# Hosoya and Harary Polynomials ofTOX(n),RTOX(n),TSL(n) and RTSL(n)

Authors: Lian Chen; Abid Mehboob; Haseeb Ahmad; Waqas Nazeer; Muhammad Hussain; M. Reza Farahani
Journal: Discrete Dynamics in Nature and Society
 (2019)

---

## Abstract

In the fields of chemical graph theory, topological index is a type of a molecular descriptor that is calculated based on the graph of a chemical compound. In 1947, Wiener introduced “path number” which is now known as Wiener index and is the oldest topological index related to molecular branching. Hosoya polynomial plays a vital role in determining Wiener index. In this report, we computed the Hosoya and the Harary polynomials forTOX(n),RTOX(n),TSL(n), and RTSL(n) networks. Moreover, we computed serval distance based topological indices, for example, Wiener index, Harary index, and multiplicative version of wiener index.


---

## Body

## 1. Introduction

In network theory, a network can be taken as graph in which nodes are taken as vertices and links between nodes are taken as edges. Network theory has applications in numerous fields including computer sciences, biology, particle physics, electronics, statistical physics, operational research, finance, sociology, and/or ecology. Uses of network theory incorporate the World Wide Web, strategic systems, internet, quality administrative systems, metabolic systems, epistemological networks, social networks, and so forth.In the fields of chemical graph theory, chemistry, and molecular topology, a topological index otherwise called a connectivity index is a kind of a molecular descriptor that is computed dependent on the molecular graph of a chemical compound [1, 2]. Topological indices are numerical parameters of a graph which describe its topology and are typically graph invariant. Topological indices are utilized for instance in the improvement of quantitative structure-activity relationships (QSARs) in which the organic action or different properties of compounds are related to their molecular graph [3, 4].Topological descriptors are obtained from hydrogen-stifled molecular graph, in which the atoms are spoken to by vertices and the bonds by edges. The associations between the atoms can be portrayed by different sorts of topological matrices (e.g., adjacency or distance matric), which can be scientifically controlled in order to infer a solitary number, normally known as graph invariant or topological index [5, 6]. Subsequently, the topological index can be characterized as two-dimensional descriptors that can be effortlessly figured from the molecular graph and do not rely upon the manner in which the graph is delineated or labeled without the energy minimization of the compound structure.The most simple topological indices do not perceive double bonds and atom types (O, C, Ca, and so forth) and disregard hydrogen atoms and characterized for associated undirected molecular graph only. Polynomials have additionally valuable applications in science; for example, the Hosoya polynomial of a graph is a generating function about distance distributing, introduced byHaruo Hosoya in 1988 [7] and for a connected graph G is defined as(1)HG,x=12∑v∈VG∑u∈VGxdv,u,where d(u,v) denotes the distance between vertices u and v. The Hosoya polynomial has many chemical applications [8], especially, almost all distance based topological indices can be computed from this polynomial [9–11].The Wiener index of a connected graphG is denoted by W(G) and defined as the sum of distances between all pairs of vertices in G; i.e., it can be formulated as (2)WG=12∑v∈VG∑u∈VGdu,v. The Wiener index was firstly introduced by Harold Wiener in 1947 to study the boiling points of paraffin [12]. It plays an important role in the so-called inverse structure-property relationship problems [13]. For more details about this topological polynomial and index, please see the paper series and the references therein [14–22]. Note that the first derivative of the Hosoya polynomial at x=1 is equal to the Wiener index:(3)WG=∂HG∂xx=1The modified Wiener index of a connected graph G is denoted by Wλ(G) and defined as the sum of λ power distances between all pairs of vertices in G, where λ=1,2,3,4,…; i.e., it can be formulated as (4)WG=12∑v∈VG∑u∈VGdu,vλ. Hyper-Wiener index is another distance-based graph invariants used for predicting physicochemical properties of organic compounds [23]. The hyper Wiener index was introduced by Randić [24] as(5)WWG=12∑v∈VG∑u∈VGdu,v+du,v2.Modified hyper Wiener index of a connected graph G is denoted by WWλ(G) and is defined as (6)WWλG=12∑v∈VG∑u∈VGdu,vλ+du,v2λ.The Harary polynomial of a connected graph G is denoted by h(G) and is defined as (7)hG=∑v∈VG∑u∈VG1du,vxdu,v.The generalized Harary index of a connected graph G is denoted by ht(G) and is defined as (8)htG=∑v∈VG∑u∈VG1du,v+twhere t=1,2,3,4,…. The multiplicative Wiener index of a connected graph G is denoted by π(G) and is defined as (9)πG=∏u,v⊆VGdu,v.In this study, we compute some topological indices of TOX(n), RTOX(n), TSL(n), and RTSL(n) networks. For details about topological indices and applications of topological indices we refer [25–32].


## 2. Methodology

A simple graphG=(V,E) is a finite nonempty set V(G) of objects called vertices together with a (possibly empty) set E(G) of unordered pairs of distinct vertices of G called edges. To compute Hosoya polynomial of a graph G, we need to compute the number of pairs of vertices at distance 1,2,3,…,d(G), where d(G)=max{d(u,v);u,v∈V(G)}. For this purpose, we use mathematical induction. The general view of Hosoya polynomial is as below:(10)HG;x=a0nx0+a1nx1+a2nx2+⋯+adnxd,where ain= number of pairs of vertices at distance i. By applying fundamental calculus, one can compute Wiener index and hyper Wiener index. Moreover, we use MATLAB for mathematical calculations and verifications (see https://en.wikipedia.org/wiki/MATLAB).


## 3. Computational Results

In this section, we present our computational results.


