# Reverse Zagreb and Reverse Hyper-Zagreb Indices for Crystallographic Structure of Molecules

Authors: Zhen Wang; Faryal Chaudhry; Maria Naseem; Adnan Asghar
Journal: Journal of Chemistry
 (2020)

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## 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. Topological indices help us collect information about algebraic graphs and give us mathematical approach to understand the properties of algebraic structures. With the help of topological indices, we can guess the properties of chemical compounds without performing experiments in wet lab. There are more than 148 topological indices in the literature, but none of them completely give all properties of under study compounds. Together, they do it to some extent; hence, there is always room to introduce new indices. In this paper, we present first and second reserve Zagreb indices and first reverse hyper-Zagreb indices, reverse GA index, and reverse atomic bond connectivity index for the crystallographic structure of molecules. We also present first and second reverse Zagreb polynomials and first and second reverse hyper-Zagreb polynomials for the crystallographic structure of molecules.


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## Body

## 1. Introduction

Topological indices enable us to collect information about algebraic structures and give us a mathematical approach to understand the properties of algebraic structures. Here, we will discuss some newly introduced first and second reverse Zagreb indices, hyper-Zagreb indices, and their polynomials for the crystallographic structure of molecules [1–9].A graph having no loop or multiple edges is known as simple graph. A molecular graph is a simple graph in which atoms and bonds are represented by vertex and edge sets, respectively. The vertex degree is the number of edges attached to that vertex [10–16]. The maximum degree of vertex among the vertices of a graph is denoted by Δ(G). Kulli et al. [17] introduce the concept of reverse vertex degree Cy, defined as Cy=ΔG−dgv+1.In discrete mathematics, graph theory in general is not only the study of different properties of objects but it also tells us about objects having same properties as investigating object. These properties of different objects are of main interest. In particular, graph polynomials related to graph are rich in information. Mathematical tools like polynomials and topological-based numbers have significant importance to collect information about the properties of chemical compounds. We can find out many hidden information about compounds through these tools. Multifold graph polynomials are present in the literature. Actually, topological indices are numeric quantities that tell us about the whole structure of graph. There are many topological indices [18, 19] that help us to study physical, chemical reactivities, and biological properties. Wiener, in 1947 [20], firstly introduce the concept of topological index while working on boiling point. In particular, Hosoya polynomial [21] plays an important in the area of distance-based topological indices; we can find out Wiener index, hyper-Wiener index, and Tratch–Stankevich–Zefirov index by Hosoya polynomial [22, 23]. Other well-established polynomials are Zagreb and hyper-Zagreb polynomials introduced by Gao.The first and second reverse Zagreb indices are as follows:(1)CM1G=∑uvϵEGcu+cv,CM2G=∑uvϵEGcu.cv.Now, the first and second reverse hyper-Zagreb indices are given by(2)HCM1G=∑uvϵEGcu+cv2,HCM2G=∑uvϵEGcu.cv2.Atom-bond connectivity index can be abbreviated as ABC index. It is defined as follows:(3)ABCG=∑uvϵEGduG+dvG−2duG.dvG.Another degree-based topological index that utilizes the difference between the geometric and arithmetic means was invented by Vukicevic and Furtula, namely, geometric-arithmetic index and is defined as follows:(4)GAG=∑uvϵEGduG.dvG1/2duG+dvG.With the help of reverse Zagreb and hyper-Zagreb indices, we are now able to write the reverse Zagreb and hyper Zagreb polynomials:(5)CM1G,x=∑uvϵEGxcu+cv,CM2G,x=∑uvϵEGxcu.cv,HCM1G,x=∑uvϵEGxcu+cv2,HCM2G,x=∑uvϵEGxcu.cv2.We introduce the idea of reverse atom-bond connectivity index and reverse geometric-arithmetic index, and it is defined as follows:(6)CABCG=∑uvϵEGcuG+cvG−2cuG.cvG,CGAG=∑uvϵEGcuG.cvG1/2cuG+cvG.


## 2. Main Results

Here, we will compute reverse Zagreb and reverse hyper-Zagreb indices for the crystallographic structure of molecules.


