# Topological Characterization of the Crystallographic Structure of Titanium Difluoride and Copper (I) Oxide

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

## Abstract

**:**

_{2}) and the crystallographic structure of Cu

_{2}O have attracted a great deal of attention in the field of quantitative structure–property relationships (QSPRs) in recent years. A topological index of a diagram (G) is a numerical quantity identified with G which portrays the sub-atomic chart G. In 1972, Gutman and Trinajstić resented the first and second Zagreb topological files of atomic diagrams. In this paper, we determine a hyper-Zagreb list, a first multiple Zagreb file, a second different Zagreb record, and Zagreb polynomials for titanium difluoride (TiF

_{2}) and the crystallographic structure of Cu

_{2}O.

## 1. Introduction

_{1}(G), the second numerous Zagreb list PM

_{2}(G), and the following files are characterized below:

_{1}(G), and PM

_{2}(G) indices, as well as M

_{1}(G, x) and M

_{2}(G,x) polynomials and their variants (see also [7,8,9,10,11,12,13]). For further research regarding the topological indices of various graph families, see [14,15,16,17,18,19,20,21,22,23,24,25,26].

## 2. Main Results and Methods

_{2}O and titanium difluoride (TiF

_{2}).

## 3. Crystallographic Structure of **Cu**_{2}O

_{2}O

_{2}O has recently attracted a great deal of attention due to its recognized properties, non-poisonous nature, ease, abundance, and straightforward creation process. Currently, the promising applications of Cu

_{2}O mostly center around concoction sensors, sun-oriented cells, photocatalysis, lithium-particle batteries, and catalysis. The concoction chart of the crystallographic structure of Cu

_{2}O is depicted in Figure 1 and Figure 2. For more data about this structure, see [3,22].

#### Methodology of Cu_{2}O Formulas

_{2}O[m, n, t] be the substance diagram of Cu

_{2}O with $m\times n$ unit cells in the plane and t layers. We first develop this chart by taking $m\times n$ joins in the $m\times n$-plane and then accumulating it in t layers. The quantity of vertices and edges of Cu

_{2}O[m, n, t] are (m + 1)(n + 1)(t + 1) + 5mnt and 8mnt, individually. In Cu

_{2}O[m, n, t], the quantity of vertices of degree 0 is 4, the quantity of vertices of degree 1 is 4m + 4n + 4t − 8, the quantity of vertices of degree 2 is 4mnt + 2mn + 2mt + 2nt − 4n − 4m − 4t + 6, and the quantity of vertices of degree 4 is 2nmt − nm – nt − mt + n + m + t − 1.

_{2}O[m, n, t].

_{2}O[m, n, t] in the following theorem.

**Theorem**

**1.**

_{2}O[m, n, t] with m, n, t ≥ 1, then its hyper-Zagreb index is equal to,

**Proof.**

_{2}O[m, n, t]. The hyper-Zagreb index is computed using Table 1 and Equation (1).

**Theorem**

**2.**

_{2}O[m, n, t] with m, n, t ≥ 1. Its first and second multiplicative Zagreb indices are equal to:

**Proof.**

_{2}O[m, n, t]. The first multiplicative Zagreb index is computed using Table 1 and Equation (2):

**Theorem**

**3.**

_{2}O[m, n, t] with m, n, t ≥ 1. Its first and second Zagreb polynomials are equal to:

_{1}(G,x) = (4m + v4n + 4t − 8)x

^{3}+ (4mn + 4mt + v4n- 8m − 8n − 8t + 12)x

^{4}

^{6},

_{2}(G,x) = (4m + 4n + 4t − 8)x

^{2}+ (4mn + 4mt + 4nt − 8m − 8n − 8t + 12)x

^{4}+ (8mnt − 4mn − 4mt − 4nt + 4m + 4n + 4t − 4)x

^{8}.

