Topological Properties of Crystallographic Structure of Molecules

Abstract

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 $Ti{F}_2$ and the crystallographic structure of cuprite $C{u}_{2}O$. Furthermore, we compute degree-based topological indices, mainly $ABC$, $GA$, $AB{C}_4$, $G{A}_5$ and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride $Ti{F}_2$ and the crystallographic structure of cuprite $C{u}_{2}O$.

1. 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 $\left(QSAR\right)$ and quantitative structure-property relationship $\left(QSPR\right)$, 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 $deg\left(t\right)$ (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.

$$R_{-\frac{1}{2}}\left(G\right)=\sum _{st\in E\left(G\right)}\frac{1}{\sqrt{d_s{d}_t}}$$

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:

$$R_{\alpha }\left(G\right)=\sum _{st\in E\left(G\right)}{(d_s{d}_t)}^{\alpha }$$

The atom bond connectivity index is of vital importance and was introduced by Estrada et al. [16]. It is defined as:

$$ABC\left(G\right)=\sum _{st\in E\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}$$

The geometric arithmetic index $GA$ of a graph G was introduced by Vukičević et al. [17]. It is defined as:

$$GA\left(G\right)=\sum _{st\in E\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}$$

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:

$$M_1\left(G\right)=\sum _{st\in E\left(G\right)}(d_s+d_t)$$

$$M_2\left(G\right)=\sum _{st\in E\left(G\right)}(d_s{d}_t)$$

The fourth version of the atom bond connectivity index $AB{C}_4$ of a graph G was introduced by Ghorbhani et al. [20]. It is defined as:

$$AB{C}_4\left(G\right)=\sum _{st\in E\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}$$

where $S_s=\sum _{st\in E\left(G\right)}{d}_t$ and $S_t=\sum _{st\in E\left(G\right)}{d}_s$.

Another molecular descriptor was the fifth version of the geometric arithmetic index $G{A}_5$ of a graph G introduced by Graovac et al. [21]. It is defined as:

$$G{A}_5\left(G\right)=\sum _{st\in E\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}$$

2. 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 $AB{C}_4$, fifth geometric arithmetic index $G{A}_5$ and general Randić index $R_{\alpha }$, for $\alpha =\{-1,1,\frac{1}{2},-\frac{1}{2}\}$ for $C{u}_{2}O[m,n,t]$ and $Ti{F}_2[m,n,t]$. Moreover, the graphical representation of these exact result is depicted for further explanation of the behavior of these topological indices.

3. Applications of Topological Indices

The atom-bond connectivity $\left(ABC\right)$ 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 $\pi$-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].

4. Crystallographic Structure of the Molecule ${\mathit{Cu}}_{\mathbf{2}}\mathit{O}$

Among various transition metal oxides, $C{u}_{2}O$ 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 $C{u}_{2}O$ mainly focus on chemical sensors, solar cells, photocatalysis, lithium-ion batteries and catalysis [28]. The chemical graph of the crystallographic structure of $C{u}_{2}O$ is described in Figure 1 and Figure 2; see details in [29]. Let $G\cong C{u}_{2}O[m,n,t]$ be the chemical graph of $C{u}_{2}O$ with $m\times n$ unit cells in the plane and t layers. We construct this graph first by taking $m\times n$ units in the $mn$-plane and then storing it up in t layers. The number of vertices and edges of $C{u}_{2}O[m,n,t]$ is $(m+1)(n+1)(t+1)+5mnt$ and $8mnt$, respectively. In $C{u}_{2}O[m,n,t]$, 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 $C{u}_{2}O[m,n,t]$ based on the degrees of end vertices of each edge is depicted in Table 1.

Figure 1. Crystallographic structure of the molecule $C{u}_{2}O$. (a) Structural characteristics of $Cu$ and O atoms in the $C{u}_{2}O$ lattice. The $C{u}_{2}O$ lattice is formed by interpenetrating the $Cu$ and O lattices with each other. (b) Unit cell of $C{u}_{2}O$. Copper atoms are shown as small blue spheres, and oxygen atoms are shown as large red spheres. In the $C{u}_{2}O$ lattice, each $Cu$ atom is coordinated with two O atoms, and each O atom is coordinated with four $Cu$ atoms.

Figure 2. (a) Unit cell of $C{u}_{2}O[1,1,1]$ (b) Crystallographic structure of $C{u}_{2}O[3,2,3].$

Table 1. Edge partition of $C{u}_{2}O[m,n,t]$ based on the degrees of end vertices of each edge.