### 2.1. Crystallographic Structure of the Molecule Cu2O

The unit cell of the crystallographic structure of the molecule Cu2O is given in Figure 1 and the crystal structure of Cu2O [3, 3, 3] is given in Figure 2.Figure 1
Unit cell of Cu2O [1, 1, 1].Figure 2
Crystal structure of Cu2O [3, 3, 3].Theorem 1.
LetG be the chemical graph of Cu2O, with m, α, t ≥ 1. The first and second reverse Zagreb indices are as follows:(1)
CM1(Cu2O) = 32mαt + 20mα + 20mt + 20αt + 36 − 28m − 28α − 28t(2)
CM2(Cu2O) = 24mαt + 24mα + 24mt + 24αt − 12m − 12α − 12tProof.
From Figure2, we can say that there are 3 types of edges in Cu2O:(7)E1Cu2O=uvϵECu2O;du=1,dv=2,E2Cu2O=uvϵECu2O;du=2,dv=2,E3Cu2O=uvϵECu2O;du=2,dv=4.
We have |E1(Cu2O)| = 4α + 4m + 4t − 8, |E2(Cu2O)| = 4αm + 4αt + 4mt − 8α − 8m − 8t + 12, and |E3(Cu2O)| = 4(2αmt − αm − αt − mt + α + m + t − 1). In this structure, the maximum edge degree is 4, and then, the reverse edges are given as follows:(8)cu=ΔG−dGu+1=5−dGu.
The reverse edge set of Cu2O is given as follows:(9)CE1Cu2O=uvϵECu2O;cu=4,cv=3,CE2Cu2O=uvϵECu2O;cu=3,cv=3,CE3Cu2O=uvϵECu2O;cu=3,cv=1.
We have |CE1(Cu2O)| = 4α + 4m + 4t − 8, |CE2(Cu2O)| = 4αm + 4αt + 4mt − 8α − 8m − 8t + 12, and |CE3(Cu2O)| = 4(2αmt − αm − αt − mt + α + m + t − 1).(i)
The first reverse ZI for Cu2O is given by(10)CM1Cu2O=∑uvϵEGcu+cv=4+34α+4m+4t−8+3+34αm+4αt+4mt−8α−8m−8t+12+3+142αmt−αm−αt−mt+α+m+t−1=32mαt+20mα+20mt+20αt+36−28m−28α−28t.(ii)
The second reverse ZI for Cu2O is given by(11)CM2Cu2O=∑uvϵEGcu.cv=4×34α+4m+4t−8+3×34αm+4αt+4mt−8α−8m−8t+12+3×142αmt−αm−αt−mt+α+m+t−1=24mαt+24mα+24mt+24αt−12m−12α−12t.Theorem 2.
The first and second reverse Zagreb polynomials for Cu2O with m, n, t ≥ 1 are as follows:(12)1.CM1Cu2O,x=x74α+4m+4t−8+x64αm+4αt+4mt−8α−8m−8t+12+x442αmt−αm−αt−mt+α+m+t−1,2.CM2Cu2O,x=x124α+4m+4t−8+x94αm+4αt+4mt−8α−8m−8t+12+x342αmt−αm−αt−mt+α+m+t−1.Proof.
Now, by the reverse edge partitions of Cu2O, we have the following results:(i)
The first reverse Zagreb polynomial for Cu2O is given as follows:(13)CM1Cu2O,x=∑uvϵEGxcu+cv=4α+4m+4t−8x3+4+4αm+4αt+4mt−8α−8m−8t+12x3+3+42αmt−αm−αt−mt+α+m+t−1x3+1=x74α+4m+4t−8+x64αm+4αt+4mt−8α−8m−8t+12+x442αmt−αm−αt−mt+α+m+t−1.(ii)
The second reverse Zagreb polynomial for Cu2O, with m, α, t ≥ 1, is given as follows:(14)CM2Cu2O,x=∑uvϵEGxcu.cv=4α+4m+4t−8x3×4+4αm+4αt+4mt−8α−8m−8t+12x3×3+42αmt−αm−αt−mt+α+m+t−1x3×1=x124α+4m+4t−8+x94αm+4αt+4mt−8α−8m−8t+12+x342αmt−αm−αt−mt+α+m+t−1.Theorem 3.
The first and second reverse hyper-Zagreb indices of silicon-carbon Cu2O with m, α, t ≥ 1 are as follows:(1)
HCM1(Cu2O) = 32mαt + 128mα + 128mt + 128αt − 76m − 76α − 76t + 24(2)
HCM2(Cu2O) = 18mαt + 315mα + 315mt + 315αt − 63m − 63α − 63t − 189Proof.
LetG be a graph of Cu2O. Then, by reverse edge partition and definition of reverse hyper-Zagreb indices, we have the following results:(i)
The first reverse hyper-ZI for Cu2O is given by(15)CM1Cu2O=∑uvϵEGcu+cv2=4+324α+4m+4t−8+3+324αm+4αt+4mt−8α−8m−8t+12+3+1242αmt−αm−αt−mt+α+m+t−1=32mαt+128mα+128mt+128αt−76m−76α−76t+24.(ii)
The second reverse hyper-ZI for Cu2O is given by(16)CM2Cu2O=∑uvϵEGcu.cv2=4×324α+4m+4t−8+3×324αm+4αt+4mt−8α−8m−8t+12+3×1242αmt−αm−αt−mt+α+m+t−1=18mαt+315mα+315mt+315αt−63m−63α−63t−189.Theorem 4.
The first and second reverse hyper-Zagreb polynomials of Cu2O with m, α, t ≥ 1 are as follows:(17)1.HCM1Cu2O,x=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1,2.HCM1Cu2O,x=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1.Proof.
Now, by the reverse edge partitions for Cu2O, we have the following results:(i)
The first reverse Zagreb polynomial for Cu2O is given as follows:(18)HCM1Cu2O,x=∑uvϵEGxcu+cv2=4α+4m+4t−8x3+42+4αm+4αt+4mt−8α−8m−8t+12x3+32+42αmt−αm−αt−mt+α+m+t−1x3+12=x494α+4m+4t−8+x364αm+4αt+4mt−8α−8m−8t+12+x1642αmt−αm−αt−mt+α+m+t−1.(ii)
The second reverse Zagreb polynomial for Cu2O is given as follows:(19)HCM2Cu2O,x=∑uvϵEGxcu.cv2=4α+4m+4t−8x3×42+4αm+4αt+4mt−8α−8m−8t+12x3×32+42αmt−αm−αt−mt+α+m+t−1x3×12=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1.Theorem 5.
LetG be the graph of Cu2O with m, α, t ≥ 1. The reverse atom-bond connectivity index and reverse geometric-arithmetic index for Cu2O with m, α, t ≥ 1 are as follows:(20)1.CABCCu2O=1386mαt+8−326mα+mt+αt+215−16+326m+α+t+24−1615−326,2.CGACu2O=114563mαt+56−283mα+αt+mt+−112+393m+α+t+168−713.Proof.
By the reverse edge partition, we have the following results:(i)
The reverse atom-bond connectivity index for Cu2O is given by(21)CABCCu2O=∑uvϵEGcuG+cvG−2cuG.cvG=4m+4α+4t−84+3−24×3+4mα+4αt+4mt−8m−8α−8t+123+3−23×3+42mαt−mα−mt−αt+m+α+t−13+1−23×1=1386mαt+8−326mα+mt+αt+215−16+326m+α+t+24−1615−326.(ii)
The reverse geometric-arithmetic index for Cu2O is given by(22)CGACu2O=∑uvϵEGcuG.cvG1/2cuG+cvG=4m+4α+4t−84×31/24+3+4mα+4αt+4mt−8m−8α−8t+123×31/23+3+42mαt−mα−mt−αt+m+α+t−13×11/23+1=114563mαt+56−283mα+αt+mt+−112+393m+α+t+168−713.The values of calculated topological indices of Cu2O at different levels are given in Table 1.Table 1
Values of calculated topological indices of Cu2O at different levels.
m = 1α = 1t = 1m = 2α = 2t = 2m = 3α = 3t = 3m = 1α = 2t = 3m = 3α = 2t = 1m = 3α = 4m = 5m = 2α = 4t = 6First reverse ZI44364118828028025602116Second reverse ZI60408118833633624242064First reverse hyper-ZI212136036601168116870486280Second reverse hyper-ZI58533578235300630061494013779Reverse ABC index−252.9−117.5−285.5−117.4−117.4−469.1−430.1Reverse GA index22246.02176.1731.6331.63405.96321.24


