Topological Properties of Crystallographic Structure of Molecules

Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of the crystal structure of titanium difluoride TiF2 and the crystallographic structure of cuprite Cu2O. Furthermore, we compute degree-based topological indices, mainly ABC, GA, ABC4, GA5 and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride TiF2 and the crystallographic structure of cuprite Cu2O.


Introduction
Graph theory is one of the most special and unique branches of mathematics by which the demonstration of any structure is made conceivable. Recently, it has attained much attention among researchers because of its wide range of applications in computer science, electrical networks, interconnected networks, biological networks, chemistry, etc. The chemical graph theory CGT is a fast growing area among researchers. It helps in understanding the structural properties of a molecular graph. There are many chemical compounds that possess a variety of applications in the fields of commercial, industrial, pharmaceutical chemistry and daily life and in the laboratory.
A relationship exists between chemical compounds and their molecular structures. The manipulation and examination of chemical structural information is made conceivable using molecular descriptors. Chemical graph theory is a branch of mathematical chemistry in which the tools of graph theory are applied to model the chemical phenomenon mathematically. Furthermore, it relates to the nontrivial applications of graph theory for solving molecular problems. This theory contributes to a prominent role in the field of chemical sciences; see for details [1][2][3].
Chem-informatics is a new subject, which is a combination of chemistry, mathematics and information science. It examines the quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR), which are utilized to predict the bioactivity and physicochemical properties of chemical compounds [4]. The field of chemical graph theory has attained much attention and consideration among researchers [5,6].
In solid state physics, the electrons of a single, isolated atom occupy atomic orbitals, each of which has a discrete energy level. When two atoms join together to form a molecule, their atomic orbitals overlap [7]. The Pauli exclusion principle dictates that no two electrons can have the same quantum numbers in a molecule. Therefore, if two identical atoms combine to form a diatomic molecule, each atomic orbital splits into two molecular orbitals of different energy, allowing the electrons in the former atomic orbitals to occupy the new orbital structure without any having the same energy. Similarly if a large number N of identical atoms come together to form a solid, such as a crystal lattice, the atoms' atomic orbitals overlap [8]. Since the Pauli exclusion principle dictates that no two electrons in the solid have the same quantum numbers, each atomic orbital splits into N discrete molecular orbitals, each with a different energy.
In chemical graph, the vertices represent atoms, and edges refer to the chemical bonds in the underlying chemical structure. A topological index is a numerical value that is computed mathematically from the molecular graph. It is associated with the chemical constitution indicating the correlation of the chemical structure with many physical, chemical properties and biological activities. The exact formulas of topological indices of certain chemical graphs have been computed and plotted in [9,10].
Let G = (V, E) be a graph where V is the vertex set and E is the edge set of G. The degree deg(t) (or d t ) of v is the number of edges of G incident with t. The length of the shortest path in a graph G is a distance d(s, t) between s and t. A graph can be represented by a polynomial, a numerical value or by matrix form. There are certain types of topological indices, mainly eccentric-based, degree-based, distance-based indices, etc. In this paper, we deal with degree-based topological indices.
The first and oldest degree-based index was introduced by Milan Randić [11] in 1975 and is defined in the following equation.
In 1988, Bollobás et al. [12] and Amic et al. [13] proposed the general Randić index independently. For more details about the Randić index, its properties and important results, see [14,15]. The general Randić index is defined as: The atom bond connectivity index is of vital importance and was introduced by Estrada et al. [16]. It is defined as: The geometric arithmetic index GA of a graph G was introduced by Vukičević et al. [17]. It is defined as: The first Zagreb index was introduced in 1972 by [18]. Later on, the second Zagreb index was introduced by [19]. The first and second Zagreb indices are formulated as: The fourth version of the atom bond connectivity index ABC 4 of a graph G was introduced by Ghorbhani et al. [20]. It is defined as: Another molecular descriptor was the fifth version of the geometric arithmetic index GA 5 of a graph G introduced by Graovac et al. [21]. It is defined as:

Research Aim
Our aim in this article is to compute the additive topological indices, mainly the atom bond connectivity index, geometric arithmetic index, fourth atom bond connectivity index ABC 4 , fifth geometric arithmetic index GA 5 and general Randić index R α , for α = {−1, 1, 1 2 , − 1 2 } for Cu 2 O[m, n, t] and TiF 2 [m, n, t]. Moreover, the graphical representation of these exact result is depicted for further explanation of the behavior of these topological indices.

Applications of Topological Indices
The atom-bond connectivity (ABC) index provides a very good correlation for the stability of linear alkanes, as well as the branched alkanes and for computing the strain energy of cyclo alkanes [22]. The Randi/'c index is a topological descriptor that has been correlated with many chemical characteristics of molecules and has been found to the parallel to computing the boiling point and Kovats constants of the molecules. To correlate with certain physicochemical properties, the GA index has much better predictive power than the predictive power of the Randić connectivity index [23,24]. The first and second Zagreb index were found to occur for the computation of the total π-electron energy of the molecules within specific approximate expressions [25]. These are among the graph invariants, which were proposed for the measurement of the skeleton of the branching of the carbon-atom [26].