$({\mathit{d}}_{\mathit{s}},{\mathit{d}}_{\mathit{t}})$	Frequency	Set of Edges
$(1,2)$	$4n+4m+4t-8$	$E_1$
$(2,2)$	$4nm+4nt+4mt-8n-8m-8t+12$	$E_2$
$(2,4)$	$4(2nmt-nm-nt-mt+n+m+t-1)$	$E_3$

Theorem 1.

Consider the graph of $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 1$, then its general Randić index is equal to,

$$R_{\alpha }G=\left\{\begin{array}{cc}8\left[8mnt-2(mn+mt+nt)+m+n+t\right],\hfill & if \alpha =1,\hfill \\ \frac{1}{2}(2mnt+mn+mt+nt+m+n+t-3),\hfill & if \alpha =-1,\hfill \\ 4(4\sqrt{2}mnt+2(1-\sqrt{2})(mn+mt+nt)\hfill & \\ +(3\sqrt{2}-4)(m+n+t)-4\sqrt{2}+6),\hfill & if \alpha =\frac{1}{2},\hfill \\ 2\sqrt{2}mnt+(2-\sqrt{2})(mn+mt+nt)\hfill & \\ +(3\sqrt{2}-4)(m+n+t)-5\sqrt{2}+6,\hfill & if \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The general Randić index,

For $\alpha =1$.

$$\begin{array}{ccc}\hfill R_1\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s\times d_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s\times d_t)+\sum _{st\in E_2\left(G\right)}(d_s\times d_t)+\sum _{st\in E_3\left(G\right)}(d_s\times d_t)\hfill \\ & =& (4m+4n+4t-8)(1\times 2)+(4mn+4mt+4nt-8m-8n-8t+12)(2\times 2)\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)(2\times 4)\hfill \\ & =& 8\left[8mnt-2(mn+mt+nt)+m+n+t\right]\hfill \end{array}$$

For $\alpha =-1$,

$$\begin{array}{ccc}\hfill R_{-1}\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{1}{(d_s\times d_t)}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{1}{(d_s\times d_t)}+\sum _{st\in E_2\left(G\right)}\frac{1}{(d_s\times d_t)}+\sum _{st\in E_3\left(G\right)}\frac{1}{(d_s\times d_t)}\hfill \\ & =& (4m+4n+4t-8)\frac{1}{(1\times 2)}+(4mn+4mt+4nt-8m-8n-8t+12)\frac{1}{(2\times 2)}\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)\frac{1}{(2\times 4)}\hfill \\ & =& \frac{1}{2}(2mnt+mn+mt+nt+m+n+t-3)\hfill \end{array}$$

For $\alpha =\frac{1}{2}$,

$$\begin{array}{ccc}\hfill R_{\frac{1}{2}}\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{(d_s\times d_t)}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{(d_s\times d_t)}+\sum _{st\in E_2\left(G\right)}\sqrt{(d_s\times d_t)}+\sum _{st\in E_3\left(G\right)}\sqrt{(d_s\times d_t)}\hfill \\ & =& (4m+4n+4t-8)\sqrt{(1\times 2)}+(4mn+4mt+4nt-8m-8n-8t+12)\sqrt{(2\times 2)}\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)\sqrt{(2\times 4)}\hfill \\ & =& 4\left(4\sqrt{2}mnt+2(1-\sqrt{2})(mn+mt+nt)+(3\sqrt{2}-4)(m+n+t)-4\sqrt{2}+6\right)\hfill \end{array}$$

For $\alpha =-\frac{1}{2}$,

$$\begin{array}{ccc}\hfill R_{-\frac{1}{2}}\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}+\sum _{st\in E_2\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}+\sum _{st\in E_3\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}\hfill \\ & =& (4m+4n+4t-8)\frac{1}{\sqrt{(1\times 2)}}+(4mn+4mt+4nt-8m-8n-8t+12)\frac{1}{\sqrt{(2\times 2)}}\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)\frac{1}{\sqrt{(2\times 4)}}\hfill \\ & =& 2\sqrt{2}mnt+(2-\sqrt{2})(mn+mt+nt)+(3\sqrt{2}-4)(m+n+t)-5\sqrt{2}+6\hfill \end{array}$$

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Theorem 2.