### 2.2. Titanium Difluoride TiF2[m, α, t]

The unit cell of crystallographic structure of titanium difluoride TiF2[m, α, t] is given in Figure 3 and the crystal structure of TiF2 [1, 2, 4] is given in Figure 4.Figure 3
Unit cell of TiF2[m, α, t].Figure 4
Crystal structure of TiF2 [1, 2, 4].Theorem 6.
LetG be the graph of titanium difluoride TiF2[m, α, t], with m, α, t ≥ 1. The first and second reverse Zagreb indices are as follows:(1)
CM1(TiF2[m, α, t]) = 192mαt + 64mα + 64mt + 64αt − 16m − 16α − 16t + 8(2)
CM2(TiF2[m, α, t]) = 160mαt + 320mα + 320mt + 320αt − 80m − 80α − 80t + 40Proof.
LetG be a graph of titanium difluoride TiF2[m, α, t]. The vertex and edge sets of titanium difluoride TiF2[m, α, t] are |V(TiF2[m, α, t])| = 12mαt + 2mα + 2mt + 2αt + m + α + t + 1 and |E(TiF2[m, α, t])| = 32mαt, respectively. From Figure 4, we can say that there are five type of edges in TiF2[m, α, t]. The edge set of TiF2m,α,t=32mαt is portioned into four edge sets:(23)E1TiF2m,α,t=uvϵETiF2m,α,t;du=1,dv=4,E2TiF2m,α,t=uvϵETiF2m,α,t;du=2,dv=4,E3TiF2m,α,t=uvϵETiF2m,α,t;du=4,dv=4,E4TiF2m,α,t=uvϵETiF2m,α,t;du=4,dv=8.
We have |E1(TiF2[m, α, t])| = 8, |E2(TiF2[m, α, t])| = 8(m + α + t − 3), |E3(TiF2[m, α, t])| = 16(mα + αt + mt) − 16(m + α + t) + 24, and |E4(TiF2[m, α, t])| = 32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8. The maximum edge degree is 8; then, the reverse edges are given as follows:(24)cu=ΔG−dGu+1=9−dGu.
The reverse edge set of TiF2[m, α, t] is given as follows:(25)CE1TiF2m,α,t=uvεETiF2m,α,t;cu=8,cv=5,CE2TiF2m,α,t=uvεETiF2m,α,t;cu=7,cv=5,CE3TiF2m,α,t=uvεETiF2m,α,t;cu=5,cv=5,CE4TiF2m,α,t=uvεETiF2m,α,t;cu=5,cv=1.
We have |E1(TiF2[m, α, t])| = 8, |E2(TiF2[m, α, t])| = 8(m + α + t − 3), |E3(TiF2[m, α, t])| = 16(mα + αt + mt) − 16(m + α + t) + 24, and |E4(TiF2[m, α, t])| = 32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8.(i)
The first reverse ZI for TiF2[m, α, t] is given by(26)CM1TiF2m,α,t=∑uvεEGcu+cv=8+58+7+58m+α+t−3+5+516mn+αt+mt−16m+α+t+24+5+132mαt−16mt+mα+αt+8m+α+t−8=192mαt+64mα+64mt+64αt−16m−16α−16t+8.(ii)
The second reverse ZI for TiF2[m, α, t] is given by(27)CM2TiF2m,α,t=∑uvϵEGcu.cv=8×58+7×58m+α+t−3+5×516mα+αt+mt−16m+α+t+24+5×132mαt−16mt+mα+αt+8m+α+t−8=160mαt+320mα+320mt+320αt−80m−80α−80t+40.Theorem 7.
The first and second reverse Zagreb polynomials for TiF2[m, α, t] are as follows:(1)
CM1(TiF2[m, α, t], x) = 8x13 + [8(m + α + t − 3)]x12 + [16(mα + αt + mt) − 16(m + α + t) + 24]x10 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x6(2)
CM2(TiF2[m, α, t], x) = 8x40 + [8(m + α + t − 3)]x35 + [16(mα + αt + mt) − 16(m + α + t) + 24]x25 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x5Proof.
Now, by the reverse edge partitions for TiF2[m, α, t], we have the following results:(i)
The first reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(28)CM1TiF2m,α,t,x=∑uvϵEGxcu+cv=8x8+5+8m+α+t−3x7+5+16mα+αt+mt−16m+α+t+24x5+5+32mαt−16mt+mα+αt+8m+α+t−8x5+1=8x13+8m+α+t−3x12+16mα+αt+mt−16m+α+t+24x10+32mαt−16mt+mα+αt+8m+α+t−8x6.(ii)
The second reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(29)CM2TiF2m,α,t,x=∑uvϵEGxcu.cv=8x8×5+8m+α+t−3x7×5+16mα+αt+mt−16m+α+t+24x5×5+32mαt−16mt+mα+αt+8m+α+t−8x5×1=8x40+8m+α+t−3x35+16mα+αt+mt−16m+α+t+24x25+32mαt−16mt+mα+αt+8m+α+t−8x5.Theorem 8.
The first and second reverse hyper-Zagreb indices of TiF2[m, α, t] are as follows:(1)
