# On Computation Degree-Based Topological Descriptors for Planar Octahedron Networks

Authors: Wang Zhen; Parvez Ali; Haidar Ali; Ghulam Dustigeer; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

A molecular graph is used to represent a chemical molecule in chemical graph theory, which is a branch of graph theory. A graph is considered to be linked if there is at least one link between its vertices. A topological index is a number that describes a graph’s topology. Cheminformatics is a relatively young discipline that brings together the field of sciences. Cheminformatics helps in establishing QSAR and QSPR models to find the characteristics of the chemical compound. We compute the first and second modified K-Banhatti indices, harmonic K-Banhatti index, symmetric division index, augmented Zagreb index, and inverse sum index and also provide the numerical results.


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## Body

## 1. Introduction

Graph theory provides topological indices, which are a useful tool. Cheminformatics is a contemporary academic discipline that brings together chemistry, mathematics, and information science. It investigates the connections between quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), which are used to predict biological activities and chemical compound characteristics.The silicate structures [1] formed from the POH network, TP network, and hex POH network [2] are discussed in this article.The following is the procedure for makingPOH networks .Step 1: consider am-dimensional silicate network.Step 2: connect new vertices in the centre of each triangular face to existing vertices in the adjacent triangular face.Step 3: all of the new centre vertices in the same silicate cell must be connected.Step 4: for them dimension, the resultant graph is known as the planar octahedron network as shown in Figure 1. Remove all silicon vertices from the graph. The triangle prism network as shown in Figure 2 and the hex POH network as shown in Figure 3 are also possible.Figure 1
Planar octahedral networkPOH2.Figure 2
Triangular prism networkTP2.Figure 3
Hexagonal planar octahedral networkHPOH2.Letψ represent a graph. Then, modified first and second K-Banhatti indices [3] can be defined as(1)MK1Bψ=∑ab∈Eψ1da+db,(2)MK2Bψ=∑ab∈Eψ1da×db.Harmonic K-Banhatti index [4] of a graph ψ is defined as(3)HKBψ=∑ab∈Eψ2da+db.Symmetric division index of a graph [5] is defined as(4)SDψ=∑ab∈Eψdadb+dbda.Augmented Zagreb index of a graphψ [4] is defined as(5)AGψ=∑ab∈Eψda×dbda+db−23.Inverse sum index of a graphψ is defined as(6)Iψ=∑ab∈Eψda×dbda+db.


## 2. Main Results

We research different indices on different kinds of planar octahedron networks. Nowadays, extensive research studies are being conducted in the field of chemical graph theory for further studying topological indices of various graphs [6–13]. For the basic notations and definitions, see [14, 15].