Consider the graph of $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 1$, then its atom bond connectivity index is equal to,

$$ABC\left(G\right)=4\sqrt{2}mnt.$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The result for the atom bond connectivity index is as follows:

$$\begin{array}{ccc}\hfill ABC\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}+\sum _{st\in E_2\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}+\sum _{st\in E_3\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}\hfill \\ & =& (4m+4n+4t-8)\sqrt{\frac{1+2-2}{1\times 2}}+(4mn+4mt+4nt-8m-8n-8t+12)\sqrt{\frac{2+2-2}{2\times 2}}\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)\sqrt{\frac{2+4-2}{2\times 4}}\hfill \\ & =& 4\sqrt{2}mnt.\hfill \end{array}$$

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Theorem 3.

Consider the graph of $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 1$, then its geometric arithmetic index is equal to,

$$GA\left(G\right)=4\left[\frac{4\sqrt{2}mnt}{3}-\left(\frac{2\sqrt{2}-3}{3}\right)(mn+mt+nt-2m-2n-2t)-2\sqrt{2}+3\right].$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The geometric arithmetic index is computed as below:

$$\begin{array}{ccc}\hfill GA\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}+\sum _{st\in E_2\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}+\sum _{st\in E_3\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}\hfill \\ & =& (4m+4n+4t-8)\frac{2\sqrt{1\times 2}}{1+2}+(4mn+4mt+4nt-8m-8n-8t+12)\frac{2\sqrt{2\times 2}}{2+2}\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)\frac{2\sqrt{2\times 4}}{2+4}\hfill \\ & =& 4\left[\frac{4\sqrt{2}mnt}{3}-\left(\frac{2\sqrt{2}-3}{3}\right)(mn+mt+nt-2m-2n-2t)-2\sqrt{2}+3\right]\hfill \end{array}$$

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Theorem 4.

Consider the graph of $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 1$, then its first and second Zagreb indices are equal to,

$$\begin{array}{ccc}\hfill M_1\left(G\right)& =& 4\left(12mnt-2(mn+mt+nt)+m+n+t\right)\hfill \\ \hfill M_2\left(G\right)& =& 8\left(8mnt-2(mn+mt+nt)+m+n+t\right).\hfill \end{array}$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The first Zagreb index is computed as below:

$$\begin{array}{ccc}\hfill M_1\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s+d_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s+d_t)+\sum _{st\in E_2\left(G\right)}(d_s+d_t)+\sum _{st\in E_3\left(G\right)}(d_s+d_t)\hfill \\ & =& (4m+4n+4t-8)(1+2)+(4mn+4mt+4nt-8m-8n-8t+12)(2+2)\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)(2+4)\hfill \\ & =& 4\left(12mnt-2(mn+mt+nt)+m+n+t\right)\hfill \end{array}$$

The second Zagreb index is computed as below:

$$\begin{array}{ccc}\hfill M_2\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s{d}_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s{d}_t)+\sum _{st\in E_2\left(G\right)}(d_s{d}_t)+\sum _{st\in E_3\left(G\right)}(d_s{d}_t)\hfill \\ & =& (4m+4n+4t-8)(1\times 2)+(4mn+4mt+4nt-8m-8n-8t+12)(2\times 2)\hfill \\ & +& (8mnt-4mn-4mt-4nt+4m+4n+4t-4)(2\times 4)\hfill \\ & =& 8\left(8mnt-2(mn+mt+nt)+m+n+t\right)\hfill \end{array}$$

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Table 2 shows the edge partition of the chemical graph $C{u}_{2}O[m,n,t]$ based on the degree sum of end vertices of each edge.

Table 2. Edge partition of $Cu2O[m,n,t]$ with $m,n,t\ge 2$ based on the degree sum of end vertices of each edge.

$({\mathit{S}}_{\mathit{s}},{\mathit{S}}_{\mathit{t}})$	Frequency	Set of Edges
$(2,4)$	$4m+4n+4t-8$	$E_1$
$(4,6)$	$4mn+4mt+4nt-8m-8n-8t+12$	$E_2$
$(5,8)$	$4n+4m+4t-8$	$E_3$
$(6,8)$	$4mn+4mt+4nt-8m-8n-8t+12$	$E_4$
$(8,8)$	$8mnt-8mn-8mt-8nt+8m+8n+8t-8$	$E_5$

Theorem 5.