**Proof.**

_{2}O[m, n, t]. The first Zagreb polynomial is computed using Table 1 and Equation (4):

_{1}(G,x) = (4m + 4n + 4t − 8)x

^{(1+2)}+ (4mn + 4mt + v4nt − 8m − 8n − 8t + 12)x

^{(2+2)}+ (8mnt − 4mn − 4mt − 4nt + 4m + 4n + 4t − 4)x

^{(2+4)}

^{3}+ (4mn + 4mt + 4n − 8m − 8n − 8t + 12)x

^{4}+ (8mnt − 4mn − 4mt − 4nt + 4m + 4n + 4t − 4)x

^{6}.

_{2}(G,x) = (4m + 4n + 4t − 8)x(1×2) + (4mn + 4mt + 4nt − 8m −8n − 8t + 12)x(2×2) + (8mnt − 4mn − 4mt − 4nt + 4m + 4n + 4t − 4)x(2×4)

## 4. Crystal Structure of Titanium Difluoride TiF_{2}[m, n, t]

_{2}[m, n, t] is depicted in Figure 5. For more subtleties, see [12,20].

#### Methodology of TiF_{2}[m, n, t] Formulas

_{2}[m, n, t] be the synthetic chart of TiF

_{2}with $m\times n$ unit cells in the plane and t layers. We develop this chart by first by taking $m\times n$ joins in the $m\times n$-plane and then putting it away in the upper part of the t layers. The quantity of vertices and edges of TiF

_{2}[m, n, t] are 12mntv + 2mn + 2mt + 2nt + m + n +1 + 1 and 32mnt, individually. In TiF

_{2}[m, n, t], the quantity of vertices of degree 1 is 8, the quantity of vertices of degree 2 is 4m + 4n + 4t − 12, the quantity of vertices of degree 4 is 8mnt + 4mn + 4mt + 4nt − 4n − 4m − 4t + 6, and the quantity of vertices of degree 8 is 4mnt − 2(mn + mt + nt) + m + n + 1 − 1.

_{2}[m, n, t]

_{2}[m, n, t] in the following theorems.

**Theorem**

**4.**

_{2}[m, n, t] with m, n, t ≥ 1. Its hyper-Zagreb index is equal to:

**Proof.**

_{2}[m, n, t]. The hyper-Zagreb index is computed using Table 2 and Equation (1):

^{2}+ (8m + 8n + 8t − 24) (2 + 4)

^{2}+ [l6(mn + mt + nt) − 16(m + n + t) + 24] (4 + 4)

^{2}+ [32mnt − 16(mn + mt + nt) + 8(m + n +1) − 8] (4 + 8)

^{2}

**Theorem**

**5.**

_{2}[m, n, t] with m, n, t ≥ 1. Its first and second multiplicative-Zagreb indices are equal to:

**Proof.**

_{2}[m, n, t]. The first multiplicative Zagreb index is computed using Table 2 and Equation (2):

**Theorem**

**6.**

_{2}[m, n, t] with m, n, t ≥ 1. Its first and second Zagreb polynomials are equal to:

_{1}(G,x) = (8)x

^{(5)}+ (8m + 8n + 8t − 24)x

^{(6)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(8)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(12}

^{)},

_{2}(G,x) = (8)x

^{(4)}+ (8m + 8n + 8t − 24)x

^{(8)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(16)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(23)}.

**Proof.**

_{2}[m, n, t]. The first Zagreb polynomial is computed using Table 2 and Equation (4):

_{1}(G,x) = (8)x

^{(1+4)}+ (8m + 8n + 8t − 24)x

^{(2+4)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(4+4)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(4+8)}

^{(5)}+ (8m + 8n + 8t − 24)x

^{(6)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(8)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(12)}.

_{2}(G,x)= (8)x

^{(1×4)}+ (8m + 8n + 8t − 24)x

^{(2×4)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(4×4)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(4×8)}

^{(4)}+ (8m + 8n + 8t − 24)x

^{(8)}+ (16mn + 16mt + 16nt − 16m − 16n − 16t + 24)x

^{(16)}+ (32mnt − 16mn − 16mt − 16nt + 8m + 8n + 8t − 8)x

^{(23)}.