HCM1(TiF2[m, α, t]) = 1152mαt− 160(m + α + t) + 1024(mα + αt + mt) − 2152(2)
HCM2(TiF2[m, α, t]) = 800mαt + 9600(mα + αt + mt) − 14600Proof.
LetG be a graph of silicon-carbon TiF2[m, α, t]. Then, by reverse edge partition and definition of reverse hyper-Zagreb indices, we have the following results:(i)
The first reverse hyper-ZI for TiF2[m, α, t] is given by(30)CM1TiF2m,α,t=∑uvϵEGcu+cv2=8+528+7+528m+α+t−3+5+5216mα+αt+mt−16m+α+t+24+5+1232mαt−16mt+mα+αt+8m+α+t−8=1152mαt−160m+α+t+1024mα+αt+mt−2152.(ii)
The second reverse hyper-ZI for TiF2[m, α, t] is given by(31)CM2TiF2m,α,t=∑uvϵEGcu.cv2=8×528+7×528m+α+t−3+5×5216mα+αt+mt−16m+α+t+24+5×1232mαt−16mt+mα+αt+8m+α+t−8=800mαt+9600mα+αt+mt−14600.Theorem 9.
The first and second reverse hyper-Zagreb polynomials of TiF2[m, α, t] are as follows:(1)
HCM1(TiF2[m, α, t], x) = 8x169 + [8(m + α + t − 3)]x144 + [16(mα + αt + mt) − 16(m + α + t) + 24]x100 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x36(2)
HCM1(TiF2[m, α, t], x) = 8x1600 + [8(m + α + t − 3)]x1225 + [16(mα + αt + mt) − 16(m + α + t) + 24]x625 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x25Proof.
Now, by the reverse edge partitions for TiF2[m, α, t], we have the following results:(i)
The first reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(32)CM1TiF2m,α,t,x=∑uvϵEGxcu+cv2=8x8+52+8m+α+t−3x7+52+16mα+αt+mt−16m+α+t+24x5+52+32mαt−16mt+mα+αt+8m+α+t−8x5+12=8x169+8m+α+t−3x144+16mα+αt+mt−16m+α+t+24x100+32mαt−16mt+mα+αt+8m+α+t−8x36.(ii)
The second reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(33)CM2TiF2m,α,t,x=∑uvϵEGxcu.cv2=8x8×52+8m+α+t−3x7×52+16mα+αt+mt−16m+α+t+24x5×52+32mαt−16mt+mα+αt+8m+α+t−8x5×12=8x1600+8m+α+t−3x1225+16mα+αt+mt−16m+α+t+24x625+32mαt−16mt+mα+αt+8m+α+t−8x25.Theorem 10.
LetG be the graph of Cu2O with m, α, t ≥ 1. The reverse atom-bond connectivity index and reverse geometric-arithmetic index for Cu2O with m, α, t ≥ 1 are given by(34)1CABCTiF2m,α,t=117522405mαt+12252−12255mα+mt+αt+40350−22450+5605m+α+t+70110−600014+16802−28005,2CGATiF2m,α,t=13325mαt+48−165mα+mt+αt+435+85−48m+α+t+1399610−1045+15635+936.Proof.
By the reverse edge partition, we have the following results:(i)
The reverse atom-bond connectivity index for TiF2[m, α, t] is given by(35)CABCTiF2m,α,t=∑uvϵEGcuG+cvG−2cuG.cvG=88+5−28×5+8m+α+t−37+5−27×5+16mα+αt+mt−16m+α+t+245+5−25×5+32mαt−16mα+mt+αt+8m+α+t−85+1−25×1=117522405mαt+12252−12255mα+mt+αt+40350−22450+5605m+α+t+70110−600014+16802−28005.(ii)
The reverse geometric-arithmetic index for TiF2[m, α, t] is given by(36)CGATiF2m,α,t=∑uvϵEGcuG.cvG1/2cuG+cvG=88×51/28+5+8m+α+t−37×51/27+5+16mα+αt+mt−16m+α+t+245×51/25+5+32mαt−16mα+mt+αt+8m+α+t−85×11/25+1=13325mαt+48−165mα+mt+αt+435+85−48m+α+t+1399610−1045+15635+936.The values of calculated topological indices at different levels are given in Table2.Table 2
Values of calculated topological indices of TiF2[m, α, t] at different levels.
m = 1α = 1t = 1m = 2α = 2t = 2m = 3α = 3t = 3m = 1α = 2t = 3m = 3α = 2t = 1m = 3α = 4m = 5m = 2α = 4t = 6First reverse ZI34422166776176817681434411848Second reverse ZI920468012280404040402372020840First reverse hyper-ZI15921839255160150641506411317696280Second reverse hyper-ZI150001070002662009580095800484600446200Reverse ABC index-127.827.9492.6-23.57-23.61346.4860.3Reverse GA index79.1276.3784.1224.5224.51634.11241.3