Consider the graph $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 2$, then its fourth atom bond connectivity index is equal to,

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& \sqrt{14}mnt+\left(\frac{4}{\sqrt{3}}-\sqrt{14}+2\right)(mn+mt+nt)-4\sqrt{2}+4\sqrt{3}-\sqrt{14}-\frac{2\sqrt{110}}{5}+6\hfill \\ & +& \left(2\sqrt{2}-\frac{8}{\sqrt{3}}+\sqrt{14}+\frac{\sqrt{110}}{5}-4\right)(m+n+t).\hfill \end{array}$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The fourth atom bond connectivity index is computed by using Table 2 in the following equation.

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_2\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_3\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \\ & +& \sum _{st\in E_4\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_5\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& (4m+4n+4t-8)\sqrt{\frac{2+4-2}{2\times 4}}+(4mn+4mt+4nt-8m-8n-8t+12)\sqrt{\frac{4+6-2}{4\times 6}}\hfill \\ & +& (4m+4n+4t-8)\sqrt{\frac{5+8-2}{5\times 8}}+(4mn+4mt+4nt-8m-8n-8t+12)\sqrt{\frac{6+8-2}{6\times 8}}\hfill \\ & +& (8mnt-8mn-8mt-8nt+8m+8n+8t-8)\sqrt{\frac{8+8-2}{8\times 8}}\hfill \\ & =& \sqrt{14}mnt+\left(\frac{4}{\sqrt{3}}-\sqrt{14}+2\right)(mn+mt+nt)-4\sqrt{2}+4\sqrt{3}-\sqrt{14}-\frac{2\sqrt{110}}{5}+6\hfill \\ & +& \left(2\sqrt{2}-\frac{8}{\sqrt{3}}+\sqrt{14}+\frac{\sqrt{110}}{5}-4\right)(m+n+t)\hfill \end{array}$$

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Theorem 6.

Consider the graph $G\cong C{u}_{2}O[m,n,t]$ with $m,n,t\ge 2$, then its fifth geometric arithmetic index is equal to,

$$\begin{array}{ccc}\hfill G{A}_5\left(G\right)& =& 8mnt+\left(\frac{16\sqrt{3}}{7}+\frac{8\sqrt{6}}{5}-8\right)(mn+mt+nt)-\frac{16\sqrt{2}}{3}+\frac{48\sqrt{3}}{7}+\frac{24\sqrt{6}}{5}-\frac{32\sqrt{10}}{13}-8\hfill \\ & +& \left(\frac{8\sqrt{2}}{3}-\frac{32\sqrt{3}}{7}-\frac{16\sqrt{6}}{5}+\frac{16\sqrt{10}}{13}+8\right)(m+n+t)\hfill \end{array}$$

Proof. 

Let G be the crystallographic structure of $C{u}_{2}O[m,n,t]$. The fifth geometric arithmetic index is computed as below:

$$\begin{array}{ccc}\hfill G{A}_5\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_2\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_3\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & +& \sum _{st\in E_4\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_5\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & =& (4m+4n+4t-8)\frac{2\sqrt{2\times 4}}{2+4}+(4mn+4mt+4nt-8m-8n-8t+12)\frac{2\sqrt{4\times 6}}{4+6}\hfill \\ & +& (4m+4n+4t-8)\frac{2\sqrt{5\times 8}}{5+8}+(4mn+4mt+4nt-8m-8n-8t+12)\frac{2\sqrt{6\times 8}}{6+8}\hfill \\ & +& (8mnt-8mn-8mt-8nt+8m+8n+8t-8)\frac{2\sqrt{8\times 8}}{8+8}\hfill \\ & =& 8mnt+\left(\frac{16\sqrt{3}}{7}+\frac{8\sqrt{6}}{5}-8\right)(mn+mt+nt)-\frac{16\sqrt{2}}{3}+\frac{48\sqrt{3}}{7}+\frac{24\sqrt{6}}{5}-\frac{32\sqrt{10}}{13}-8\hfill \\ & +& \left(\frac{8\sqrt{2}}{3}-\frac{32\sqrt{3}}{7}-\frac{16\sqrt{6}}{5}+\frac{16\sqrt{10}}{13}+8\right)(m+n+t)\hfill \end{array}$$

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5. 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 $Ti{F}_2[m,n,t]$ is described in Figure 3; for more details, see [30]. Let $G\cong Ti{F}_2[m,n,t]$ be the chemical graph of $Ti{F}_2$ with $m\times n$ unit cells in the plane and t layers. We construct this graph first by taking $m\times n$ units in the $mn$-plane and then storing it up in t layers. The number of vertices and edges of $Ti{F}_2[m,n,t]$ is $12mnt+2mn+2mt+2nt+m+n+t+1$ and $32mnt$, respectively.In $Ti{F}_2[m,n,t]$, 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 $Ti{F}_2[m,n,t]$ based on the degrees of end vertices of each edge is depicted in Table 3.