### 3.1. Triangular Oxide NetworkTOX(n)

A triangulated irregular network (TIN) is a portrayal of a consistent surface comprising completely of triangular aspects, utilized primarily as Discrete Global Grid in primary elevation modeling [33, 34]. Oxide networks play a vital role in the study of silicate networks [35, 36]. If we delete silicon vertices from a silicate network, we get an oxide network OX(n). A triangular oxide network is denoted as TOX(n). An oxide network TOX(n) is depicted in Figure 1, unit cell, Figure 2, for n=2, and Figure 3, for n=5.Figure 1
Triangular oxide network, unit cell.Figure 2
Triangular oxide network TOX(2).Figure 3
Triangular oxide network TOX(5).Lemma 1.
The triangular oxide network has3(n2+n)/2 edges and n2+3n+2/2 vertices.Theorem 2.
For the triangular oxide networkTOX(n) for n≥1, where n is length of triangular oxide network, we have (11)HTOXn;x=n2+3n+22+∑m=1n3m2n-m+1n-m+2xmProof.
To prove this theorem, we need to computeamn, where m=0,1,2,3,…,n. Obviously, from Lemma 1, we have (12)a0n=V=n2+3n+22 and (13)a1n=E=3n2+n2.The remaining proof is as follows.
It can be observed from Figures1–3 that there are n2+n/2 number of triangles, soa22=61a23=63a24=660a25=610a26=615a27=621
Now, one can conclude that(14)a2n=6n-12+n-12. Now, for a3n,a33=91a34=93a35=960a36=910a37=915a38=921
Now, one can conclude that(15)a3n=9n-22+n-22.Now, for a4n,a44=121a45=123a46=1260a47=1210a48=1215a49=1221
Now, one can conclude that(16)a4n=12n-32+n-32.From the above-mentioned method we can generalizea2n=6n-12+n-1/2a3n=9n-22+n-2/2a4n=12n-32+n-3/2a5n=15n-42+n-4/2a6n=18n-52+n-5/2
Now, one can conclude that(17)amn=3m2n-m+1n-m+2xm.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network TOXn. (18)HTOXn;x=n2+3n+22+∑m=1n3m2n-m+1n-m+2xm.Theorem 3.
For the triangular oxide networkTOXn, the Harary polynomial is(19)hTOXn;x=∑m=1n32n-m+1n-m+2xm.Proof.
From the calculation given in Theorem2, we have (20)hG,x=∑v∈VTOXn∑u∈VTOXn1dv,uxdv,u,=∑v∈VTOXn∑u∈VTOXn1mxm=∑m=1n32n-m+1n-m+2xmTheorem 4.
For the triangular oxide networkTOXn, we have (1)
WλTOXn=∑m=1n3/2n-m+1n-m+2mλ+1(2)
WWλTOXn=∑m=1n3/2n-m+1n-m+2mλ+1+m2λ+1(3)
HtTOXn=∑m=1n3m/2n-m+1n-m+2/m+t(4)
πTOXn=∏m=1nm3m/2n-m+1n-m+2Proof.
From the calculation given in Theorem2, we have the following.
(1) The modified Wiener index is(21)WλTOXn=12∑v∈VTOXn∑u∈VTOXndu,vλ=12∑v∈VTOXn∑u∈VTOXnmλ=∑m=1n3m2n-m+1n-m+2mλ=∑m=1n32n-m+1n-m+2mλ+1Table 1 contains values of modified Wiener index of TOX for different values of n and λ.
(2) The modified hyper Wiener index is(22)WWλTOXn=12∑v∈VTOXn∑u∈VTOXndu,vλ+∑v∈VTOXn∑u∈VTOXndu,v2λ=WλTOXn+12∑v∈VTOXn∑u∈VTOXndu,v2λ=WλTOXn+12∑v∈VTOXn∑u∈VTOXnm2λ=12∑v∈VTOXn∑u∈VTOXnmλ+m2λ=∑m=1n3m2n-m+1n-m+2mλ+m2λ=∑m=1n32n-m+1n-m+2mλ+1+m2λ+1.Table 2 contains the values of modified hyper Wiener index of TOX for different values of n and λ.
(3) The generalized Harary index is(23)HtTOXn=12∑v∈VTOXn∑u∈VTOXn1du,v+t=12∑v∈VG∑u∈VG1m+t=∑m=1n3m/2n-m+1n-m+2m+t.In Table 3, we give values of generalized Harary index of TOX for different values of n and λ.
(4) The multiplicative Wiener index is(24)πG=∏u,v⊆VTOXndu,v=∏u,v⊆VTOXnm=∏m=1nm3m/2n-m+1n-m+2.Table 1
Modified Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 3 21 81 231 546 1134  2 3 33 171 609 1722 4158  3 3 57 405 1815 6162 17334  4 3 105 1035 5865 23970 78678  5 3 201 2781 20031 98826 379134  6 3 393 7731 71169 425082 1909278Table 2
Modified hyper Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 6 54 252 840 2268 5292  2 6 138 1206 6474 25692 82836  3 6 450 8136 72984 431244 1926612  4 6 1650 64710 978690 8612340 53222076  5 6 6354 552672 14234160 187582068 1609127532  6 6 24978 4864446 215894154 4295457852 51426477396Table 3
Generalized Harary index ofTOX.
t ∖ n 1 2 3 4 5 6 0 3 12 30 60 105 168  1 3/2 13/2 69/4 723/20 657/10 7593/70  2 1 9/2 123/10 132/5 1713/35 2301/28  3 3/4 69/20 48/5 732/35 2199/56 3733/56  4 3/5 14/5 276/35 243/14 1381/42 1969/35  5 1/2 33/14 375/56 2495/168 793/28 15013/308  6 3/7 57/28 163/28 909/70 4157/167 9469/220Corollary 5.
The Wiener index of triangular oxide networkTOXn is(25)WTOXn;x=n520+3n48+n3+9n38+9n20.Proof.
This result can be followed immediately from Theorem4 by putting λ=1 in modified Wiener index.Corollary 6.
The hyper Wiener index of triangular oxide networkTOXn is(26)WTOXn;x=n640+11n540+9n48+17n38+37n220+3n5.Proof.
This result can be followed immediately from Theorem4 by putting λ=1 in modified hyper wiener index.Corollary 7.
The Harary index of triangular oxide networkTOXn is(27)HTOXn=12n3+3n2+2n.Proof.
This result can be followed immediately from Theorem4 by putting t=0 in generalized Harary index.


### 3.2. Regular Triangular Oxide NetworkRTOXn

The regular triangular oxide network is shown in Figure4, unit cell, Figure 5, for n=2 and Figure 6, for n=5.Figure 4
Regular triangular oxide network (unit cell).Figure 5
Regular triangular oxide network RTOX(2).Figure 6
Regular triangular oxide network RTOX(5).Lemma 8.
The regular triangular oxide networkRTOXn has 3n2+6n edges and 3/2n2+9/2n+1 vertices.Theorem 9.
IfRTOXn is the regular triangular oxide network n≥1, where n is length of triangular oxide network, then (28)HRTOXn;x=32n2+92n+1+3n2+6nx+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5xm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,2n+1. Obviously from Lemma 8(29)a0n=V=32n2+92n+1and (30)a1n=E=3n2+6nThe remaining proof is as follows.
It can be observed from Figures4–6 that there are n2+2n number of triangles:a21=23+2a22=28+16a23=215+38a24=224+68a25=235+106
Now, one can conclude that(31)a2n=6n2+6n-4.Now for a4na42=63+1a43=68+25a44=615+64a45=624+118a46=635+187
Now, one can conclude that(32)a4n=272n-12+272n-1-8.Now for a6na63=83+6a64=88+50a65=815+120a66=824+216a67=835+338
Now, one can conclude that(33)a6n=21n-22+21n-2-12.Now for a8na84=103+11a85=108+75a86=1015+176a87=1024+314a88=1035+489
Now, one can conclude that(34)a8n=572n-32+572n-3-16.From the above-mentioned method we can generalizea2n=6n2+6n-4a4n=27/2n-12+27/2n-1-8a6n=21n-22+21n-2-12a8n=57/2n-32+57/2n-3-16a10n=36n-42+36n-4-20
Now, one can conclude that(35)amn=15m-64n-m2+12+n-m2+1-2mxm.Now we can calculate for odd distance.
Now fora3na31=13+1a32=28+21a33=315+46a34=424+70a35=535+87
Now, one can conclude that(36)a3n=10.5n2+1.5n-8.Now for a5na52=13+5a53=28+43a54=315+98a55=424+164a56=535+235
Now, one can conclude that(37)a5n=16.5n-12+1.5n-1-10.Now for a7na73=13+9a74=28+65a75=315+150a76=424+258a77=535+383
Now, one can conclude that(38)a7n=22.5n-22+1.5n-2-12.Now for a9na94=13+13a95=28+87a96=315+202a97=424+356a98=535+531
Now, one can conclude that(39)a9n=28.5n-32+1.5n-3-14.From the above-mentioned method we can generalizea3n=10.5n2+1.5n-8a5n=16.5n-12+1.5n-1-10a7n=22.5n-22+1.5n-2-12a9n=28.5n-32+1.5n-3-14a11n=34.5n-42+1.5n-4-16
Now, one can conclude that(40)amn=6m+32n-m-12+12+32n-m-12+1-m+5xm.Here, by what have been mentioned above, we have following computations for the Hosoya polynomial of regular triangular oxide network RTOXn. (41)HRTOXn;x=32n2+92n+1+3n2+6nx+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5xm.Theorem 10.
For regular triangular oxide network systemRTOXn, the Harary polynomial is (42)hRTOXn;x=3n2+6nx+∑m≡0mod⁡21m15m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5xm.Proof.
From the calculation given in Theorem9, we have (43)hRTOXn,x=12∑v∈VRTOXn∑u∈VRTOXn1dv,uxdv,u=∑v∈VRTOXn∑u∈VRTOXn11x1+∑v∈VRTOXn∑u∈VRTOXn1mxm=3n2+6nx+∑m≡0mod⁡2,m≥21m15m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5xm.Theorem 11.
For the regular triangular oxide network systemRTOXn, we have WλRTOXn=3n2+6n + ∑m≡0mod⁡2(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)mλ + ∑m≡1mod⁡2,m>16m+3/22n-m+3/22+3/22n-m+3/2-m-5mλWWλRTOXn=3n2+6n1λ+12λ + ∑m≡0mod⁡2(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)mλ+m2λ + ∑m≡1mod⁡2,m>1(6m+3/22n-m+3/22 + 3/22n-m+3/2 − m-5)mλ+m2λHtRTOXn=3n2+6n/1+t + ∑m≡0mod⁡21/m+t(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)15m-6/4(2n-m+2/22 + 2n-m+2/2)  − 2m + ∑m≡1mod⁡2,m>11/m+t(6m+3/22n-m+3/22 + 3/22n-m+3/2-m-5)πRTOXn=13n2+6n × ∏m≡0mod⁡2m15m-6/42n-m+2/22+2n-m+2/2-2m × ∏m≡1mod⁡2,m>1m6m+3/22n-m+3/22+3/22n-m+3/2-m-5mλ.Proof.
From the calculation given in Theorem9, we have the following.
(1) The modified Wiener index of regular triangular oxide network is(44)WλRTOXn=12∑v∈VRTOXn∑u∈VRTOXndu,vλ=∑v∈VRTOXn∑u∈VRTOXn1λ+∑v∈VRTOXn∑u∈VRTOXnmλ=3n2+6n+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mmλ+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5mλ.Table 4 contains different values of modified Wiener index of RTOX for different values of n and λ.
(2) The modified hyper Wiener index of regular triangular oxide network is(45)WWλRTOXn=12∑v∈VRTOXn∑u∈VRTOXndu,vλ+∑v∈VRTOXn∑u∈VRTOXndu,v2λ=WλRTOXn+12∑v∈VRTOXn∑u∈VRTOXndu,v2λ=WλRTOXn+12∑v∈VRTOXn∑u∈VRTOXn12λ+12∑v∈VOKn+2∑u∈VRTOXnm2λ=12∑v∈VRTOXn∑u∈VRTOXn1λ+12λ+12∑v∈VRTOXn∑u∈VRTOXnmλ+m2λ=3n2+6n1λ+12λ+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mmλ+m2λ+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5mλ+m2λ.Table 5 contains values of modified hyper Wiener index of RTOX for different values of n and λ.
(3) The generalized Harary index is(46)htRTOXn=12∑v∈VRTOXn∑u∈VRTOXn1du,v+t=12∑v∈VRTOXn∑u∈VRTOXn11+t+∑v∈VRTOXn∑u∈VRTOXn1m+t=3n2+6n1+t+∑m≡0mod⁡21t+m15m-642n-m+222+2n-m+22-2m+∑m≡1mod⁡2,m>11t+m6m+322n-m+322+322n-m+32-m-5.Table 6 is about the values of generalized Harary index of RTOX for different values of n and λ.
(4) The multiplicative Wiener index is(47)πRTOXn=∏u,v⊆VRTOXndu,v=∏u,v⊆VRTOXn1×∏u,v⊆VRTOXnm=13n2+6n×∏m≡0mod⁡2m15m-6/42n-m+2/22+2n-m+2/2-2m×∏m≡1mod⁡2,m>1m6m+3/22n-m+3/22+3/22n-m+3/2-m-5mλ.Table 4
Modified Wiener index ofRTOX.
λ ∖ n 1 2 3 4 5 6 1 37 315 1305 3856 9303 19575  2 77 989 5447 20062 57937 141907  3 181 3495 25689 118276 409647 1169511  4 461 13397 131759 759310 3156433 10509235  5 1237 54495 718425 5184796 25875903 100496535  6 3437 231869 4103087 37092982 222278977 1007138467Table 5
Modified hyper Wiener index ofRTOX.
λ ∖ n 1 2 3 4 5 6 1 114 1304 6752 23918 67240 161482  2 538 14386 137206 779372 3214370 10651142  3 3618 235364 4128776 37211258 222688624 1008307978  4 28762 4634554 148143358 2102868620 18172355906 112088413190  5 245634 100320044 5862567752 131158675298 1636767140320 13752061166362  6 2161978 2291918386 247080692926 8743290167492 157800041881490 1807230473840342Table 6
Generalized Harary index ofRTOX.
t ∖ n 1 2 3 4 5 6 0 43/3 3521/60 5990/41 15655/54 26687/53 17609/22  1 49/6 741/20 7949/81 14439/71 15694/43 36915/62  2 29/5 5819/210 10623/140 74267/461 30345/103 19519/40  3 271/60 4679/210 6173/99 7805/58 13463/54 39652/95  4 389/105 766/41 5576/105 47907/413 14106/65 24175/66  5 22/7 1498/93 10887/235 715/7 11169/58 52343/160  6 172/63 8146/575 3578/87 6669/73 24437/141 65417/221Corollary 12.
The Wiener index of regular triangular oxide networkRTOXn is (48)WRTOXn;x=9n510+27n44+31n32+53n24+3n5.Proof.
This result immediately follows from Theorem11 by taking λ=1 in modified Wiener index.Corollary 13.
The hyper Wiener index of regular triangular oxide networkRTOXn is (49)WRTOXn;x=910n6+9n5+1194n4+2756n3+58720n2-56n.Proof.
This result immediately follows from Theorem11 by taking λ=1 in generalized hyper Wiener index.Corollary 14.
The Harary index of regular triangular oxide networkRTOXn by putting t=o is (50)h0RTOXn=3n2+6n+∑m≡0mod⁡21m15m-642n-m+222+2n-m+22-2m+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5.Proof.
This result immediately follows from Theorem11 by taking t=1 in generalized harary index.


