uv∈EG2n6−n6.
By using Table12, we have(30)MoG2=0+6n2−6+3n2−12n+92,MoG2=15n2−12n−32,which is required.Table 12
Edge partition of chain silicate network of odd order.
Edge partitiondu,dvNumber of edgesMostar index3, 3n+403, 622n−16n2−66, 6n−23n2−12n+9/2


## 2.3. Results for the Ortho ChainSn

In this section, we discussSn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index. The single molecule of para and ortho chain has the same structure. Basically, it is a cycle graph having 4 sides denoted as C4 and represented as a four-sided regular polygon. The ortho chain has a zig-zag structure where each corner of C4 is attached linearly, as shown in Figure 3. The para chain has a structure in which each C4 is attached at corner to corner with other C4 but not linearly, as shown in Figure 4 [28].Figure 3
Ortho chain ofn vertices.Figure 4
Para chain ofn order.Theorem 13.
LetG3 be the ortho chain of n order; then, its Szeged index is 3n3+15n2−2n.Proof.
LetG3≅Sn, where n≥2; also, n is an integer.(31)SzG3=∑e∈EG3nunv,SzG3=∑e∈EG3n2n2+∑e∈EG3n2n4+∑e∈EG3n4n4.
By using Table13, we have(32)SzG3=3n3+3n2+26n−82+12n2−4n+3n3+3n2−22n+82,SzG3=3n3+15n2−2n,which is required.Table 13
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2n+23n3+3n2+26n−8/22, 42n12n2−4n4, 4n−23n3+3n2−22n+8/2Theorem 14.
LetG3 be the ortho chain of n order; then, its PI index is 12n2+4n.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(33)PIvG3=∑e∈EG3nu+nv,PIvG3=∑e∈EG3n2+n2+∑e∈EG3n2+n4+∑e∈EG3n4+n4.
By using Table14, we have(34)PIvG3=3n2+7n+2+6n2+2n+3n2−5n−2,PIvG3=12n2+4n,which is required.Table 14
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2n+23n2+7n+22, 42n6n2+2n4, 4n−23n2−5n−2Theorem 15.
LetG3 be the ortho chain of n order; then, its ABCGG index is 6n2+14n/3n3+3n2+26n−8+3n2+n−1/6n2−2n+6n2−10n−8/3n3+3n2−22n+8.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(35)ABCGGG3=∑e∈EG3nu+nv−2nunv,ABCGGG3=∑e∈EG3n2+n2−2n2n2+∑e∈EG3n2+n4−2n2n4+∑e∈EG3n4+n4−2n4n4.
By using Table15, we have(36)ABCGGG3=6n2+14n3n3+3n2+26n−8+3n2+n−16n2−2n+6n2−10n−83n3+3n2−22n+8,which is required.Table 15
Edge partition of ortho chain ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2, 2n+26n2+14n/3n3+3n2+26n−82, 42n3n2+n−1/6n2−2n4, 4n−26n2−10n−8/3n3+3n2−22n+8Theorem 16.
LetG3 be the ortho chain of n order; then, its NGGG3 index is 2/3n3+3n2+26n−8+1/23n2−n+2/3n3+3n2−22n+8.Proof.
LetG3≅Sn, where n≥2; also, n is an integer:(37)NGGG3=∑e∈EG31nunv,NGGG3=∑e∈EG31n2n2+∑e∈EG31n2n4+∑e∈EG31n4n4.
By using Table16, we have(38)NGGG3=23n3+3n2+26n−8+123n2−n+23n3+3n2−22n+8,which is required.Table 16
Edge partition of oxide network ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 2n+22/3n3+3n2+26n−82, 42n1/23n2−n4, 4n−22/3n3+3n2−22n+8Theorem 17.
LetG3 be the ortho chain of even order; then, its Mostar index is 9n2−6n.Proof.
LetG3≅Sn, where n≥2; also, n is even:(39)MoG3=∑uv∈EG3nu−nv,MoG3=∑uv∈EG3n2−n2+∑uv∈EG3n2−n4+∑uv∈EG3n4−n4.
By using Table17, we have(40)MoG3=3n2+12n−122+6n2−6n+3n2−12n+122,MoG3=9n2−6n,which is required.Table 17
Edge partition of ortho chain of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2n+23n2+12n−12/22, 42n6n2−6n4, 4n−23n2−12n+12/2Theorem 18.
LetG3 be the ortho chain of odd order; then, its Mostar index is 9n2−6n−3.Proof.
LetG3≅Sn, where n≥1; also, n is odd:(41)MoG3=∑uv∈EG3nu−nv,MoG3=∑uv∈EG3n2−n2+∑uv∈EG3n2−n4+∑uv∈EG3n4−n4.
By using Table18, we have(42)MoG3=3n2+12n−152+6n2−6n+3n2−12n+92,MoG3=9n2−6n−3,which is required.Table 18
Edge partition of ortho chain of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2n+23n2+12n−15/22, 42n6n2−6n4, 4n−23n2−12n+9/2