Figure 3. (a)The unit cell of of $Ti{F}_2[m,n,t]$ with $Ti$ atoms in red and F atoms in green; (b) the crystal structure of $Ti{F}_2[4,1,2]$.

Table 3. Edge partition of $Ti{F}_2[m,n,t]$ based on the degrees of end vertices of each edge.

$({\mathit{d}}_{\mathit{s}},{\mathit{d}}_{\mathit{t}})$	Frequency	Set of Edges
$(1,4)$	8	$E_1$
$(2,4)$	$8(m+n+t-3)$	$E_2$
$(4,4)$	$16(mn+mt+nt)-16(m+n+t)+24$	$E_3$
$(4,8)$	$32mnt-16(mt+mn+nt)+8(m+n+t)-8$	$E_4$

Theorem 7.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t\ge 1$, then its general Randić index is equal to,

$$R_{\alpha }G=\left\{\begin{array}{cc}32\left[32mnt-8(mn+mt+nt)+2(m+n+t)-1\right],\hfill & if \alpha =1,\hfill \\ \frac{1}{4}(4mnt+2(mn+mt+nt)+m+n+t+1),\hfill & if \alpha =-1,\hfill \\ 16(8\sqrt{2}mnt+4(1-\sqrt{2})(mn+mt+nt)\hfill & \\ +(3\sqrt{2}-4)(m+n+t)-5\sqrt{2}+7),\hfill & if \alpha =\frac{1}{2},\hfill \\ 4\sqrt{2}mnt+2(2-\sqrt{2})(mn+mt+nt)\hfill & \\ +(3\sqrt{2}-4)(m+n+t)-7\sqrt{2}+10,\hfill & if \alpha =-\frac{1}{2}.\hfill \end{array}\right.$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The general Randić index,

For $\alpha =1$.

$$\begin{array}{ccc}\hfill R_1\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s\times d_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s\times d_t)+\sum _{st\in E_2\left(G\right)}(d_s\times d_t)+\sum _{st\in E_3\left(G\right)}(d_s\times d_t)+\sum _{st\in E_4\left(G\right)}(d_s\times d_t)\hfill \\ & =& \left(8\right)(1\times 4)+(8m+8n+8t-24)(2\times 4)\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)(4\times 4)\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)(4\times 8)\hfill \\ & =& 32\left[32mnt-8(mn+mt+nt)+2(m+n+t)-1\right]\hfill \end{array}$$

For $\alpha =-1$,

$$\begin{array}{ccc}\hfill R_{-1}\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{1}{(d_s\times d_t)}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{1}{(d_s\times d_t)}+\sum _{st\in E_2\left(G\right)}\frac{1}{(d_s\times d_t)}+\sum _{st\in E_3\left(G\right)}\frac{1}{(d_s\times d_t)}+\sum _{st\in E_4\left(G\right)}\frac{1}{(d_s\times d_t)}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill R_{-1}\left(G\right)& =& \left(8\right)\frac{1}{(1\times 4)}+(8m+8n+8t-24)\frac{1}{(2\times 4)}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)\frac{1}{(4\times 4)}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)\frac{1}{(4\times 8)}\hfill \\ & =& \frac{1}{4}(4mnt+2(mn+mt+nt)+m+n+t+1)\hfill \end{array}$$

For $\alpha =\frac{1}{2}$,

$$\begin{array}{ccc}\hfill R_{\frac{1}{2}}\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{(d_s\times d_t)}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{(d_s\times d_t)}+\sum _{st\in E_2\left(G\right)}\sqrt{(d_s\times d_t)}+\sum _{st\in E_3\left(G\right)}\sqrt{(d_s\times d_t)}+\sum _{st\in E_4\left(G\right)}\sqrt{(d_s\times d_t)}\hfill \\ & =& \left(8\right)\sqrt{(1\times 4)}+(8m+8n+8t-24)\sqrt{(2\times 4)}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)\sqrt{(4\times 4)}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)\sqrt{(4\times 8)}\hfill \\ & =& 16\left(8\sqrt{2}mnt+4(1-\sqrt{2})(mn+mt+nt)+(3\sqrt{2}-4)(m+n+t)-5\sqrt{2}+7\right)\hfill \end{array}$$