### 3.3. Triangular Silicate NetworkTSLn

Silicates are the largest, the most complicated, and the most interesting class of minerals so far.SiO4 tetrahedron is the basic chemical unit of silicates. The silicates sheets are rings of tetrahedrons linked by shared oxygen nodes to other rings in two-dimensional planes producing a sheet like structures. A silicate can be obtained by fusing a metal oxide or a metal carbonate with sand. Essentially every silicate contains SiO4 tetrahedron. The corner and the center vertices represent oxygen and silicon ions, respectively. These vertices are called oxygen nodes and silicon node, respectively. In SiO4 tetrahedra, the corner and the center vertices represent oxygen and silicon ions, respectively (Figures 7–9). Unit cell is shown in Figure 7, for n=2 in Figure 8 and for n=5 in Figure 9.Figure 7
S i O 4.Figure 8
Triangular silicate network TsL(2).Figure 9
Triangular silicate network TsL(5).Lemma 15.
The triangular silicate network has3n2+n edges and n2+2n+1 vertices.Theorem 16.
IfTSLn is the triangular silicate network n≥1, where n is length of triangular silicate network then(51)HTSLn;x=n2+2n+1+∑m=1n6m-3n-m+12+n-m+1xm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,n. Obviously, from Lemma 15, we have (52)a0n=V=n2+2n+1, and (53)a1n=E=3n2+3n.The remaining proof is as follows.
It can be observed from Figures7–9 that there are n2+n/2 number of triangles:a22=181a23=183a24=1860a25=1810a26=1815a27=1821
Now, one can conclude that(54)a2n=18n-12+n-12. Now, for a3n,a33=301a34=303a35=3060a36=3010a37=3015a38=3021
Now, one can conclude that(55)a3n=30n-22+n-22. Now, for a4n,a44=421a45=423a46=4260a47=4210a48=4215a49=4221
Now, one can conclude that(56)a4n=42n-32+n-32.From the above-mentioned method we can generalizea2n=18n-12+n-1/2a3n=30n-22+n-2/2a4n=42n-32+n-3/2a5n=54n-42+n-4/2a6n=66n-52+n-5/2
Now, one can conclude that(57)amn=6m-3n-m+12+n-m+1.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network TOXn.(58)HTSLn;x=6m-3n-m+12+n-m+1xm.Theorem 17.
For the triangular silicate networkTSLn, the Harary polynomial is(59)hTSLn;x=∑m=1n1m6m-3n-m+12+n-m+1xm.Proof.
From the calculation given in Theorem16, we have (60)hG,x=∑v∈VTSLn∑u∈VTSLn1dv,uxdv,u=∑v∈VTSLn∑u∈VTSLn1mxm=∑m=1n1m6m-3n-m+12+n-m+1xm.Theorem 18.
For the triangular silicate networkTSLn, we have (1)
WλTSLn=∑m=1n6m-3n-m+12+n-m+1mλ.(2)
WWλTSLn=∑m=1n6m-3n-m+12+n-m+1mλ+m2λ.(3)
HtTSLn=∑m=1n6m-3n-m+12+n-m+1/m+t.(4)
πTSLn=∏m=1nm6m-3n-m+12+n-m+1.Proof.
From the calculation given in Theorem16, we have the following.
(1) The modified Wiener index of the triangular silicate networkTSLn: (61)WλTSLn=12∑v∈VTSLn∑u∈VTSLndu,vλ=12∑v∈VTSLn∑u∈VTSLnmλ=∑m=1n6m-3n-m+12+n-m+1mλ.Table 7 is about different values of modified Wiener index of the triangular silicate network TSLn for different values of n and λ.
(2) The modified hyper Wiener index of the triangular silicate networkTSLn is (62)WWλTSLn=12∑v∈VTSLn∑u∈VTSLndu,vλ+∑v∈VTSLn∑u∈VTSLndu,v2λ=WλTSLn+12∑v∈VTSLn∑u∈VTSLndu,v2λ=WλTSLn+12∑v∈VTSLn∑u∈VTSLnm2λ=12∑v∈VTSLn∑u∈VTSLnmλ+m2λ=∑m=1n6m-3n-m+12+n-m+1mλ+m2λ.Table 8 is about different values of modified hyper Wiener index of the triangular silicate network TSLn for different values of n and λ.
(3) The generalized Harary index the triangular silicate networkTSLn is (63)HtTSLn=12∑v∈VTSLn∑u∈VTSLn1du,v+t=12∑v∈VG∑u∈VG1m+t=∑m=1n6m-3n-m+12+n-m+1m+t.Table 9 is about different values of generalized Harary index of the triangular silicate network TSLn for different values of n and t.
(4) The multiplicative Wiener index the triangular silicate networkTSLn is (64)πG=∏u,v⊆VTSLndu,v=∏u,v⊆VTSLnm=∏m=1nm6m-3n-m+12+n-m+1.Table 7
Modified Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 6 54 234 714 1764 3780  2 6 90 522 1974 5796 14364  3 6 162 1278 6042 21204 61020  4 6 306 3330 19830 83556 280044  5 6 594 9054 68394 347364 1359180  6 6 1170 25362 244614 1502676 6878844Table 8
Modified hyper Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 12 144 756 2688 7560 18144  2 12 396 3852 21804 89352 294408  3 12 1332 26640 250656 1523880 6939864  4 12 4932 214020 3390540 30661992 192968568  5 12 19044 1835856 49533648 670624920 5856390504  6 12 74916 16189812 753159804 15395422392 187613691048Table 9
Generalized Harary index ofTOX.
t ∖ n 1 2 3 4 5 6 0 6 27 73 309/2 2823/10 2337/5  1 3 15 87/2 969/10 921/5 11019/35  2 2 21/2 63/2 72 978/7 6801/28  2 3/2 81/10 124/5 288/5 453/4 2389/12  4 6/5 33/5 717/35 1347/28 2673/28 11841/70  5 1 39/7 489/28 3473/84 5783/70 10309/70  6 6/7 135/28 1279/84 2539/70 11440/157 6134/47Corollary 19.
The Wiener index of triangular silicate networkTSLn is(65)WTSLn;x=n55+5n44+5n32+7n24+3n10.Proof.
This result follows immediately Theorem18 by putting λ=1 in modified Wiener index.Corollary 20.
The hyper Wiener index of triangular silicate networkTSLn is(66)WTSLn;x=n610+n5+7n42+5n3+12n25.Proof.
This result follows immediately Theorem18 by putting λ=1 in modified hyper Wiener index.Corollary 21.
The Harary index of triangular Silicate networkTSLn is(67)HtTSLn=12n23+21n+4n2-32+3n+n2Hn.where Hn is harmonic number.Proof.
This result follows immediately Theorem18 by putting t=1 in generalized Harary index.