### 2.1. Results for Planar Octahedron Network POH (m)

The planar octahedron network is the resulting graph for them dimension. All silicon vertices should be removed from the scene. There are also the triangular prism network and the hex POH network. Now, we calculate several key indices for the POH network in the following theorems.Theorem 1.
Consider the planar octahedral network POH (m); then, its first and second modified K-Banhatti indices are equal to(7)MK1Bψ1=518m2+34m,MK2Bψ1=8132m2+916m.Proof.
Letψ1≅POHm. From equation (1), we have(8)MK1Bψ=∑ab∈Eψ1da+db.
Using Table1, we have(9)MK1Bψ1=14+4E1POHm+14+8E2POHm+18+8E3POHm=18E1POHm+112E2POHm+116E3POHm=1818m2+12m+11236m2−48m+12+11618m2−36m+18.
We get the following value after calculations:(10)⇒MK1Bψ1=518m2+34m.
Letψ1≅POHm. From equation (2), we have(11)MK1Bψ=∑ab∈Eψ1da+db.
Using Table1, we have(12)MK2Bψ1=14×4E1POHm+14×8E2POHm+18×8E3POHm=116E1POHm+132E2POHm+164E3POHm=11618m2+12m+13236m2−48m+12+16418m2−36m+18.
We get the following value after calculations:(13)⇒MK2Bψ1=8132m2+916m.Table 1
Edge partition.
da,dbNumber of edgesE1=4,418m2+12mE2=4,836m2−48m+12E3=8,818m2−36m+18Theorem 2.
The harmonic K-Banhatti and symmetric division indices are equal in the POH (m) network.(14)HKBψ=514m2+32m,SDψ=162m2.Proof.
Letφ1≅POHm network; from equation (3),(15)HKBψ=∑ab∈Eψ2da+db.
Using Table1, we have(16)HKBψ1=24+4E1POHm+24+8E2POHm+28+8E3POHm=28E1POHm+212E2POHm+216E3POHm=1418m2+12m+1636m2−48m+12+1818m2−36m+18.
We get the following value after calculations:(17)⇒HKBψ1=514m2+32m.
For the symmetric division index of a graph using equation (3),(18)SDψ=∑ab∈Eψdadb+dbda.
Using Table1, we have(19)SDψ1=44+44E1POHm+48+84E2POHm+88+88E3POHm=2E1POHm+12+21E2POHm+2E3POHm=218m2+12m+5236m2−48m+12+218m2−36m+18.
After calculations,(20)⇒SDψ1=162m2.Theorem 3.
Then, augmented Zagreb and inverse sum indices are equal to thePOH network.(21)AGψ1=416820224128625m2−28364803087m,Iψ1204m2−24m.Proof.
Letφ1≅POHm network. For the augmented Zagreb index, using equation (5),(22)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table1, we have(23)AGψ1=1663E1POHm+32103E2POHm+64143E3POHm=166318m2+12m+3210336m2−48m+12+6414318m2−36m+18.
After some calculations, we get(24)⇒AGψ1=416820224128625m2−28364803087m.
For the inverse sum index and by using equation (6), we have(25)Iψ=∑ab∈Eψda×dbda+db.
Using Table1, we have(26)Iψ1=168E1POHm+3212E2POHm+6416E3POHm=218m2+12m+321236m2−48m+12+418m2−36m+18.
After calculations,(27)⇒Iψ1=204m2−24m.


### 2.2. Results for Triangular Prism Network TP (m)

In this part, we propose the theorem for theTP network.Theorem 4.
The first and second modified K-Banhatti indices are equal to theTP network:(28)MK1Bψ2=132m2+23m,MK2Bψ2=72m2+23m.Proof.
Letψ2≅TPm. From equation (1), we have(29)MK1Bψ=∑ab∈Eψ1da+db.
Using Table2, we have(30)MK1Bψ2=13+3E1TPm+13+6E2TPm+16+6E3TPm=16E1TPm+19E2TPm+112E3TPm=1618m2+6m+1918m2+6m+11218m2−36m+18.
After some calculations, we get(31)⇒MK1Bψ2=132m2+23m.
Letψ2≅TPm. From equation (2), we have(32)MK1Bψ=∑ab∈Eψ1da+db.
Using Table2, we have(33)MK2Bψ2=13×3E1TPm+13×6E2TPm+16×6E3TPm=19E1TPm+118E2TPm+136E3TPm=1918m2+6m+11818m2+6m+13618m2−36m+18.
After calculations,(34)⇒MK2Bψ2=72m2+23m.Table 2
Edge partition ofTP network.
da,dbNumber of edgesE1=3,318m2+6mE2=3,618m2+6mE3=6,618m2−36m+18Theorem 5.
In the POH (m) network, the harmonic K-Banhatti and symmetric division indices are equal to(35)HKBψ2=13m2+43m,SDψ2=117m2+3m.Proof.
Letφ2≅TPm network, and by using equation (3), we have(36)HKBψ=∑ab∈Eψ2da+db.
Using Table2, we have(37)HKBψ2=23+3E1TPm+23+6E2TPm+26+6E3TPm=26E1TPm+29E2TPm+212E3TPm=1318m2+6m+2918m2+6m+1618m2−36m+18.
After calculations,(38)⇒HKBψ2=13m2+43m.
For the symmetric division index of a graph using equation (3), we have(39)SDψ=∑ab∈Eψdadb+dbda.
Using Table2, we have(40)SDψ2=33+33E1TPm+36+63E2TPm+66+66E3TPm=2E1TPm+12+2E2TPm+2E3TPm=218m2+6m+5218m2+6m+218m2−36m+18.
After some calculations, we get(41)⇒SDψ2=117m2+3m.Theorem 6.
The augmented Zagreb and inverse sum indices are equal to theTP network.(42)AGψ2=18534234511372000m2−5344087591372000m,Iψ2=204m2−24m.Proof.
Letφ2≅POHm network. Using equation (5) for the augmented Zagreb index,(43)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table2, we have(44)AGψ2=943E1TPm+1873E2TPm+36103E3TPm=94318m2+6m+187318m2+6m+3610318m2−36m+18.
We get the following value after calculations:(45)⇒AGψ2=18534234511372000m2−5344087591372000m.
For the inverse sum index and by using equation (6), we have(46)Iψ=∑ab∈Eψda×dbda+db.
Using Table2, we have(47)Iψ2=96E1TPm+2E2TPm+3E3TPm=9618m2+6m+218m2+6m+318m2−36m+18.
After calculations,(48)⇒Iψ2=117m2−15m.