## 2.1. Crystallographic Structure of the Molecule Cu2O

The unit cell of the crystallographic structure of the molecule Cu2O is given in Figure 1 and the crystal structure of Cu2O [3, 3, 3] is given in Figure 2.Figure 1
Unit cell of Cu2O [1, 1, 1].Figure 2
Crystal structure of Cu2O [3, 3, 3].Theorem 1.
LetG be the chemical graph of Cu2O, with m, α, t ≥ 1. The first and second reverse Zagreb indices are as follows:(1)
CM1(Cu2O) = 32mαt + 20mα + 20mt + 20αt + 36 − 28m − 28α − 28t(2)
CM2(Cu2O) = 24mαt + 24mα + 24mt + 24αt − 12m − 12α − 12tProof.
From Figure2, we can say that there are 3 types of edges in Cu2O:(7)E1Cu2O=uvϵECu2O;du=1,dv=2,E2Cu2O=uvϵECu2O;du=2,dv=2,E3Cu2O=uvϵECu2O;du=2,dv=4.
We have |E1(Cu2O)| = 4α + 4m + 4t − 8, |E2(Cu2O)| = 4αm + 4αt + 4mt − 8α − 8m − 8t + 12, and |E3(Cu2O)| = 4(2αmt − αm − αt − mt + α + m + t − 1). In this structure, the maximum edge degree is 4, and then, the reverse edges are given as follows:(8)cu=ΔG−dGu+1=5−dGu.
The reverse edge set of Cu2O is given as follows:(9)CE1Cu2O=uvϵECu2O;cu=4,cv=3,CE2Cu2O=uvϵECu2O;cu=3,cv=3,CE3Cu2O=uvϵECu2O;cu=3,cv=1.
We have |CE1(Cu2O)| = 4α + 4m + 4t − 8, |CE2(Cu2O)| = 4αm + 4αt + 4mt − 8α − 8m − 8t + 12, and |CE3(Cu2O)| = 4(2αmt − αm − αt − mt + α + m + t − 1).(i)
The first reverse ZI for Cu2O is given by(10)CM1Cu2O=∑uvϵEGcu+cv=4+34α+4m+4t−8+3+34αm+4αt+4mt−8α−8m−8t+12+3+142αmt−αm−αt−mt+α+m+t−1=32mαt+20mα+20mt+20αt+36−28m−28α−28t.(ii)
The second reverse ZI for Cu2O is given by(11)CM2Cu2O=∑uvϵEGcu.cv=4×34α+4m+4t−8+3×34αm+4αt+4mt−8α−8m−8t+12+3×142αmt−αm−αt−mt+α+m+t−1=24mαt+24mα+24mt+24αt−12m−12α−12t.Theorem 2.
The first and second reverse Zagreb polynomials for Cu2O with m, n, t ≥ 1 are as follows:(12)1.CM1Cu2O,x=x74α+4m+4t−8+x64αm+4αt+4mt−8α−8m−8t+12+x442αmt−αm−αt−mt+α+m+t−1,2.CM2Cu2O,x=x124α+4m+4t−8+x94αm+4αt+4mt−8α−8m−8t+12+x342αmt−αm−αt−mt+α+m+t−1.Proof.
Now, by the reverse edge partitions of Cu2O, we have the following results:(i)
The first reverse Zagreb polynomial for Cu2O is given as follows:(13)CM1Cu2O,x=∑uvϵEGxcu+cv=4α+4m+4t−8x3+4+4αm+4αt+4mt−8α−8m−8t+12x3+3+42αmt−αm−αt−mt+α+m+t−1x3+1=x74α+4m+4t−8+x64αm+4αt+4mt−8α−8m−8t+12+x442αmt−αm−αt−mt+α+m+t−1.(ii)
The second reverse Zagreb polynomial for Cu2O, with m, α, t ≥ 1, is given as follows:(14)CM2Cu2O,x=∑uvϵEGxcu.cv=4α+4m+4t−8x3×4+4αm+4αt+4mt−8α−8m−8t+12x3×3+42αmt−αm−αt−mt+α+m+t−1x3×1=x124α+4m+4t−8+x94αm+4αt+4mt−8α−8m−8t+12+x342αmt−αm−αt−mt+α+m+t−1.Theorem 3.
The first and second reverse hyper-Zagreb indices of silicon-carbon Cu2O with m, α, t ≥ 1 are as follows:(1)
HCM1(Cu2O) = 32mαt + 128mα + 128mt + 128αt − 76m − 76α − 76t + 24(2)
HCM2(Cu2O) = 18mαt + 315mα + 315mt + 315αt − 63m − 63α − 63t − 189Proof.
