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

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. ,ere 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.


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][2][3][4][5][6][7][8][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. e vertex degree is the number of edges attached to that vertex [10][11][12][13][14][15][16]. e 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 C y , defined as C y � Δ(G) − d g (v) + 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. ese 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. ere 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. e first and second reverse Zagreb indices are as follows: (1) 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, geometricarithmetic 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:

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 Cu 2 O. e unit cell of the crystallographic structure of the molecule Cu 2 O is given in Figure 1 and the crystal structure of Cu 2 O [3, 3, 3] is given in Figure 2.
Proof. From Figure 2, we can say that there are 3 types of edges in Cu 2 O: We have |E 1 (Cu 2 O)| � 4α + 4m + 4t − 8, |E 2 (Cu 2 O)| � 4αm + 4αt + 4mt − 8α − 8m − 8t + 12, and |E 3 (Cu 2 O)| � 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: e reverse edge set of Cu 2 O is given as follows: We have |CE 1 (i) e first reverse ZI for Cu 2 O is given by 2 Journal of Chemistry (ii) e second reverse ZI for Cu 2 O is given by  e first and second reverse Zagreb polynomials for Cu 2 O with m, n, t ≥ 1 are as follows: Proof. Now, by the reverse edge partitions of Cu 2 O, we have the following results: (i) e first reverse Zagreb polynomial for Cu 2 O is given as follows: (ii) e second reverse Zagreb polynomial for Cu 2 O, with m, α, t ≥ 1, is given as follows: □ Theorem 3. e first and second reverse hyper-Zagreb indices of silicon-carbon Cu 2 O with m, α, t ≥ 1 are as follows: Proof. Let G be a graph of Cu 2 O. en, by reverse edge partition and definition of reverse hyper-Zagreb indices, we have the following results: (i) e first reverse hyper-ZI for Cu 2 O is given by Journal of Chemistry (ii) e second reverse hyper-ZI for Cu 2 O is given by □ Theorem 4. e first and second reverse hyper-Zagreb polynomials of Cu 2 O with m, α, t ≥ 1 are as follows: Proof. Now, by the reverse edge partitions for Cu 2 O, we have the following results: (i) e first reverse Zagreb polynomial for Cu 2 O is given as follows: (ii) e second reverse Zagreb polynomial for Cu 2 O is given as follows: □ Journal of Chemistry Proof. By the reverse edge partition, we have the following results: (i) e reverse atom-bond connectivity index for Cu 2 O is given by (ii) e reverse geometric-arithmetic index for Cu 2 O is given by □ e values of calculated topological indices of Cu 2 O at different levels are given in Table 1.

Titanium Difluoride TiF 2 [m, α, t].
e unit cell of crystallographic structure of titanium difluoride TiF 2 [m, α, t] is given in Figure 3 and the crystal structure of TiF 2 [1, 2, 4] is given in Figure 4.

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][27][28] and study the topology of the crystallographic structure of molecules and can be used in drug delivery [29][30][31].

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

Conflicts of Interest
e authors declare that they have no conflicts of interest.