### 2.3. Results for Hexagonal Planar Octahedron (HPOH) Network

In this part, we propose the theorem for the HPOH network.Theorem 7.
The first and second modified K-Banhatti indices are equal to the hex POH network:(49)MK1Bψ3=518m2−4m−138,MK2Bψ3=8132m2−1516m−3932.Proof.
Letψ3≅HPOHm. From equation (1), we have(50)MK1Bψ=∑ab∈Eψ1da+db.
Using Table3, we have(51)MK1Bψ3=14+4E1HPOHm+14+8E2HPOHm+18+8E3HPOHm=18E1HPOHm+112E2HPOHm+116E3HPOHm=1818m2+18m−30+11236m2−48m+12+11618m2−36m+18.
We get the following value after calculations:(52)⇒MK1Bψ3=518m2−4m−138.
Letψ2≅HPOHm. From equation (2), we have(53)MK1Bψ=∑ab∈Eψ1da+db.
Using Table3, we have(54)MK2Bψ3=14×4E1HPOHm+14×8E2HPOHm+18×8E3HPOHm=116E1HPOHm+132E2HPOHm+164E3HPOHm=11618m2+18m−30+13236m2−48m+12+16418m2−36m+18.
We get the following value after calculations:(55)⇒MK2Bψ3=8132m2−1516m−3932.Table 3
Edge partition.
da,dbNumber of edgesE1=4,418m2+18m−30E2=4,836m2−48m+12E3=8,818m2−36m+18Theorem 8.
Then, harmonic K-Banhatti and symmetric division indices are equal to the hex POH network:(56)HKBψ3=514m2−8m−134,SDψ3=162m2−156m+6.Proof.
Letφ1≅HPOHm network, and from equation (3),(57)HKBψ=∑ab∈Eψ2da+db.
Using Table3, we have(58)HKBψ3=24+4E1HPOHm+24+8E2HPOHm+28+8E3HPOHm=28E1HPOHm+212E2HPOHm+216E3HPOHm=1418m2+18m−30+1636m2−48m+12+1818m2−36m+18.
We get the following value after calculations:(59)⇒HKBψ3=514m2−8m−134.
For the symmetric division index of a graph using equation (3), we have(60)SDψ=∑ab∈Eψdadb+dbda.
Using Table3, we have(61)SDψ3=44+44E1HPOHm+48+84E2HPOHm+88+88E3HPOHm=2E1HPOHm+12+21E2HPOHm+2E3HPOHm=218m2+18m−30+5236m2−48m+12+218m2−36m+18.
After calculations,(62)⇒SDψ3=162m2−156m+6.Theorem 9.
The augmented Zagreb and inverse sum indices are equal to the hex POH network:(63)AGψ3=416820224128625m2−600773632128625m+595764224385875,Iψ3=204m2−236m+44.Proof.
Letφ1≅HPOHm network. Using equation (5) for the augmented Zagreb index,(64)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table3, we have(65)AGψ3=1663E1HPOHm+32103E2HPOHm+64143E3HPOHm=166318m2+18m−30+3210336m2−48m+12+6414318m2−36m+18.
We get the following value after calculations:(66)⇒AGψ3=416820224128625m2−600773632128625m+595764224385875.
For the inverse sum index and by using equation (6), we have(67)Iψ=∑ab∈Eψda×dbda+db.
Using Table3, we have(68)Iψ3=168E1HPOHm+3212E2HPOHm+6416E3HPOHm=218m2+18m−30+321236m2−48m+12+418m2−36m+18.
After calculations,(69)⇒Iψ3=204m2−236m+44.