LetG be a graph of Cu2O. Then, by reverse edge partition and definition of reverse hyper-Zagreb indices, we have the following results:(i)
The first reverse hyper-ZI for Cu2O is given by(15)CM1Cu2O=∑uvϵEGcu+cv2=4+324α+4m+4t−8+3+324αm+4αt+4mt−8α−8m−8t+12+3+1242αmt−αm−αt−mt+α+m+t−1=32mαt+128mα+128mt+128αt−76m−76α−76t+24.(ii)
The second reverse hyper-ZI for Cu2O is given by(16)CM2Cu2O=∑uvϵEGcu.cv2=4×324α+4m+4t−8+3×324αm+4αt+4mt−8α−8m−8t+12+3×1242αmt−αm−αt−mt+α+m+t−1=18mαt+315mα+315mt+315αt−63m−63α−63t−189.Theorem 4.
The first and second reverse hyper-Zagreb polynomials of Cu2O with m, α, t ≥ 1 are as follows:(17)1.HCM1Cu2O,x=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1,2.HCM1Cu2O,x=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1.Proof.
Now, by the reverse edge partitions for Cu2O, we have the following results:(i)
The first reverse Zagreb polynomial for Cu2O is given as follows:(18)HCM1Cu2O,x=∑uvϵEGxcu+cv2=4α+4m+4t−8x3+42+4αm+4αt+4mt−8α−8m−8t+12x3+32+42αmt−αm−αt−mt+α+m+t−1x3+12=x494α+4m+4t−8+x364αm+4αt+4mt−8α−8m−8t+12+x1642αmt−αm−αt−mt+α+m+t−1.(ii)
The second reverse Zagreb polynomial for Cu2O is given as follows:(19)HCM2Cu2O,x=∑uvϵEGxcu.cv2=4α+4m+4t−8x3×42+4αm+4αt+4mt−8α−8m−8t+12x3×32+42αmt−αm−αt−mt+α+m+t−1x3×12=x1444α+4m+4t−8+x814αm+4αt+4mt−8α−8m−8t+12+x942αmt−αm−αt−mt+α+m+t−1.Theorem 5.
LetG be the graph of Cu2O with m, α, t ≥ 1. The reverse atom-bond connectivity index and reverse geometric-arithmetic index for Cu2O with m, α, t ≥ 1 are as follows:(20)1.CABCCu2O=1386mαt+8−326mα+mt+αt+215−16+326m+α+t+24−1615−326,2.CGACu2O=114563mαt+56−283mα+αt+mt+−112+393m+α+t+168−713.Proof.
By the reverse edge partition, we have the following results:(i)
The reverse atom-bond connectivity index for Cu2O is given by(21)CABCCu2O=∑uvϵEGcuG+cvG−2cuG.cvG=4m+4α+4t−84+3−24×3+4mα+4αt+4mt−8m−8α−8t+123+3−23×3+42mαt−mα−mt−αt+m+α+t−13+1−23×1=1386mαt+8−326mα+mt+αt+215−16+326m+α+t+24−1615−326.(ii)
The reverse geometric-arithmetic index for Cu2O is given by(22)CGACu2O=∑uvϵEGcuG.cvG1/2cuG+cvG=4m+4α+4t−84×31/24+3+4mα+4αt+4mt−8m−8α−8t+123×31/23+3+42mαt−mα−mt−αt+m+α+t−13×11/23+1=114563mαt+56−283mα+αt+mt+−112+393m+α+t+168−713.The values of calculated topological indices of Cu2O at different levels are given in Table 1.Table 1
Values of calculated topological indices of Cu2O at different levels.
m = 1α = 1t = 1m = 2α = 2t = 2m = 3α = 3t = 3m = 1α = 2t = 3m = 3α = 2t = 1m = 3α = 4m = 5m = 2α = 4t = 6First reverse ZI44364118828028025602116Second reverse ZI60408118833633624242064First reverse hyper-ZI212136036601168116870486280Second reverse hyper-ZI58533578235300630061494013779Reverse ABC index−252.9−117.5−285.5−117.4−117.4−469.1−430.1Reverse GA index22246.02176.1731.6331.63405.96321.24