### 3.4. Regular Triangular Silicate NetworkRTSLn

The regular triangular silicate network is shown in Figures10–12, unit cell in Figure 10, for n=2 in Figure 11 and for n=5 in Figure 12.Figure 10
Regular triangular silicate network (unit cell).Figure 11
Regular triangular silicate network RTsL(2).Figure 12
Regular triangular silicate network RTsL(5).Lemma 22.
The regular triangular silicate network has6n2+12n edges and 5/2n2+13/2n+1 vertices.Theorem 23.
IfRTSLn is the regular triangular silicate network n≥1, where n is length of triangular silicate network then (68)HRTSLn;x=52n2+132n+1+6n2+12nx+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9xm+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mxm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,2n+1. Obviously, from Lemma 22, we have (69)a0n=V=52n2+132n+1and (70)a1n=E=6n2+12n.The remaining proof is as follows.
It can be observed from Figures10–12 thata21=18a22=72a23=153a24=261a25=396.
Now, one can conclude that(71)a2n=272n2+272n-9.
Now fora4na42=48a43=180a44=378a45=642a46=972.
Now, one can conclude that(72)a4n=33n-12+33n-1-18.Now for a6na63=78a64=288a65=603a66=1023a67=1548.
Now, one can conclude that(73)a6n=1052n-22+1052n-2-27.Now for a8na84=108a85=396a86=828a87=1404a88=2124.
Now, one can conclude that(74)a8n=72n-32+72n-3-36.From the above-mentioned method we can generalizea2n=27/2n2+27/2n-9a4n=33n-12+33n-1-18a6n=105/2n-22+105/2n-2-27a8n=72n-32+72n-3-36a10n=183/2n-42+183/2n-4-45.
Now, one can conclude that(75)amn=39m-2442n-m+222+2n-m+22-92m.Now we can calculate for odd distance.
Fora3na31=9a32=87a33=216a34=396a35=627.
Now, one can conclude that(76)a3n=512n2+32n-18.Now for a5na52=21a53=153a54=372a55=678a56=1071.
Now, one can conclude that(77)a5n=872n-12+32n-1-24.Now for a7na73=33a74=219a75=528a76=960a77=1515.
Now, one can conclude that(78)a7n=1232n-22+32n-2-30.Now for a9na94=45a95=285a96=684a97=1292a98=1959.
Now, one can conclude that(79)a9n=1592n-32+32n-3-36.From the above-mentioned method we can generalizea3n=51/2n2+3/2n-18a5n=87/2n-12+3/2n-1-24a7n=123/2n-22+3/2n-2-30a9n=159/2n-32+3/2n-3-36a11n=195/2n-32+3/2n-3-42.
Now, one can conclude that(80)amn=18m-322n-m+322+322n-m+32-3m+9.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of regular triangular silicate network RTSLn.(81)HRTSLn;x=52n2+132n+1+6n2+12nx+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9xm+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mxm.Theorem 24.
For regular triangular silicate network systemRTSLn, the Harary polynomial is (82)hRTSLn;x=6n2+12nx+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92mxm+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9xm.Proof.
From the information given in Theorem23, we have (83)hRTSLn,x=12∑v∈VRTSLn∑u∈VRTSLn1dv,uxdv,u=∑v∈VRTSLn∑u∈VRTSLn11x1+∑v∈VRTSLn∑u∈VRTSLn1mxm=6n2+12nx+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92mxm+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9xm.Theorem 25.
For the regular triangular silicate network systemRTSLn, we haveProof.
From the information given in Theorem23, we have the following.
(1) The modified Wiener index of regular triangular silicate network is(84)WλRTSLn=12∑v∈VRTSLn∑u∈VRTSLndu,vλ=∑v∈VRTSLn∑u∈VRTSLn1λ+∑v∈VRTSLn∑u∈VRTSLnmλ=6n2+12n+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mmλ+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9mλ.Table 10 contains the values of modified Wiener index of regular triangular silicate network for different values of n and λ.
(2) The modified hyper Wiener index of regular triangular silicate network is(85)WWλRTSLn=12∑v∈VRTSLn∑u∈VRTSLndu,vλ+∑v∈VRTSLn∑u∈VRTSLndu,v2λ=WλRTSLn+12∑v∈VRTSLn∑u∈VRTSLndu,v2λ=WλRTSLn+12∑v∈VRTSLn∑u∈VRTSLn12λ+12∑v∈VOKn+2∑u∈VRTSLnm2λ=12∑v∈VRTSLn∑u∈VRTSLn1λ+12λ+12∑v∈VRTSLn∑u∈VRTSLnmλ+m2λ=6n2+12n1λ+12λ+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mmλ+m2λ+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9mλ+m2λ.Table 11 contains the values of modified hyper Wiener index of regular triangular silicate network for different values of n and λ.
(3) The generalized Harary index of regular triangular silicate network is(86)htRTSLn=12∑v∈VRTSLn∑u∈VRTSLn1du,v+t=12∑v∈VRTSLn∑u∈VRTSLn11+t+∑v∈VRTSLn∑u∈VRTSLn1m+t=6n2+12n1+t+∑m≡0mod⁡21m+t39m-2442n-m+222+2n-m+22-92m+∑m≡1mod⁡2,m>11m+t18m-322n-m+322+322n-m+32-3m+9.Table 12 contains the values of generalized Harary index of regular triangular silicate network for different values of n and t.
(4) The multiplicative Wiener index of regular triangular silicate network is(87)πRTSLn=∏u,v⊆VRTSLndu,v=∏u,v⊆VRTSLn1×∏u,v⊆VRTSLnm=16n2+12n×∏m≡0mod⁡2m39m-24/42n-m+2/22+2n-m+2/2-9/2m×∏m≡1mod⁡2,m>1m18m-3/22n-m+3/22+3/22n-m+3/2-3m+9mλ.Table 10
Modified Wiener index ofRTSL.
λ ∖ n 1 2 3 4 5 6 1 81 750 3228 9756 23895 50826  2 171 2412 13776 51756 151365 374016  3 405 8670 65958 309042 1081881 3111276  4 1035 33660 342060 2002344 8401089 28144380  5 2781 138270 1881078 13770666 69289425 270544116  6 7731 592812 10816836 99094776 598206945 2723160396Table 11
Modified hyper Wiener index ofRTSL.
λ ∖ n 1 2 3 4 5 6 1 252 3162 17004 61512 175260 424842  2 1206 36072 355836 2054100 8552454 28518396  3 8136 601482 10882794 99403818 599288826 2726271672  4 64710 11971800 394609080 5670015168 49306706898 305266571880  5 552672 260759082 15735721674 356212348962 4470377552850 37677244051392  6 4864446 5979382632 667051149216 23886660553680 433393778907474 4976938113309816Table 12
Generalized Harary index ofRTSL.
t ∖ n 1 2 3 4 5 6 0 30 646/5 23227/70 28947/43 20216/17 47899/25  1 69/4 1657/20 12699/56 16341/34 29821/34 44938/31  2 123/10 312/5 18556/105 31473/82 9278/13 31133/26  3 48/5 6197/123 17071/117 36405/113 55228/91 8233/8  4 276/35 4448/105 4363/35 89996/323 15371/29 26277/29  5 375/56 3696/101 55916/513 8116/33 22631/48 23526/29  6 163/28 2353/73 12407/128 1763/8 22532/53 19856/27Corollary 26.
The Wiener index of regular triangular silicate networkRTSLn is (88)WRTSLn;x=5n52+35n42+36n3+26n2-n.Proof.
This result immediately follows from Theorem25 by putting λ=1 in modified Wiener index.Corollary 27.
The hyper Wiener index of regular triangular silicate networkRTSLn is (89)WRTSLn;x=52n6+24n5+75n4+104n3+1092n2+-8n.Proof.
This result immediately follows from Theorem25 by putting λ=1 in modified hyper Wiener index.Corollary 28.
The Harary index of regular triangular silicate networkRTSLn is (90)h0RTSLn=6n2+12n+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92m+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9.Proof.
This result immediately follows from Theorem25 by putting t=1 in generalized Harary index.