Crystallographic Structure of the Molecule Cu 2 O
Among various transition metal oxides, Cu 2 O has attracted much attention in recent years owing to its distinguished properties and non-toxic nature, low-cost, abundance and simple fabrication process [27]. Nowadays, the promising applications of Cu 2 O mainly focus on chemical sensors, solar cells, photocatalysis, lithium-ion batteries and catalysis [28]. The chemical graph of the crystallographic structure of Cu 2 O is described in Figures 1 and 2; see details in [29]. Let G ∼ = Cu 2 O[m, n, t] be the chemical graph of Cu 2 O with m × n unit cells in the plane and t layers. We construct this graph first by taking m × n units in the mn-plane and then storing it up in t layers. The number of vertices and edges of Cu 2 O[m, n, t] is (m + 1)(n + 1)(t + 1) + 5mnt and 8mnt, respectively. In Cu 2 O[m, n, t], the number of vertices of degree zero is four; the number of vertices of degree one is 4m + 4n + 4t − 8; the number of vertices of degree two is 4mnt + 2mn + 2mt + 2nt − 4n − 4m − 4t + 6; and the number of vertices of degree four is 2nmt − nm − nt − mt + n + m + t − 1. Furthermore, the edge partition of Cu 2 O[m, n, t] based on the degrees of end vertices of each edge is depicted in Table 1.
Proof. Let G be the crystallographic structure of Cu 2 O[m, n, t]. The general Randić index, The result for the atom bond connectivity index is as follows: Theorem 3. Consider the graph of G ∼ = Cu 2 O[m, n, t] with m, n, t ≥ 1, then its geometric arithmetic index is equal to, Proof. Let G be the crystallographic structure of Cu 2 O[m, n, t]. The geometric arithmetic index is computed as below: Proof. Let G be the crystallographic structure of Cu 2 O[m, n, t]. The first Zagreb index is computed as below: The second Zagreb index is computed as below: Theorem 5. Consider the graph G ∼ = Cu 2 O[m, n, t] with m, n, t ≥ 2, then its fourth atom bond connectivity index is equal to, Proof. Let G be the crystallographic structure of Cu 2 O[m, n, t]. The fourth atom bond connectivity index is computed by using Table 2 in the following equation.
Theorem 6. Consider the graph G ∼ = Cu 2 O[m, n, t] with m, n, t ≥ 2, then its fifth geometric arithmetic index is equal to, Proof. Let G be the crystallographic structure of Cu 2 O[m, n, t]. The fifth geometric arithmetic index is computed as below:

Crystal Structure of Titanium Difluoride
Titanium difluoride is a water-insoluble titanium source for use in oxygen-sensitive applications, such as metal production. Fluoride compounds have diverse applications in current technologies and science, from oil refining and etching to synthetic organic chemistry and the manufacture of pharmaceuticals. The chemical graph of the crystal structure of titanium difluoride TiF 2 [m, n, t] is described in Figure 3; for more details, see [30]. Let G ∼ = TiF 2 [m, n, t] be the chemical graph of TiF 2 with m × n unit cells in the plane and t layers. We construct this graph first by taking m × n units in the mn−plane and then storing it up in t layers. The number of vertices and edges of TiF 2 [m, n, t] is 12mnt + 2mn + 2mt + 2nt + m + n + t + 1 and 32mnt, respectively.In TiF 2 [m, n, t], the number of vertices of degree one is eight; the number of vertices of degree two is 4m + 4n + 4t − 12; the number of vertices of degree four is 8mnt + 4mn + 4mt + 4nt − 4n − 4m − 4t + 6; and the number of vertices of degree eight is 4mnt − 2(mn + mt + nt) + m + n + t − 1. The edge partition of TiF 2 [m, n, t] based on the degrees of end vertices of each edge is depicted in Table 3.
Theorem 7. Consider the graph G ∼ = TiF 2 [m, n, t] with m, n, t ≥ 1, then its general Randić index is equal to, Proof. Let G ∼ = TiF 2 [m, n, t] be the crystal structure of titanium difluoride. The general Randić index, Theorem 8. Consider the graph G ∼ = TiF 2 [m, n, t] with m, n, t ≥ 1, then its atom bond connectivity index is equal to, Proof. Let G ∼ = TiF 2 [m, n, t] be the crystal structure of titanium difluoride. The atom bond connectivity index can be calculated by using Table 3 in the following equation.
Proof. Let G ∼ = TiF 2 [m, n, t] be the crystal structure of titanium difluoride. The first and second Zagreb indices are computed as below: Table 4 shows the edge partition of the chemical graph TiF 2 [m, n, t] based on the degree sum of the end vertices of each edge.
Theorem 11. Consider the graph G ∼ = TiF 2 [m, n, t] with m, n, t > 1, then its fourth atom bond connectivity index is equal to, Proof. Let G ∼ = TiF 2 [m, n, t] be the crystal structure of titanium difluoride. The fourth atom bond connectivity index is computed as below: − 12192 325

Discussion
Since the topological indices have many applications in different branches of science, namely pharmaceutical, chemistry and biological drugs, the graphical representation of these calculated results is helpful to scientists. The graphical representations of topological indices for      The graphical representations of topological indices for titanium difluoride TiF 2 are depicted for Randić indices in Figures 9 and 10. The atomic bond connectivity index and geometric arithmetic index for titanium difluoride TiF 2 are depicted in Figure 11. The first and second Zagreb indices for titanium difluoride TiF 2 are depicted in Figure 12. The fourth atomic bond connectivity index and the fifth geometric arithmetic index for titanium difluoride TiF 2 are depicted in Figure 13.

Conclusions
In this paper, we have computed some degree-based topological indices, namely the atom bond connectivity index ABC, the geometric arithmetic index GA, the general Randić index, the GA 5 index, the ABC 4 index and the first and second Zagreb indices for the chemical graph of the crystal structure of titanium difluoride TiF 2 and crystallographic structure of cuprite Cu 2 O.
In the future, we are interested in computing the distance-based and counting-related topological indices for these structures.