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

Wang, Zhen; Chaudhry, Faryal; Naseem, Maria; Asghar, Adnan

doi:https://doi.org/10.1155/2020/9805829

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Volume 2020 | Article ID 9805829 | https://doi.org/10.1155/2020/9805829

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

Zhen Wang,¹Faryal Chaudhry,²Maria Naseem,³and Adnan Asghar²

Guest Editor: Jia-Bao Liu

Received09 Oct 2019

Accepted08 Feb 2020

Published13 Mar 2020

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.

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 , defined as .

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:

Now, the first and second reverse hyper-Zagreb indices are given by

Atom-bond connectivity index can be abbreviated as ABC index. It is defined as follows:

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:

With the help of reverse Zagreb and hyper-Zagreb indices, we are now able to write the reverse Zagreb and hyper Zagreb polynomials:

We introduce the idea of reverse atom-bond connectivity index and reverse geometric-arithmetic index, and it is defined as follows:

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.

Theorem 1. Let G 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 − 12t

Proof. From Figure 2, we can say that there are 3 types of edges in Cu₂O: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:The reverse edge set of Cu₂O is given as follows: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(ii)The second reverse ZI for Cu₂O is given by

Theorem 2. The first and second reverse Zagreb polynomials for Cu₂O with m, n, t ≥ 1 are as follows:

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:(ii)The second reverse Zagreb polynomial for Cu₂O, with m, α, t ≥ 1, is given as follows:

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 − 189

Proof. Let G 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(ii)The second reverse hyper-ZI for Cu₂O is given by

Theorem 4. The first and second reverse hyper-Zagreb polynomials of Cu₂O with m, α, t ≥ 1 are as follows:

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:(ii)The second reverse Zagreb polynomial for Cu₂O is given as follows:

Theorem 5. Let G 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:

Proof. By the reverse edge partition, we have the following results:(i)The reverse atom-bond connectivity index for Cu₂O is given by(ii)The reverse geometric-arithmetic index for Cu₂O is given by

The values of calculated topological indices of Cu₂O at different levels are given in Table 1.

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.

Theorem 6. Let G 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 + 40

Proof. Let G 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 is portioned into four edge sets: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:The reverse edge set of TiF₂[m, α, t] is given as follows: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(ii)The second reverse ZI for TiF₂[m, α, t] is given by

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]x⁵

Proof. 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:(ii)The second reverse Zagreb polynomial for TiF₂[m, α, t] is given as follows:

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) − 14600

Proof. Let G 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(ii)The second reverse hyper-ZI for TiF₂[m, α, t] is given by

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]x²⁵

Proof. 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:(ii)The second reverse Zagreb polynomial for TiF₂[m, α, t] is given as follows:

Theorem 10. Let G 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

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(ii)The reverse geometric-arithmetic index for TiF₂[m, α, t] is given by

The values of calculated topological indices at different levels are given in Table 2.

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].

Data Availability

All data required for this paper are included within this paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to this paper.

Acknowledgments

This research was supported by the Natural Science Fund of Anhui Wenda University of Information Engineering (Grant no. XZR2019A10).

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Journal of Chemistry

Mathematical Tools for Solving Problems of Chemical Structure Generation

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

Abstract

1. Introduction

2. Main Results

2.1. Crystallographic Structure of the Molecule Cu2O

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

3. Conclusion

Data Availability

Conflicts of Interest

Authors’ Contributions

Acknowledgments

References

Copyright

2.1. Crystallographic Structure of the Molecule Cu₂O

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