## 3.1. Triangular Oxide NetworkTOX(n)

A triangulated irregular network (TIN) is a portrayal of a consistent surface comprising completely of triangular aspects, utilized primarily as Discrete Global Grid in primary elevation modeling [33, 34]. Oxide networks play a vital role in the study of silicate networks [35, 36]. If we delete silicon vertices from a silicate network, we get an oxide network OX(n). A triangular oxide network is denoted as TOX(n). An oxide network TOX(n) is depicted in Figure 1, unit cell, Figure 2, for n=2, and Figure 3, for n=5.Figure 1
Triangular oxide network, unit cell.Figure 2
Triangular oxide network TOX(2).Figure 3
Triangular oxide network TOX(5).Lemma 1.
The triangular oxide network has3(n2+n)/2 edges and n2+3n+2/2 vertices.Theorem 2.
For the triangular oxide networkTOX(n) for n≥1, where n is length of triangular oxide network, we have (11)HTOXn;x=n2+3n+22+∑m=1n3m2n-m+1n-m+2xmProof.
To prove this theorem, we need to computeamn, where m=0,1,2,3,…,n. Obviously, from Lemma 1, we have (12)a0n=V=n2+3n+22 and (13)a1n=E=3n2+n2.The remaining proof is as follows.
It can be observed from Figures1–3 that there are n2+n/2 number of triangles, soa22=61a23=63a24=660a25=610a26=615a27=621
Now, one can conclude that(14)a2n=6n-12+n-12. Now, for a3n,a33=91a34=93a35=960a36=910a37=915a38=921
Now, one can conclude that(15)a3n=9n-22+n-22.Now, for a4n,a44=121a45=123a46=1260a47=1210a48=1215a49=1221
Now, one can conclude that(16)a4n=12n-32+n-32.From the above-mentioned method we can generalizea2n=6n-12+n-1/2a3n=9n-22+n-2/2a4n=12n-32+n-3/2a5n=15n-42+n-4/2a6n=18n-52+n-5/2
Now, one can conclude that(17)amn=3m2n-m+1n-m+2xm.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network TOXn. (18)HTOXn;x=n2+3n+22+∑m=1n3m2n-m+1n-m+2xm.Theorem 3.
For the triangular oxide networkTOXn, the Harary polynomial is(19)hTOXn;x=∑m=1n32n-m+1n-m+2xm.Proof.
From the calculation given in Theorem2, we have (20)hG,x=∑v∈VTOXn∑u∈VTOXn1dv,uxdv,u,=∑v∈VTOXn∑u∈VTOXn1mxm=∑m=1n32n-m+1n-m+2xmTheorem 4.
For the triangular oxide networkTOXn, we have (1)
WλTOXn=∑m=1n3/2n-m+1n-m+2mλ+1(2)
WWλTOXn=∑m=1n3/2n-m+1n-m+2mλ+1+m2λ+1(3)
HtTOXn=∑m=1n3m/2n-m+1n-m+2/m+t(4)
πTOXn=∏m=1nm3m/2n-m+1n-m+2Proof.
From the calculation given in Theorem2, we have the following.
(1) The modified Wiener index is(21)WλTOXn=12∑v∈VTOXn∑u∈VTOXndu,vλ=12∑v∈VTOXn∑u∈VTOXnmλ=∑m=1n3m2n-m+1n-m+2mλ=∑m=1n32n-m+1n-m+2mλ+1Table 1 contains values of modified Wiener index of TOX for different values of n and λ.
(2) The modified hyper Wiener index is(22)WWλTOXn=12∑v∈VTOXn∑u∈VTOXndu,vλ+∑v∈VTOXn∑u∈VTOXndu,v2λ=WλTOXn+12∑v∈VTOXn∑u∈VTOXndu,v2λ=WλTOXn+12∑v∈VTOXn∑u∈VTOXnm2λ=12∑v∈VTOXn∑u∈VTOXnmλ+m2λ=∑m=1n3m2n-m+1n-m+2mλ+m2λ=∑m=1n32n-m+1n-m+2mλ+1+m2λ+1.Table 2 contains the values of modified hyper Wiener index of TOX for different values of n and λ.
(3) The generalized Harary index is(23)HtTOXn=12∑v∈VTOXn∑u∈VTOXn1du,v+t=12∑v∈VG∑u∈VG1m+t=∑m=1n3m/2n-m+1n-m+2m+t.In Table 3, we give values of generalized Harary index of TOX for different values of n and λ.
(4) The multiplicative Wiener index is(24)πG=∏u,v⊆VTOXndu,v=∏u,v⊆VTOXnm=∏m=1nm3m/2n-m+1n-m+2.Table 1
Modified Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 3 21 81 231 546 1134  2 3 33 171 609 1722 4158  3 3 57 405 1815 6162 17334  4 3 105 1035 5865 23970 78678  5 3 201 2781 20031 98826 379134  6 3 393 7731 71169 425082 1909278Table 2
Modified hyper Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 6 54 252 840 2268 5292  2 6 138 1206 6474 25692 82836  3 6 450 8136 72984 431244 1926612  4 6 1650 64710 978690 8612340 53222076  5 6 6354 552672 14234160 187582068 1609127532  6 6 24978 4864446 215894154 4295457852 51426477396Table 3
Generalized Harary index ofTOX.
t ∖ n 1 2 3 4 5 6 0 3 12 30 60 105 168  1 3/2 13/2 69/4 723/20 657/10 7593/70  2 1 9/2 123/10 132/5 1713/35 2301/28  3 3/4 69/20 48/5 732/35 2199/56 3733/56  4 3/5 14/5 276/35 243/14 1381/42 1969/35  5 1/2 33/14 375/56 2495/168 793/28 15013/308  6 3/7 57/28 163/28 909/70 4157/167 9469/220Corollary 5.
The Wiener index of triangular oxide networkTOXn is(25)WTOXn;x=n520+3n48+n3+9n38+9n20.Proof.
This result can be followed immediately from Theorem4 by putting λ=1 in modified Wiener index.Corollary 6.
The hyper Wiener index of triangular oxide networkTOXn is(26)WTOXn;x=n640+11n540+9n48+17n38+37n220+3n5.Proof.
This result can be followed immediately from Theorem4 by putting λ=1 in modified hyper wiener index.Corollary 7.
The Harary index of triangular oxide networkTOXn is(27)HTOXn=12n3+3n2+2n.Proof.
This result can be followed immediately from Theorem4 by putting t=0 in generalized Harary index.