## 2.2. Titanium Difluoride TiF2[m, α, t]

The unit cell of crystallographic structure of titanium difluoride TiF2[m, α, t] is given in Figure 3 and the crystal structure of TiF2 [1, 2, 4] is given in Figure 4.Figure 3
Unit cell of TiF2[m, α, t].Figure 4
Crystal structure of TiF2 [1, 2, 4].Theorem 6.
LetG be the graph of titanium difluoride TiF2[m, α, t], with m, α, t ≥ 1. The first and second reverse Zagreb indices are as follows:(1)
CM1(TiF2[m, α, t]) = 192mαt + 64mα + 64mt + 64αt − 16m − 16α − 16t + 8(2)
CM2(TiF2[m, α, t]) = 160mαt + 320mα + 320mt + 320αt − 80m − 80α − 80t + 40Proof.
LetG be a graph of titanium difluoride TiF2[m, α, t]. The vertex and edge sets of titanium difluoride TiF2[m, α, t] are |V(TiF2[m, α, t])| = 12mαt + 2mα + 2mt + 2αt + m + α + t + 1 and |E(TiF2[m, α, t])| = 32mαt, respectively. From Figure 4, we can say that there are five type of edges in TiF2[m, α, t]. The edge set of TiF2m,α,t=32mαt is portioned into four edge sets:(23)E1TiF2m,α,t=uvϵETiF2m,α,t;du=1,dv=4,E2TiF2m,α,t=uvϵETiF2m,α,t;du=2,dv=4,E3TiF2m,α,t=uvϵETiF2m,α,t;du=4,dv=4,E4TiF2m,α,t=uvϵETiF2m,α,t;du=4,dv=8.
We have |E1(TiF2[m, α, t])| = 8, |E2(TiF2[m, α, t])| = 8(m + α + t − 3), |E3(TiF2[m, α, t])| = 16(mα + αt + mt) − 16(m + α + t) + 24, and |E4(TiF2[m, α, t])| = 32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8. The maximum edge degree is 8; then, the reverse edges are given as follows:(24)cu=ΔG−dGu+1=9−dGu.
The reverse edge set of TiF2[m, α, t] is given as follows:(25)CE1TiF2m,α,t=uvεETiF2m,α,t;cu=8,cv=5,CE2TiF2m,α,t=uvεETiF2m,α,t;cu=7,cv=5,CE3TiF2m,α,t=uvεETiF2m,α,t;cu=5,cv=5,CE4TiF2m,α,t=uvεETiF2m,α,t;cu=5,cv=1.
We have |E1(TiF2[m, α, t])| = 8, |E2(TiF2[m, α, t])| = 8(m + α + t − 3), |E3(TiF2[m, α, t])| = 16(mα + αt + mt) − 16(m + α + t) + 24, and |E4(TiF2[m, α, t])| = 32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8.(i)
The first reverse ZI for TiF2[m, α, t] is given by(26)CM1TiF2m,α,t=∑uvεEGcu+cv=8+58+7+58m+α+t−3+5+516mn+αt+mt−16m+α+t+24+5+132mαt−16mt+mα+αt+8m+α+t−8=192mαt+64mα+64mt+64αt−16m−16α−16t+8.(ii)
The second reverse ZI for TiF2[m, α, t] is given by(27)CM2TiF2m,α,t=∑uvϵEGcu.cv=8×58+7×58m+α+t−3+5×516mα+αt+mt−16m+α+t+24+5×132mαt−16mt+mα+αt+8m+α+t−8=160mαt+320mα+320mt+320αt−80m−80α−80t+40.Theorem 7.
The first and second reverse Zagreb polynomials for TiF2[m, α, t] are as follows:(1)
CM1(TiF2[m, α, t], x) = 8x13 + [8(m + α + t − 3)]x12 + [16(mα + αt + mt) − 16(m + α + t) + 24]x10 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x6(2)
CM2(TiF2[m, α, t], x) = 8x40 + [8(m + α + t − 3)]x35 + [16(mα + αt + mt) − 16(m + α + t) + 24]x25 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x5Proof.
Now, by the reverse edge partitions for TiF2[m, α, t], we have the following results:(i)
The first reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(28)CM1TiF2m,α,t,x=∑uvϵEGxcu+cv=8x8+5+8m+α+t−3x7+5+16mα+αt+mt−16m+α+t+24x5+5+32mαt−16mt+mα+αt+8m+α+t−8x5+1=8x13+8m+α+t−3x12+16mα+αt+mt−16m+α+t+24x10+32mαt−16mt+mα+αt+8m+α+t−8x6.(ii)
The second reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(29)CM2TiF2m,α,t,x=∑uvϵEGxcu.cv=8x8×5+8m+α+t−3x7×5+16mα+αt+mt−16m+α+t+24x5×5+32mαt−16mt+mα+αt+8m+α+t−8x5×1=8x40+8m+α+t−3x35+16mα+αt+mt−16m+α+t+24x25+32mαt−16mt+mα+αt+8m+α+t−8x5.Theorem 8.
The first and second reverse hyper-Zagreb indices of TiF2[m, α, t] are as follows:(1)
HCM1(TiF2[m, α, t]) = 1152mαt− 160(m + α + t) + 1024(mα + αt + mt) − 2152(2)
HCM2(TiF2[m, α, t]) = 800mαt + 9600(mα + αt + mt) − 14600Proof.
LetG be a graph of silicon-carbon TiF2[m, α, t]. Then, by reverse edge partition and definition of reverse hyper-Zagreb indices, we have the following results:(i)