## 2.1. Results for Planar Octahedron Network POH (m)

The planar octahedron network is the resulting graph for them dimension. All silicon vertices should be removed from the scene. There are also the triangular prism network and the hex POH network. Now, we calculate several key indices for the POH network in the following theorems.Theorem 1.
Consider the planar octahedral network POH (m); then, its first and second modified K-Banhatti indices are equal to(7)MK1Bψ1=518m2+34m,MK2Bψ1=8132m2+916m.Proof.
Letψ1≅POHm. From equation (1), we have(8)MK1Bψ=∑ab∈Eψ1da+db.
Using Table1, we have(9)MK1Bψ1=14+4E1POHm+14+8E2POHm+18+8E3POHm=18E1POHm+112E2POHm+116E3POHm=1818m2+12m+11236m2−48m+12+11618m2−36m+18.
We get the following value after calculations:(10)⇒MK1Bψ1=518m2+34m.
Letψ1≅POHm. From equation (2), we have(11)MK1Bψ=∑ab∈Eψ1da+db.
Using Table1, we have(12)MK2Bψ1=14×4E1POHm+14×8E2POHm+18×8E3POHm=116E1POHm+132E2POHm+164E3POHm=11618m2+12m+13236m2−48m+12+16418m2−36m+18.
We get the following value after calculations:(13)⇒MK2Bψ1=8132m2+916m.Table 1
Edge partition.
da,dbNumber of edgesE1=4,418m2+12mE2=4,836m2−48m+12E3=8,818m2−36m+18Theorem 2.
The harmonic K-Banhatti and symmetric division indices are equal in the POH (m) network.(14)HKBψ=514m2+32m,SDψ=162m2.Proof.
Letφ1≅POHm network; from equation (3),(15)HKBψ=∑ab∈Eψ2da+db.
Using Table1, we have(16)HKBψ1=24+4E1POHm+24+8E2POHm+28+8E3POHm=28E1POHm+212E2POHm+216E3POHm=1418m2+12m+1636m2−48m+12+1818m2−36m+18.
We get the following value after calculations:(17)⇒HKBψ1=514m2+32m.
For the symmetric division index of a graph using equation (3),(18)SDψ=∑ab∈Eψdadb+dbda.
Using Table1, we have(19)SDψ1=44+44E1POHm+48+84E2POHm+88+88E3POHm=2E1POHm+12+21E2POHm+2E3POHm=218m2+12m+5236m2−48m+12+218m2−36m+18.
After calculations,(20)⇒SDψ1=162m2.Theorem 3.
Then, augmented Zagreb and inverse sum indices are equal to thePOH network.(21)AGψ1=416820224128625m2−28364803087m,Iψ1204m2−24m.Proof.
Letφ1≅POHm network. For the augmented Zagreb index, using equation (5),(22)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table1, we have(23)AGψ1=1663E1POHm+32103E2POHm+64143E3POHm=166318m2+12m+3210336m2−48m+12+6414318m2−36m+18.
After some calculations, we get(24)⇒AGψ1=416820224128625m2−28364803087m.
For the inverse sum index and by using equation (6), we have(25)Iψ=∑ab∈Eψda×dbda+db.
Using Table1, we have(26)Iψ1=168E1POHm+3212E2POHm+6416E3POHm=218m2+12m+321236m2−48m+12+418m2−36m+18.
After calculations,(27)⇒Iψ1=204m2−24m.