For $\alpha =-\frac{1}{2}$,

$$\begin{array}{ccc}\hfill R_{-\frac{1}{2}}\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}+\sum _{st\in E_2\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}+\sum _{st\in E_3\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}+\sum _{st\in E_4\left(G\right)}\frac{1}{\sqrt{(d_s\times d_t)}}\hfill \\ & =& \left(8\right)\frac{1}{\sqrt{(1\times 4)}}+(8m+8n+8t-24)\frac{1}{\sqrt{(2\times 4)}}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)\frac{1}{\sqrt{(4\times 4)}}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)\frac{1}{\sqrt{(4\times 8)}}\hfill \\ & =& 4\sqrt{2}mnt+2(2-\sqrt{2})(mn+mt+nt)+(3\sqrt{2}-4)(m+n+t)-7\sqrt{2}+10\hfill \end{array}$$

 ☐

Theorem 8.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t\ge 1$, then its atom bond connectivity index is equal to,

$$\begin{array}{ccc}\hfill ABC\left(G\right)& =& 2\left[4\sqrt{5}mnt-2(\sqrt{5}-\sqrt{6})(mn+mt+nt)+(2\sqrt{2}+\sqrt{5}-2\sqrt{6})(m+n+t)\right]\hfill \\ & +& 2\left[-6\sqrt{2}+2\sqrt{3}-\sqrt{5}+3\sqrt{6}\right]\hfill \end{array}$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The atom bond connectivity index can be calculated by using Table 3 in the following equation.

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill ABC\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}+\sum _{st\in E_2\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}+\sum _{st\in E_3\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}+\sum _{st\in E_4\left(G\right)}\sqrt{\frac{d_s+d_t-2}{d_s{d}_t}}\hfill \\ & =& \left(8\right)\sqrt{\frac{1+4-2}{1\times 4}}+(8m+8n+8t-24)\sqrt{\frac{2+4-2}{2\times 4}}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)\sqrt{\frac{4+4-2}{4\times 4}}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)\sqrt{\frac{4+8-2}{4\times 8}}\hfill \\ & =& 2\left[4\sqrt{5}mnt-2(\sqrt{5}-\sqrt{6})(mn+mt+nt)+(2\sqrt{2}+\sqrt{5}-2\sqrt{6})(m+n+t)\right]\hfill \\ & +& 2\left[-6\sqrt{2}+2\sqrt{3}-\sqrt{5}+3\sqrt{6}\right]\hfill \end{array}\end{array}$$

 ☐

Theorem 9.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t\ge 1$, then its geometric arithmetic index is equal to,

$$GA\left(G\right)=8\left[\frac{8\sqrt{2}(mnt-1)}{3}-\left(\frac{4\sqrt{2}}{3}-2\right)(mn+mt+nt-m-n-t)+\frac{19}{5}\right]$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The geometric arithmetic index is computed as below:

$$\begin{array}{ccc}\hfill GA\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}+\sum _{st\in E_2\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}+\sum _{st\in E_3\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}+\sum _{st\in E_4\left(G\right)}\frac{2\sqrt{d_s{d}_t}}{d_s+d_t}\hfill \\ & =& \left(8\right)\frac{2\sqrt{1\times 4}}{1+4}+(8m+8n+8t-24)\frac{2\sqrt{2\times 4}}{2+4}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+24\right)\frac{2\sqrt{4\times 4}}{4+4}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)\frac{2\sqrt{4\times 8}}{4+8}\hfill \\ & =& 8\left[\frac{8\sqrt{2}(mnt-1)}{3}-\left(\frac{4\sqrt{2}}{3}-2\right)(mn+mt+nt-m-n-t)+\frac{19}{5}\right]\hfill \end{array}$$

 ☐

Theorem 10.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t\ge 1$, then its first and second Zagreb indices are equal to,

$$M_1\left(G\right)=8\left[48mnt-8(mn+mt+nt)+2(m+n+t)-1\right],$$

$$M_2\left(G\right)=32\left[32mnt-8(mn+mt+nt)+2(m+n+t)-1\right].$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The first and second Zagreb indices are computed as below:

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill M_1\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s+d_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s+d_t)+\sum _{st\in E_2\left(G\right)}(d_s+d_t)+\sum _{st\in E_3\left(G\right)}(d_s+d_t)+\sum _{st\in E_4\left(G\right)}(d_s+d_t)\hfill \\ & =& \left(8\right)(1+4)+(8m+8n+8t-24)(2+4)+\left(16(mn+mt+nt)-16(m+n+t)+24\right)(4+4)\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)(4+8)\hfill \\ & =& 8\left(48mnt-8(mn+mt+nt)+2(m+n+t)-1\right)\hfill \end{array}\end{array}$$