## 2.4. Results for the Para ChainQn

In this section, we discussQn and compute the exact results for Szeged, PI, ABCGG, NGG, and Mostar index.Theorem 19.
LetG4 be the para chain of n order; then, its Szeged index is 6n3+6n2+4n.Proof.
LetG4≅Qn, where n≥2; also, n is an integer.(43)SzG4=∑e∈EG4nunv,SzG4=∑e∈EG4n2n2+∑e∈EG4n2n4.
By using Table19, we have(44)SzG4=24n−8+6n3+6n2−20n+8,SzG4=6n3+6n2+4n,which is required.Table 19
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesSzeged index2, 2424n−82, 44n−46n3+6n2−20n+8Theorem 20.
LetG4 be the para chain of n order; then, its PI index is 12n2+4n.Proof.
LetG4≅Qn, where n≥2; also, n is an integer:(45)PIvG4=∑e∈EG4nu+nv,PIvG4=∑e∈EG4n2+n2+∑e∈EG4n2+n4.
By using Table20, we have(46)PIvG4=12n+4+12n2−8n−4,PIvG4=12n2+4n,which is required.Table 20
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesPI index2, 2412n+42, 44n−412n2−8n−4Theorem 21.
LetG4 be the para chain of n order; then, its ABCGG index is 6n+1/12n−4+6n2−4n−3/3n3+3n2−10n+4.Proof.
LetG4≅OXn, where n≥2; also, n is an integer:(47)ABCGGG4=∑e∈EG4nu+nv−2nunv,ABCGGG4=∑e∈EG4n2+n2−2n2n2+∑e∈EG4n2+n4−2n2n4.
By using Table21, we have(48)ABCGGG4=6n+112n−4+6n2−4n−33n3+3n2−10n+4,which is required.Table 21
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesABCGG index2, 246n+1/12n−42, 44n−46n2−4n−3/3n3+3n2−10n+4Theorem 22.
LetG4 be the para chain of n order; then, its NGGG4 index is 1/26n−2+1/6n3+6n2−20n+8.Proof.
LetG4≅Qn, where n≥2 also n is an integer:(49)NGGG4=∑e∈EG41nunv,NGGG4=∑e∈EG41n2n2+∑e∈EG41n2n4.
By using Table22, we have(50)NGGG4=126n−2+16n3+6n2−20n+8,which is required.Table 22
Edge partition of para chain ofn order.
Edge partitiondu,dvNumber of edgesNGG index2, 241/26n−22, 44n−41/6n3+6n2−20n+8Theorem 23.
LetG4 be the para chain of even order; then, its Mostar index is 6n2.Proof.
LetG4≅Qn, where n≥2; also, n is even:(51)MoG4=∑uv∈EG4nu−nv,MoG4=∑uv∈EG4n2−n2+∑uv∈EG4n2−n4.
By using Table23, we have(52)MoG4=12n−12+6n2−12n+12,MoG4=6n2,which is required.Table 23
Edge partition of para chain of even order.
Edge partitiondu,dvNumber of edgesMostar index2, 2412n−122, 44n−46n2−12n+12Theorem 24.
LetG4 be the para chain of odd order; then, its Mostar index is 6n2−6.Proof.
LetG4≅Qn, where n≥1; also, n is odd:(53)MoG4∑uv∈EG4nu−nv,MoG4=∑uv∈EG4n2−n2+∑uv∈EG4n2−n4.
By using Table24, we have(54)MoG4=12n−12+6n2+12n+6,MoG4=6n2−6,which is required.For the comparison of Szeged, PI,ABCGG, and NGG index of COXn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 25 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of CSn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 26 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of Sn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 27 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.For the comparison of Szeged, PI,ABCGG, and NGG index of Qn, we computed the indices for different values of n. By increasing the values of n, we can clearly check from Table 28 that the order of Szeged and PI index is increasing while that of ABCGG and NGG is decreasing.Table 24
Edge partition of para chain of odd order.
Edge partitiondu,dvNumber of edgesMostar index2, 2412n−122, 44n−46n2+12n+6Table 25
Comparison table for chain oxide network.
OXnSzGPIABCNGG136——21420——33742——476722.751.0451351102.710.9562181562.660.9073292102.620.8684722722.590.8496513422.560.83108704202.530.81Table 26
Comparison table for chain silicate.
CSnSzGPIABCNGG1612——23042——38190——41681562.910.5853002402.890.4964863422.880.4577354622.860.41810566002.830.38914587562.810.351019509302.800.33Table 27
Comparison table for ortho chain.
SnSzGPIABCNGG11616——28056——3210120——44242081.980.2657403201.880.19611764561.800.15717506161.740.12824808001.680.109338410061.630.0910448012401.580.08Table 28
Comparison table for para chain.
QnSzGPIABCNGG11616——280561.610.3232281201.460.2044962081.370.1659203201.300.13615364561.260.11723806161.220.10834888001.190.099489610081.160.0810664012401.140.07


## 3. Conclusion

In this article, we have figured out several bond-additive TIs such as Szeged, PI,ABC, NGG, and Mostar index. We calculated the closed formulae for abovementioned TIs of chain silicate, oxide network, para, and ortho chain. The above outcomes contribute in the field of natural sciences and pharmaceutical science. Our exploration kept on determining new consequences of these graphs.



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