## 2.2. Results for Triangular Prism Network TP (m)

In this part, we propose the theorem for theTP network.Theorem 4.
The first and second modified K-Banhatti indices are equal to theTP network:(28)MK1Bψ2=132m2+23m,MK2Bψ2=72m2+23m.Proof.
Letψ2≅TPm. From equation (1), we have(29)MK1Bψ=∑ab∈Eψ1da+db.
Using Table2, we have(30)MK1Bψ2=13+3E1TPm+13+6E2TPm+16+6E3TPm=16E1TPm+19E2TPm+112E3TPm=1618m2+6m+1918m2+6m+11218m2−36m+18.
After some calculations, we get(31)⇒MK1Bψ2=132m2+23m.
Letψ2≅TPm. From equation (2), we have(32)MK1Bψ=∑ab∈Eψ1da+db.
Using Table2, we have(33)MK2Bψ2=13×3E1TPm+13×6E2TPm+16×6E3TPm=19E1TPm+118E2TPm+136E3TPm=1918m2+6m+11818m2+6m+13618m2−36m+18.
After calculations,(34)⇒MK2Bψ2=72m2+23m.Table 2
Edge partition ofTP network.
da,dbNumber of edgesE1=3,318m2+6mE2=3,618m2+6mE3=6,618m2−36m+18Theorem 5.
In the POH (m) network, the harmonic K-Banhatti and symmetric division indices are equal to(35)HKBψ2=13m2+43m,SDψ2=117m2+3m.Proof.
Letφ2≅TPm network, and by using equation (3), we have(36)HKBψ=∑ab∈Eψ2da+db.
Using Table2, we have(37)HKBψ2=23+3E1TPm+23+6E2TPm+26+6E3TPm=26E1TPm+29E2TPm+212E3TPm=1318m2+6m+2918m2+6m+1618m2−36m+18.
After calculations,(38)⇒HKBψ2=13m2+43m.
For the symmetric division index of a graph using equation (3), we have(39)SDψ=∑ab∈Eψdadb+dbda.
Using Table2, we have(40)SDψ2=33+33E1TPm+36+63E2TPm+66+66E3TPm=2E1TPm+12+2E2TPm+2E3TPm=218m2+6m+5218m2+6m+218m2−36m+18.
After some calculations, we get(41)⇒SDψ2=117m2+3m.Theorem 6.
The augmented Zagreb and inverse sum indices are equal to theTP network.(42)AGψ2=18534234511372000m2−5344087591372000m,Iψ2=204m2−24m.Proof.
Letφ2≅POHm network. Using equation (5) for the augmented Zagreb index,(43)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table2, we have(44)AGψ2=943E1TPm+1873E2TPm+36103E3TPm=94318m2+6m+187318m2+6m+3610318m2−36m+18.
We get the following value after calculations:(45)⇒AGψ2=18534234511372000m2−5344087591372000m.
For the inverse sum index and by using equation (6), we have(46)Iψ=∑ab∈Eψda×dbda+db.
Using Table2, we have(47)Iψ2=96E1TPm+2E2TPm+3E3TPm=9618m2+6m+218m2+6m+318m2−36m+18.
After calculations,(48)⇒Iψ2=117m2−15m.