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill M_2\left(G\right)& =& \sum _{st\in E\left(G\right)}(d_s{d}_t)\hfill \\ & =& \sum _{st\in E_1\left(G\right)}(d_s{d}_t)+\sum _{st\in E_2\left(G\right)}(d_s{d}_t)+\sum _{st\in E_3\left(G\right)}(d_s{d}_t)+\sum _{st\in E_4\left(G\right)}(d_s{d}_t)\hfill \\ & =& \left(8\right)(1\times 4)+(8m+8n+8t-24)(2\times 4)+\left(16(mn+mt+nt)-16(m+n+t)+24\right)(4\times 4)\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8(m+n+t)-8\right)(4\times 8)\hfill \\ & =& 32\left[32mnt-8(mn+mt+nt)+2(m+n+t)-1\right]\hfill \end{array}\end{array}$$

 ☐

Table 4 shows the edge partition of the chemical graph $Ti{F}_2[m,n,t]$ based on the degree sum of the end vertices of each edge.

Table 4. Edge partition of $Ti{F}_2[m,n,t]$, $m,n,s\ge 2$ based on the degree sum of the end vertices of each edge.

$({\mathit{S}}_{\mathit{s}},{\mathit{S}}_{\mathit{t}})$	Frequency	Set of Edges
$(4,13)$	8	$E_1$
$(8,18)$	$8(m+n+t-3)$	$E_2$
$(13,16)$	16	$E_3$
$(16,18)$	$16(mn+mt+nt)-16(m+n+t)+8$	$E_4$
$(16,24)$	$32mnt-16(mn+mt+nt)+8$	$E_5$
$(18,32)$	$8(m+n+t-2)$	$E_6$

Theorem 11.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t>1$, then its fourth atom bond connectivity index is equal to,

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& \frac{4\sqrt{57}mnt}{3}-\left(\frac{2\sqrt{57}}{3}-\frac{16}{3}\right)(mn+mt+nt)+\left(\frac{4}{\sqrt{3}}+\frac{4\sqrt{6}}{3}-\frac{16}{3}\right)(m+n+t)\hfill \\ & -& 4\sqrt{6}-\frac{8}{\sqrt{3}}+\frac{12\sqrt{39}}{13}+\frac{\sqrt{57}}{3}+\frac{4\sqrt{195}}{13}+\frac{8}{3}\hfill \end{array}$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The fourth atom bond connectivity index is computed as below:

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& \sum _{st\in E\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_2\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_3\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \\ & +& \sum _{st\in E_4\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_5\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}+\sum _{st\in E_6\left(G\right)}\sqrt{\frac{S_s+S_t-2}{S_s{S}_t}}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill AB{C}_4\left(G\right)& =& \left(8\right)\sqrt{\frac{4+13-2}{4\times 13}}+(8m+8n+8t-24)\sqrt{\frac{8+18-2}{8\times 18}}+\left(16\right)\sqrt{\frac{13+16-2}{13\times 16}}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+8\right)\sqrt{\frac{16+18-2}{16\times 18}}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8\right)\sqrt{\frac{16+24-2}{16\times 24}}+(8m+8n+8t-16)\sqrt{\frac{18+32-2}{18\times 32}}\hfill \\ & =& \frac{4\sqrt{57}mnt}{3}-\left(\frac{2\sqrt{57}}{3}-\frac{16}{3}\right)(mn+mt+nt)+\left(\frac{4}{\sqrt{3}}+\frac{4\sqrt{6}}{3}-\frac{16}{3}\right)(m+n+t)\hfill \\ & -& 4\sqrt{6}-\frac{8}{\sqrt{3}}+\frac{12\sqrt{39}}{13}+\frac{\sqrt{57}}{3}+\frac{4\sqrt{195}}{13}+\frac{8}{3}\hfill \end{array}$$

 ☐

Theorem 12.

Consider the graph $G\cong Ti{F}_2[m,n,t]$ with $m,n,t\ge 2$, then its fifth geometric arithmetic index is equal to,

$$\begin{array}{ccc}\hfill G{A}_5\left(G\right)& =& \frac{64\sqrt{6}mnt}{5}+\left(\frac{192\sqrt{2}}{17}-\frac{32\sqrt{6}}{5}\right)(mn+mt+nt)-\left(\frac{192\sqrt{2}}{17}-\frac{4896}{325}\right)(m+n+t)\hfill \\ & +& \frac{96\sqrt{2}}{17}+\frac{16\sqrt{6}}{5}+\frac{3104\sqrt{13}}{493}-\frac{12192}{325}\hfill \end{array}$$

Proof. 