## 3.2. Regular Triangular Oxide NetworkRTOXn

The regular triangular oxide network is shown in Figure4, unit cell, Figure 5, for n=2 and Figure 6, for n=5.Figure 4
Regular triangular oxide network (unit cell).Figure 5
Regular triangular oxide network RTOX(2).Figure 6
Regular triangular oxide network RTOX(5).Lemma 8.
The regular triangular oxide networkRTOXn has 3n2+6n edges and 3/2n2+9/2n+1 vertices.Theorem 9.
IfRTOXn is the regular triangular oxide network n≥1, where n is length of triangular oxide network, then (28)HRTOXn;x=32n2+92n+1+3n2+6nx+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5xm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,2n+1. Obviously from Lemma 8(29)a0n=V=32n2+92n+1and (30)a1n=E=3n2+6nThe remaining proof is as follows.
It can be observed from Figures4–6 that there are n2+2n number of triangles:a21=23+2a22=28+16a23=215+38a24=224+68a25=235+106
Now, one can conclude that(31)a2n=6n2+6n-4.Now for a4na42=63+1a43=68+25a44=615+64a45=624+118a46=635+187
Now, one can conclude that(32)a4n=272n-12+272n-1-8.Now for a6na63=83+6a64=88+50a65=815+120a66=824+216a67=835+338
Now, one can conclude that(33)a6n=21n-22+21n-2-12.Now for a8na84=103+11a85=108+75a86=1015+176a87=1024+314a88=1035+489
Now, one can conclude that(34)a8n=572n-32+572n-3-16.From the above-mentioned method we can generalizea2n=6n2+6n-4a4n=27/2n-12+27/2n-1-8a6n=21n-22+21n-2-12a8n=57/2n-32+57/2n-3-16a10n=36n-42+36n-4-20
Now, one can conclude that(35)amn=15m-64n-m2+12+n-m2+1-2mxm.Now we can calculate for odd distance.
Now fora3na31=13+1a32=28+21a33=315+46a34=424+70a35=535+87
Now, one can conclude that(36)a3n=10.5n2+1.5n-8.Now for a5na52=13+5a53=28+43a54=315+98a55=424+164a56=535+235
Now, one can conclude that(37)a5n=16.5n-12+1.5n-1-10.Now for a7na73=13+9a74=28+65a75=315+150a76=424+258a77=535+383
Now, one can conclude that(38)a7n=22.5n-22+1.5n-2-12.Now for a9na94=13+13a95=28+87a96=315+202a97=424+356a98=535+531
Now, one can conclude that(39)a9n=28.5n-32+1.5n-3-14.From the above-mentioned method we can generalizea3n=10.5n2+1.5n-8a5n=16.5n-12+1.5n-1-10a7n=22.5n-22+1.5n-2-12a9n=28.5n-32+1.5n-3-14a11n=34.5n-42+1.5n-4-16
Now, one can conclude that(40)amn=6m+32n-m-12+12+32n-m-12+1-m+5xm.Here, by what have been mentioned above, we have following computations for the Hosoya polynomial of regular triangular oxide network RTOXn. (41)HRTOXn;x=32n2+92n+1+3n2+6nx+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5xm.Theorem 10.
For regular triangular oxide network systemRTOXn, the Harary polynomial is (42)hRTOXn;x=3n2+6nx+∑m≡0mod⁡21m15m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5xm.Proof.
From the calculation given in Theorem9, we have (43)hRTOXn,x=12∑v∈VRTOXn∑u∈VRTOXn1dv,uxdv,u=∑v∈VRTOXn∑u∈VRTOXn11x1+∑v∈VRTOXn∑u∈VRTOXn1mxm=3n2+6nx+∑m≡0mod⁡2,m≥21m15m-642n-m+222+2n-m+22-2mxm+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5xm.Theorem 11.
For the regular triangular oxide network systemRTOXn, we have WλRTOXn=3n2+6n + ∑m≡0mod⁡2(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)mλ + ∑m≡1mod⁡2,m>16m+3/22n-m+3/22+3/22n-m+3/2-m-5mλWWλRTOXn=3n2+6n1λ+12λ + ∑m≡0mod⁡2(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)mλ+m2λ + ∑m≡1mod⁡2,m>1(6m+3/22n-m+3/22 + 3/22n-m+3/2 − m-5)mλ+m2λHtRTOXn=3n2+6n/1+t + ∑m≡0mod⁡21/m+t(15m-6/4(2n-m+2/22 + 2n-m+2/2)-2m)15m-6/4(2n-m+2/22 + 2n-m+2/2)  − 2m + ∑m≡1mod⁡2,m>11/m+t(6m+3/22n-m+3/22 + 3/22n-m+3/2-m-5)πRTOXn=13n2+6n × ∏m≡0mod⁡2m15m-6/42n-m+2/22+2n-m+2/2-2m × ∏m≡1mod⁡2,m>1m6m+3/22n-m+3/22+3/22n-m+3/2-m-5mλ.Proof.
From the calculation given in Theorem9, we have the following.
(1) The modified Wiener index of regular triangular oxide network is(44)WλRTOXn=12∑v∈VRTOXn∑u∈VRTOXndu,vλ=∑v∈VRTOXn∑u∈VRTOXn1λ+∑v∈VRTOXn∑u∈VRTOXnmλ=3n2+6n+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mmλ+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5mλ.Table 4 contains different values of modified Wiener index of RTOX for different values of n and λ.
(2) The modified hyper Wiener index of regular triangular oxide network is(45)WWλRTOXn=12∑v∈VRTOXn∑u∈VRTOXndu,vλ+∑v∈VRTOXn∑u∈VRTOXndu,v2λ=WλRTOXn+12∑v∈VRTOXn∑u∈VRTOXndu,v2λ=WλRTOXn+12∑v∈VRTOXn∑u∈VRTOXn12λ+12∑v∈VOKn+2∑u∈VRTOXnm2λ=12∑v∈VRTOXn∑u∈VRTOXn1λ+12λ+12∑v∈VRTOXn∑u∈VRTOXnmλ+m2λ=3n2+6n1λ+12λ+∑m≡0mod⁡215m-642n-m+222+2n-m+22-2mmλ+m2λ+∑m≡1mod⁡2,m>16m+322n-m+322+322n-m+32-m-5mλ+m2λ.Table 5 contains values of modified hyper Wiener index of RTOX for different values of n and λ.
(3) The generalized Harary index is(46)htRTOXn=12∑v∈VRTOXn∑u∈VRTOXn1du,v+t=12∑v∈VRTOXn∑u∈VRTOXn11+t+∑v∈VRTOXn∑u∈VRTOXn1m+t=3n2+6n1+t+∑m≡0mod⁡21t+m15m-642n-m+222+2n-m+22-2m+∑m≡1mod⁡2,m>11t+m6m+322n-m+322+322n-m+32-m-5.Table 6 is about the values of generalized Harary index of RTOX for different values of n and λ.
(4) The multiplicative Wiener index is(47)πRTOXn=∏u,v⊆VRTOXndu,v=∏u,v⊆VRTOXn1×∏u,v⊆VRTOXnm=13n2+6n×∏m≡0mod⁡2m15m-6/42n-m+2/22+2n-m+2/2-2m×∏m≡1mod⁡2,m>1m6m+3/22n-m+3/22+3/22n-m+3/2-m-5mλ.Table 4
Modified Wiener index ofRTOX.
λ ∖ n 1 2 3 4 5 6 1 37 315 1305 3856 9303 19575  2 77 989 5447 20062 57937 141907  3 181 3495 25689 118276 409647 1169511  4 461 13397 131759 759310 3156433 10509235  5 1237 54495 718425 5184796 25875903 100496535  6 3437 231869 4103087 37092982 222278977 1007138467Table 5
Modified hyper Wiener index ofRTOX.
λ ∖ n 1 2 3 4 5 6 1 114 1304 6752 23918 67240 161482  2 538 14386 137206 779372 3214370 10651142  3 3618 235364 4128776 37211258 222688624 1008307978  4 28762 4634554 148143358 2102868620 18172355906 112088413190  5 245634 100320044 5862567752 131158675298 1636767140320 13752061166362  6 2161978 2291918386 247080692926 8743290167492 157800041881490 1807230473840342Table 6
Generalized Harary index ofRTOX.
t ∖ n 1 2 3 4 5 6 0 43/3 3521/60 5990/41 15655/54 26687/53 17609/22  1 49/6 741/20 7949/81 14439/71 15694/43 36915/62  2 29/5 5819/210 10623/140 74267/461 30345/103 19519/40  3 271/60 4679/210 6173/99 7805/58 13463/54 39652/95  4 389/105 766/41 5576/105 47907/413 14106/65 24175/66  5 22/7 1498/93 10887/235 715/7 11169/58 52343/160  6 172/63 8146/575 3578/87 6669/73 24437/141 65417/221Corollary 12.
The Wiener index of regular triangular oxide networkRTOXn is (48)WRTOXn;x=9n510+27n44+31n32+53n24+3n5.Proof.
This result immediately follows from Theorem11 by taking λ=1 in modified Wiener index.Corollary 13.
The hyper Wiener index of regular triangular oxide networkRTOXn is (49)WRTOXn;x=910n6+9n5+1194n4+2756n3+58720n2-56n.Proof.
This result immediately follows from Theorem11 by taking λ=1 in generalized hyper Wiener index.Corollary 14.
The Harary index of regular triangular oxide networkRTOXn by putting t=o is (50)h0RTOXn=3n2+6n+∑m≡0mod⁡21m15m-642n-m+222+2n-m+22-2m+∑m≡1mod⁡2,m>11m6m+322n-m+322+322n-m+32-m-5.Proof.
This result immediately follows from Theorem11 by taking t=1 in generalized harary index.