## 2.3. Results for Hexagonal Planar Octahedron (HPOH) Network

In this part, we propose the theorem for the HPOH network.Theorem 7.
The first and second modified K-Banhatti indices are equal to the hex POH network:(49)MK1Bψ3=518m2−4m−138,MK2Bψ3=8132m2−1516m−3932.Proof.
Letψ3≅HPOHm. From equation (1), we have(50)MK1Bψ=∑ab∈Eψ1da+db.
Using Table3, we have(51)MK1Bψ3=14+4E1HPOHm+14+8E2HPOHm+18+8E3HPOHm=18E1HPOHm+112E2HPOHm+116E3HPOHm=1818m2+18m−30+11236m2−48m+12+11618m2−36m+18.
We get the following value after calculations:(52)⇒MK1Bψ3=518m2−4m−138.
Letψ2≅HPOHm. From equation (2), we have(53)MK1Bψ=∑ab∈Eψ1da+db.
Using Table3, we have(54)MK2Bψ3=14×4E1HPOHm+14×8E2HPOHm+18×8E3HPOHm=116E1HPOHm+132E2HPOHm+164E3HPOHm=11618m2+18m−30+13236m2−48m+12+16418m2−36m+18.
We get the following value after calculations:(55)⇒MK2Bψ3=8132m2−1516m−3932.Table 3
Edge partition.
da,dbNumber of edgesE1=4,418m2+18m−30E2=4,836m2−48m+12E3=8,818m2−36m+18Theorem 8.
Then, harmonic K-Banhatti and symmetric division indices are equal to the hex POH network:(56)HKBψ3=514m2−8m−134,SDψ3=162m2−156m+6.Proof.
Letφ1≅HPOHm network, and from equation (3),(57)HKBψ=∑ab∈Eψ2da+db.
Using Table3, we have(58)HKBψ3=24+4E1HPOHm+24+8E2HPOHm+28+8E3HPOHm=28E1HPOHm+212E2HPOHm+216E3HPOHm=1418m2+18m−30+1636m2−48m+12+1818m2−36m+18.
We get the following value after calculations:(59)⇒HKBψ3=514m2−8m−134.
For the symmetric division index of a graph using equation (3), we have(60)SDψ=∑ab∈Eψdadb+dbda.
Using Table3, we have(61)SDψ3=44+44E1HPOHm+48+84E2HPOHm+88+88E3HPOHm=2E1HPOHm+12+21E2HPOHm+2E3HPOHm=218m2+18m−30+5236m2−48m+12+218m2−36m+18.
After calculations,(62)⇒SDψ3=162m2−156m+6.Theorem 9.
The augmented Zagreb and inverse sum indices are equal to the hex POH network:(63)AGψ3=416820224128625m2−600773632128625m+595764224385875,Iψ3=204m2−236m+44.Proof.
Letφ1≅HPOHm network. Using equation (5) for the augmented Zagreb index,(64)AGψ=∑ab∈Eψda×dbda+db−23.
Using Table3, we have(65)AGψ3=1663E1HPOHm+32103E2HPOHm+64143E3HPOHm=166318m2+18m−30+3210336m2−48m+12+6414318m2−36m+18.
We get the following value after calculations:(66)⇒AGψ3=416820224128625m2−600773632128625m+595764224385875.
For the inverse sum index and by using equation (6), we have(67)Iψ=∑ab∈Eψda×dbda+db.
Using Table3, we have(68)Iψ3=168E1HPOHm+3212E2HPOHm+6416E3HPOHm=218m2+18m−30+321236m2−48m+12+418m2−36m+18.
After calculations,(69)⇒Iψ3=204m2−236m+44.


## 3. Comparison of Indices through Graphs

The comparison of the first and second K-Banhatti, harmonic K-Banhatti, symmetric division, augmented Zagreb, and inverse sum indices for thePOH network, TP network, and HPOH network is conducted for different values. The comparison graphs are shown in Figures 4–9 .Figure 4
For POH 1.Figure 5
For POH 2.Figure 6
For TP 1.Figure 7
For TP 2.Figure 8
For hex POH 1.Figure 9
For hex POH 2.


## 4. Conclusion

In this paper, first and the second K-Banhatti, harmonic K-Banhatti, symmetric division, augmented Zagreb, and inverse sum indices have been computed for the planar octahedron networks. From a chemical standpoint, these findings might be useful for computer scientists and chemists, who come across these networks. Additional multiplicative degree-based indices should be computed soon.



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