Let $G\cong Ti{F}_2[m,n,t]$ be the crystal structure of titanium difluoride. The fifth geometric arithmetic index is computed as below:

$$\begin{array}{ccc}\hfill G{A}_5\left(G\right)& =& \sum _{st\in E\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & =& \sum _{st\in E_1\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_2\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_3\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & +& \sum _{st\in E_4\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_5\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}+\sum _{st\in E_6\left(G\right)}\frac{2\sqrt{S_s{S}_t}}{S_s+S_t}\hfill \\ & =& \left(8\right)\frac{2\sqrt{4\times 13}}{4+13}+(8m+8n+8t-24)\frac{2\sqrt{8\times 18}}{8+18}+\left(16\right)\frac{2\sqrt{13\times 16}}{13+16}\hfill \\ & +& \left(16(mn+mt+nt)-16(m+n+t)+8\right)\frac{2\sqrt{16\times 18}}{16+18}\hfill \\ & +& \left(32mnt-16(mn+mt+nt)+8\right)\frac{2\sqrt{16\times 24}}{16+24}+(8m+8n+8t-16)\frac{2\sqrt{18\times 32}}{18+32}\hfill \\ & =& \frac{64\sqrt{6}mnt}{5}+\left(\frac{192\sqrt{2}}{17}-\frac{32\sqrt{6}}{5}\right)(mn+mt+nt)-\left(\frac{192\sqrt{2}}{17}-\frac{4896}{325}\right)(m+n+t)\hfill \\ & +& \frac{96\sqrt{2}}{17}+\frac{16\sqrt{6}}{5}+\frac{3104\sqrt{13}}{493}-\frac{12192}{325}\hfill \end{array}$$

 ☐

6. 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 $C{u}_{2}O[m,n,t]$ are depicted for Randić indices in Figure 4 and Figure 5. The atomic bond connectivity index and geometric arithmetic index for $C{u}_{2}O[m,n,t]$ are depicted in Figure 6. The first and second Zagreb indices for $C{u}_{2}O[m,n,t]$ are depicted in Figure 7. The fourth atomic bond connectivity index and the fifth geometric arithmetic index for $C{u}_{2}O[m,n,t]$ are depicted in Figure 8.

Figure 4. The graphical representation of the Randić index for (a) $\alpha =1$ and (b) for $\alpha =-1$.

Figure 5. The graphical representation of the Randić index for (a) $\alpha =\frac{1}{2}$ and (b) for $\alpha =\frac{-1}{2}$.

Figure 6. The graphical representation of the (a) $ABC$ index and (b) $GA$ index.

Figure 7. The graphical representation of the (a) first Zagreb index and (b) second Zagreb index.

Figure 8. The graphical representation of the (a) $AB{C}_4$ index and (b) $G{A}_5$ index.

The graphical representations of topological indices for titanium difluoride $Ti{F}_2$ are depicted for Randić indices in Figure 9 and Figure 10. The atomic bond connectivity index and geometric arithmetic index for titanium difluoride $Ti{F}_2$ are depicted in Figure 11. The first and second Zagreb indices for titanium difluoride $Ti{F}_2$ are depicted in Figure 12. The fourth atomic bond connectivity index and the fifth geometric arithmetic index for titanium difluoride $Ti{F}_2$ are depicted in Figure 13.

Figure 9. The graphical representation of the Randić index for (a) $\alpha =1$ and (b) for $\alpha =-1$.

Figure 10. The graphical representation of the Randić index for (a) $\alpha =\frac{1}{2}$ and (b) for $\alpha =\frac{-1}{2}$.

Figure 11. The graphical representation of the (a) $ABC$ index and (b) $GA$ index.

Figure 12. The graphical representation of the (a) first Zagreb index and (b) second Zagreb index.

Figure 13. The graphical representation of the (a) $AB{C}_4$ index and (b) $G{A}_5$ index.

7. 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 $G{A}_5$ index, the $AB{C}_4$ index and the first and second Zagreb indices for the chemical graph of the crystal structure of titanium difluoride $Ti{F}_2$ and crystallographic structure of cuprite $C{u}_{2}O$.

In the future, we are interested in computing the distance-based and counting-related topological indices for these structures.