## 3.3. Triangular Silicate NetworkTSLn

Silicates are the largest, the most complicated, and the most interesting class of minerals so far.SiO4 tetrahedron is the basic chemical unit of silicates. The silicates sheets are rings of tetrahedrons linked by shared oxygen nodes to other rings in two-dimensional planes producing a sheet like structures. A silicate can be obtained by fusing a metal oxide or a metal carbonate with sand. Essentially every silicate contains SiO4 tetrahedron. The corner and the center vertices represent oxygen and silicon ions, respectively. These vertices are called oxygen nodes and silicon node, respectively. In SiO4 tetrahedra, the corner and the center vertices represent oxygen and silicon ions, respectively (Figures 7–9). Unit cell is shown in Figure 7, for n=2 in Figure 8 and for n=5 in Figure 9.Figure 7
S i O 4.Figure 8
Triangular silicate network TsL(2).Figure 9
Triangular silicate network TsL(5).Lemma 15.
The triangular silicate network has3n2+n edges and n2+2n+1 vertices.Theorem 16.
IfTSLn is the triangular silicate network n≥1, where n is length of triangular silicate network then(51)HTSLn;x=n2+2n+1+∑m=1n6m-3n-m+12+n-m+1xm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,n. Obviously, from Lemma 15, we have (52)a0n=V=n2+2n+1, and (53)a1n=E=3n2+3n.The remaining proof is as follows.
It can be observed from Figures7–9 that there are n2+n/2 number of triangles:a22=181a23=183a24=1860a25=1810a26=1815a27=1821
Now, one can conclude that(54)a2n=18n-12+n-12. Now, for a3n,a33=301a34=303a35=3060a36=3010a37=3015a38=3021
Now, one can conclude that(55)a3n=30n-22+n-22. Now, for a4n,a44=421a45=423a46=4260a47=4210a48=4215a49=4221
Now, one can conclude that(56)a4n=42n-32+n-32.From the above-mentioned method we can generalizea2n=18n-12+n-1/2a3n=30n-22+n-2/2a4n=42n-32+n-3/2a5n=54n-42+n-4/2a6n=66n-52+n-5/2
Now, one can conclude that(57)amn=6m-3n-m+12+n-m+1.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of triangular oxide network TOXn.(58)HTSLn;x=6m-3n-m+12+n-m+1xm.Theorem 17.
For the triangular silicate networkTSLn, the Harary polynomial is(59)hTSLn;x=∑m=1n1m6m-3n-m+12+n-m+1xm.Proof.
From the calculation given in Theorem16, we have (60)hG,x=∑v∈VTSLn∑u∈VTSLn1dv,uxdv,u=∑v∈VTSLn∑u∈VTSLn1mxm=∑m=1n1m6m-3n-m+12+n-m+1xm.Theorem 18.
For the triangular silicate networkTSLn, we have (1)
WλTSLn=∑m=1n6m-3n-m+12+n-m+1mλ.(2)
WWλTSLn=∑m=1n6m-3n-m+12+n-m+1mλ+m2λ.(3)
HtTSLn=∑m=1n6m-3n-m+12+n-m+1/m+t.(4)
πTSLn=∏m=1nm6m-3n-m+12+n-m+1.Proof.
From the calculation given in Theorem16, we have the following.
(1) The modified Wiener index of the triangular silicate networkTSLn: (61)WλTSLn=12∑v∈VTSLn∑u∈VTSLndu,vλ=12∑v∈VTSLn∑u∈VTSLnmλ=∑m=1n6m-3n-m+12+n-m+1mλ.Table 7 is about different values of modified Wiener index of the triangular silicate network TSLn for different values of n and λ.
(2) The modified hyper Wiener index of the triangular silicate networkTSLn is (62)WWλTSLn=12∑v∈VTSLn∑u∈VTSLndu,vλ+∑v∈VTSLn∑u∈VTSLndu,v2λ=WλTSLn+12∑v∈VTSLn∑u∈VTSLndu,v2λ=WλTSLn+12∑v∈VTSLn∑u∈VTSLnm2λ=12∑v∈VTSLn∑u∈VTSLnmλ+m2λ=∑m=1n6m-3n-m+12+n-m+1mλ+m2λ.Table 8 is about different values of modified hyper Wiener index of the triangular silicate network TSLn for different values of n and λ.
(3) The generalized Harary index the triangular silicate networkTSLn is (63)HtTSLn=12∑v∈VTSLn∑u∈VTSLn1du,v+t=12∑v∈VG∑u∈VG1m+t=∑m=1n6m-3n-m+12+n-m+1m+t.Table 9 is about different values of generalized Harary index of the triangular silicate network TSLn for different values of n and t.
(4) The multiplicative Wiener index the triangular silicate networkTSLn is (64)πG=∏u,v⊆VTSLndu,v=∏u,v⊆VTSLnm=∏m=1nm6m-3n-m+12+n-m+1.Table 7
Modified Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 6 54 234 714 1764 3780  2 6 90 522 1974 5796 14364  3 6 162 1278 6042 21204 61020  4 6 306 3330 19830 83556 280044  5 6 594 9054 68394 347364 1359180  6 6 1170 25362 244614 1502676 6878844Table 8
Modified hyper Wiener index ofTOX.
λ ∖ n 1 2 3 4 5 6 1 12 144 756 2688 7560 18144  2 12 396 3852 21804 89352 294408  3 12 1332 26640 250656 1523880 6939864  4 12 4932 214020 3390540 30661992 192968568  5 12 19044 1835856 49533648 670624920 5856390504  6 12 74916 16189812 753159804 15395422392 187613691048Table 9
Generalized Harary index ofTOX.
t ∖ n 1 2 3 4 5 6 0 6 27 73 309/2 2823/10 2337/5  1 3 15 87/2 969/10 921/5 11019/35  2 2 21/2 63/2 72 978/7 6801/28  2 3/2 81/10 124/5 288/5 453/4 2389/12  4 6/5 33/5 717/35 1347/28 2673/28 11841/70  5 1 39/7 489/28 3473/84 5783/70 10309/70  6 6/7 135/28 1279/84 2539/70 11440/157 6134/47Corollary 19.
The Wiener index of triangular silicate networkTSLn is(65)WTSLn;x=n55+5n44+5n32+7n24+3n10.Proof.
This result follows immediately Theorem18 by putting λ=1 in modified Wiener index.Corollary 20.
The hyper Wiener index of triangular silicate networkTSLn is(66)WTSLn;x=n610+n5+7n42+5n3+12n25.Proof.
This result follows immediately Theorem18 by putting λ=1 in modified hyper Wiener index.Corollary 21.
The Harary index of triangular Silicate networkTSLn is(67)HtTSLn=12n23+21n+4n2-32+3n+n2Hn.where Hn is harmonic number.Proof.
This result follows immediately Theorem18 by putting t=1 in generalized Harary index.