The first reverse hyper-ZI for TiF2[m, α, t] is given by(30)CM1TiF2m,α,t=∑uvϵEGcu+cv2=8+528+7+528m+α+t−3+5+5216mα+αt+mt−16m+α+t+24+5+1232mαt−16mt+mα+αt+8m+α+t−8=1152mαt−160m+α+t+1024mα+αt+mt−2152.(ii)
The second reverse hyper-ZI for TiF2[m, α, t] is given by(31)CM2TiF2m,α,t=∑uvϵEGcu.cv2=8×528+7×528m+α+t−3+5×5216mα+αt+mt−16m+α+t+24+5×1232mαt−16mt+mα+αt+8m+α+t−8=800mαt+9600mα+αt+mt−14600.Theorem 9.
The first and second reverse hyper-Zagreb polynomials of TiF2[m, α, t] are as follows:(1)
HCM1(TiF2[m, α, t], x) = 8x169 + [8(m + α + t − 3)]x144 + [16(mα + αt + mt) − 16(m + α + t) + 24]x100 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x36(2)
HCM1(TiF2[m, α, t], x) = 8x1600 + [8(m + α + t − 3)]x1225 + [16(mα + αt + mt) − 16(m + α + t) + 24]x625 + [32mαt − 16(mt + mα + αt) + 8(m + α + t) − 8]x25Proof.
Now, by the reverse edge partitions for TiF2[m, α, t], we have the following results:(i)
The first reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(32)CM1TiF2m,α,t,x=∑uvϵEGxcu+cv2=8x8+52+8m+α+t−3x7+52+16mα+αt+mt−16m+α+t+24x5+52+32mαt−16mt+mα+αt+8m+α+t−8x5+12=8x169+8m+α+t−3x144+16mα+αt+mt−16m+α+t+24x100+32mαt−16mt+mα+αt+8m+α+t−8x36.(ii)
The second reverse Zagreb polynomial for TiF2[m, α, t] is given as follows:(33)CM2TiF2m,α,t,x=∑uvϵEGxcu.cv2=8x8×52+8m+α+t−3x7×52+16mα+αt+mt−16m+α+t+24x5×52+32mαt−16mt+mα+αt+8m+α+t−8x5×12=8x1600+8m+α+t−3x1225+16mα+αt+mt−16m+α+t+24x625+32mαt−16mt+mα+αt+8m+α+t−8x25.Theorem 10.
LetG be the graph of Cu2O with m, α, t ≥ 1. The reverse atom-bond connectivity index and reverse geometric-arithmetic index for Cu2O with m, α, t ≥ 1 are given by(34)1CABCTiF2m,α,t=117522405mαt+12252−12255mα+mt+αt+40350−22450+5605m+α+t+70110−600014+16802−28005,2CGATiF2m,α,t=13325mαt+48−165mα+mt+αt+435+85−48m+α+t+1399610−1045+15635+936.Proof.
By the reverse edge partition, we have the following results:(i)
The reverse atom-bond connectivity index for TiF2[m, α, t] is given by(35)CABCTiF2m,α,t=∑uvϵEGcuG+cvG−2cuG.cvG=88+5−28×5+8m+α+t−37+5−27×5+16mα+αt+mt−16m+α+t+245+5−25×5+32mαt−16mα+mt+αt+8m+α+t−85+1−25×1=117522405mαt+12252−12255mα+mt+αt+40350−22450+5605m+α+t+70110−600014+16802−28005.(ii)
The reverse geometric-arithmetic index for TiF2[m, α, t] is given by(36)CGATiF2m,α,t=∑uvϵEGcuG.cvG1/2cuG+cvG=88×51/28+5+8m+α+t−37×51/27+5+16mα+αt+mt−16m+α+t+245×51/25+5+32mαt−16mα+mt+αt+8m+α+t−85×11/25+1=13325mαt+48−165mα+mt+αt+435+85−48m+α+t+1399610−1045+15635+936.The values of calculated topological indices at different levels are given in Table2.Table 2
Values of calculated topological indices of TiF2[m, α, t] at different levels.
m = 1α = 1t = 1m = 2α = 2t = 2m = 3α = 3t = 3m = 1α = 2t = 3m = 3α = 2t = 1m = 3α = 4m = 5m = 2α = 4t = 6First reverse ZI34422166776176817681434411848Second reverse ZI920468012280404040402372020840First reverse hyper-ZI15921839255160150641506411317696280Second reverse hyper-ZI150001070002662009580095800484600446200Reverse ABC index-127.827.9492.6-23.57-23.61346.4860.3Reverse GA index79.1276.3784.1224.5224.51634.11241.3


## 3. Conclusion

In this paper, we computed first and second reverse Zagreb indices, first and second reverse hyper-Zagreb indices, reverse GA index, reverse atomic bond connectivity index, first and second reverse Zagreb polynomials, and first and second reverse hyper-Zagreb polynomials for the crystallographic structure of molecules [24, 25]. Our results are important to guess the properties [26–28] and study the topology of the crystallographic structure of molecules and can be used in drug delivery [29–31].



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