## 3.4. Regular Triangular Silicate NetworkRTSLn

The regular triangular silicate network is shown in Figures10–12, unit cell in Figure 10, for n=2 in Figure 11 and for n=5 in Figure 12.Figure 10
Regular triangular silicate network (unit cell).Figure 11
Regular triangular silicate network RTsL(2).Figure 12
Regular triangular silicate network RTsL(5).Lemma 22.
The regular triangular silicate network has6n2+12n edges and 5/2n2+13/2n+1 vertices.Theorem 23.
IfRTSLn is the regular triangular silicate network n≥1, where n is length of triangular silicate network then (68)HRTSLn;x=52n2+132n+1+6n2+12nx+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9xm+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mxm.Proof.
To prove this theorem, we need to computeamn where m=0,1,2,3,…,2n+1. Obviously, from Lemma 22, we have (69)a0n=V=52n2+132n+1and (70)a1n=E=6n2+12n.The remaining proof is as follows.
It can be observed from Figures10–12 thata21=18a22=72a23=153a24=261a25=396.
Now, one can conclude that(71)a2n=272n2+272n-9.
Now fora4na42=48a43=180a44=378a45=642a46=972.
Now, one can conclude that(72)a4n=33n-12+33n-1-18.Now for a6na63=78a64=288a65=603a66=1023a67=1548.
Now, one can conclude that(73)a6n=1052n-22+1052n-2-27.Now for a8na84=108a85=396a86=828a87=1404a88=2124.
Now, one can conclude that(74)a8n=72n-32+72n-3-36.From the above-mentioned method we can generalizea2n=27/2n2+27/2n-9a4n=33n-12+33n-1-18a6n=105/2n-22+105/2n-2-27a8n=72n-32+72n-3-36a10n=183/2n-42+183/2n-4-45.
Now, one can conclude that(75)amn=39m-2442n-m+222+2n-m+22-92m.Now we can calculate for odd distance.
Fora3na31=9a32=87a33=216a34=396a35=627.
Now, one can conclude that(76)a3n=512n2+32n-18.Now for a5na52=21a53=153a54=372a55=678a56=1071.
Now, one can conclude that(77)a5n=872n-12+32n-1-24.Now for a7na73=33a74=219a75=528a76=960a77=1515.
Now, one can conclude that(78)a7n=1232n-22+32n-2-30.Now for a9na94=45a95=285a96=684a97=1292a98=1959.
Now, one can conclude that(79)a9n=1592n-32+32n-3-36.From the above-mentioned method we can generalizea3n=51/2n2+3/2n-18a5n=87/2n-12+3/2n-1-24a7n=123/2n-22+3/2n-2-30a9n=159/2n-32+3/2n-3-36a11n=195/2n-32+3/2n-3-42.
Now, one can conclude that(80)amn=18m-322n-m+322+322n-m+32-3m+9.Here, by what have been mentioned above, we have the following computations for the Hosoya polynomial of regular triangular silicate network RTSLn.(81)HRTSLn;x=52n2+132n+1+6n2+12nx+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9xm+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mxm.Theorem 24.
For regular triangular silicate network systemRTSLn, the Harary polynomial is (82)hRTSLn;x=6n2+12nx+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92mxm+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9xm.Proof.
From the information given in Theorem23, we have (83)hRTSLn,x=12∑v∈VRTSLn∑u∈VRTSLn1dv,uxdv,u=∑v∈VRTSLn∑u∈VRTSLn11x1+∑v∈VRTSLn∑u∈VRTSLn1mxm=6n2+12nx+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92mxm+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9xm.Theorem 25.
For the regular triangular silicate network systemRTSLn, we haveProof.
From the information given in Theorem23, we have the following.
(1) The modified Wiener index of regular triangular silicate network is(84)WλRTSLn=12∑v∈VRTSLn∑u∈VRTSLndu,vλ=∑v∈VRTSLn∑u∈VRTSLn1λ+∑v∈VRTSLn∑u∈VRTSLnmλ=6n2+12n+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mmλ+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9mλ.Table 10 contains the values of modified Wiener index of regular triangular silicate network for different values of n and λ.
(2) The modified hyper Wiener index of regular triangular silicate network is(85)WWλRTSLn=12∑v∈VRTSLn∑u∈VRTSLndu,vλ+∑v∈VRTSLn∑u∈VRTSLndu,v2λ=WλRTSLn+12∑v∈VRTSLn∑u∈VRTSLndu,v2λ=WλRTSLn+12∑v∈VRTSLn∑u∈VRTSLn12λ+12∑v∈VOKn+2∑u∈VRTSLnm2λ=12∑v∈VRTSLn∑u∈VRTSLn1λ+12λ+12∑v∈VRTSLn∑u∈VRTSLnmλ+m2λ=6n2+12n1λ+12λ+∑m≡0mod⁡239m-2442n-m+222+2n-m+22-92mmλ+m2λ+∑m≡1mod⁡2,m>118m-322n-m+322+322n-m+32-3m+9mλ+m2λ.Table 11 contains the values of modified hyper Wiener index of regular triangular silicate network for different values of n and λ.
(3) The generalized Harary index of regular triangular silicate network is(86)htRTSLn=12∑v∈VRTSLn∑u∈VRTSLn1du,v+t=12∑v∈VRTSLn∑u∈VRTSLn11+t+∑v∈VRTSLn∑u∈VRTSLn1m+t=6n2+12n1+t+∑m≡0mod⁡21m+t39m-2442n-m+222+2n-m+22-92m+∑m≡1mod⁡2,m>11m+t18m-322n-m+322+322n-m+32-3m+9.Table 12 contains the values of generalized Harary index of regular triangular silicate network for different values of n and t.
(4) The multiplicative Wiener index of regular triangular silicate network is(87)πRTSLn=∏u,v⊆VRTSLndu,v=∏u,v⊆VRTSLn1×∏u,v⊆VRTSLnm=16n2+12n×∏m≡0mod⁡2m39m-24/42n-m+2/22+2n-m+2/2-9/2m×∏m≡1mod⁡2,m>1m18m-3/22n-m+3/22+3/22n-m+3/2-3m+9mλ.Table 10
Modified Wiener index ofRTSL.
λ ∖ n 1 2 3 4 5 6 1 81 750 3228 9756 23895 50826  2 171 2412 13776 51756 151365 374016  3 405 8670 65958 309042 1081881 3111276  4 1035 33660 342060 2002344 8401089 28144380  5 2781 138270 1881078 13770666 69289425 270544116  6 7731 592812 10816836 99094776 598206945 2723160396Table 11
Modified hyper Wiener index ofRTSL.
λ ∖ n 1 2 3 4 5 6 1 252 3162 17004 61512 175260 424842  2 1206 36072 355836 2054100 8552454 28518396  3 8136 601482 10882794 99403818 599288826 2726271672  4 64710 11971800 394609080 5670015168 49306706898 305266571880  5 552672 260759082 15735721674 356212348962 4470377552850 37677244051392  6 4864446 5979382632 667051149216 23886660553680 433393778907474 4976938113309816Table 12
Generalized Harary index ofRTSL.
t ∖ n 1 2 3 4 5 6 0 30 646/5 23227/70 28947/43 20216/17 47899/25  1 69/4 1657/20 12699/56 16341/34 29821/34 44938/31  2 123/10 312/5 18556/105 31473/82 9278/13 31133/26  3 48/5 6197/123 17071/117 36405/113 55228/91 8233/8  4 276/35 4448/105 4363/35 89996/323 15371/29 26277/29  5 375/56 3696/101 55916/513 8116/33 22631/48 23526/29  6 163/28 2353/73 12407/128 1763/8 22532/53 19856/27Corollary 26.
The Wiener index of regular triangular silicate networkRTSLn is (88)WRTSLn;x=5n52+35n42+36n3+26n2-n.Proof.
This result immediately follows from Theorem25 by putting λ=1 in modified Wiener index.Corollary 27.
The hyper Wiener index of regular triangular silicate networkRTSLn is (89)WRTSLn;x=52n6+24n5+75n4+104n3+1092n2+-8n.Proof.
This result immediately follows from Theorem25 by putting λ=1 in modified hyper Wiener index.Corollary 28.
The Harary index of regular triangular silicate networkRTSLn is (90)h0RTSLn=6n2+12n+∑m≡0mod⁡21m39m-2442n-m+222+2n-m+22-92m+∑m≡1mod⁡2,m>11m18m-322n-m+322+322n-m+32-3m+9.Proof.
This result immediately follows from Theorem25 by putting t=1 in generalized Harary index.



---

*Source: 8696982-2019-07-16.xml*