## 5. Comparisons and Discussion

_{2}O[m, n, t] are graphical portrayals, in Figure 4, that can be used for specific estimations of m, n, and t. By fluctuating the estimation of m, n, and t in the given area, the principal, second various Zagreb files and Zagreb polynomials carried on in an unexpected way.

_{2}[m, n, t] are graphical portrayals in Figure 7 for specific estimations of m, n, and t. By changing the estimations of m, n, and t in the given space, the primary, second numerous Zagreb lists and Zagreb polynomials carried on in an unexpected way.

_{2}O[m, n, t] and TiF

_{2}[m, n, t], their qualities gave complete π electron vitality in expanding the request for higher estimations of m, n, and t.

## 6. Conclusions

_{2}**)**and the crystallographic structure of Cu

_{2}O, and studied their topological indices. We determined the hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for titanium difluoride, as well as the crystallographic structure of Cu

_{2}O. Additionally, by using MATLAB, we plotted these computed results numerically and discussed their behavior regarding their monotonicity.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Crystallographic structure of Cu

_{2}O. (

**a**) Structural attributes of Cu and O particles in the Cu

_{2}O cross section. The Cu

_{2}O cross section is shaped by interpenetrating the Cu and O grids with one another. (

**b**) Unit cell of Cu

_{2}O. Copper particles appear as small blue circles, and oxygen iotas appear as large red circles. In the Cu

_{2}O cross section, every Cu iota is composed of two O particles, and every O molecule is facilitated by four Cu iotas.

**Figure 3.**First and second multiplicative Zagreb indices PM

_{1}(G) and PM

_{2}(G) of G, equivalent to Cu

_{2}O[m, n, t] for t = 2. Blue and red colors represent PM

_{1}(G) and PM

_{2}(G), respectively.

**Figure 4.**Comparison of the first and second Zagreb polynomials M

_{1}(G, x) and M

_{2}(G, x) of G = Cu

_{2}O[m, n, t] for t = 10 = m = n. Blue and red lines represent M

_{1}(G, x) and M

_{2}(G, x), respectively. M

_{2}(G, x) is shown to grow more rapidly than M

_{1}(G, x).

**Figure 5.**Crystal structure titanium difluoride TiF

_{2}[m, n, t]. (

**a**) Unit cell of TiF

_{2}[m, n, t] with Ti atoms in red and F atoms in green. (

**b**) Crystal structure of TiF

_{2}[4, 1, 2].

**Figure 6.**First and second multiplicative Zagreb indices PM

_{1}(G) and PM

_{2}(G) of G, equivalent to TiF

_{2}[m, n, t] for t = 2. The blue and red colors represent PM

_{1}(G) and PM

_{2}(G), respectively. We can see that, in the given domain, PM

_{1}(G) is more dominant than PM

_{2}(G).

**Figure 7.**Comparison of the first and second Zagreb polynomials M

_{1}(G, x) and M

_{2}(G, x) of G = TiF

_{2}[m, n, t], for t = 10 = m = n. The blue and red represent M

_{1}(G, x) and M

_{2}(G, x), respectively. We can see that M

_{2}(G, x) grows more rapidly than M

_{1}(G, x).

Number of edges | 4n + 4m + 4t − 8 | 4nm + 4nt + 4mt − 8n − 8m − 8t + 12 | 4(2nmt − nm − nt − mt + n + m + t − 1) |

Set of Edges | E_{1} | E_{2} | E_{3} |

Number of edges | 8 | 8(m + n + t − 3) | 16(mn + mt + nt) − 16(m + n + t) + 24 | 32mnt − 16(mt + mn + nt) + 8(m + n + t) − 8 |
---|---|---|---|---|

Set of Edges | E_{1} | E_{2} | E_{3} |

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**MDPI and ACS Style**

Yang, H.; Muhammad, M.H.; Rashid, M.A.; Ahmad, S.; Siddiqui, M.K.; Naeem, M. Topological Characterization of the Crystallographic Structure of Titanium Difluoride and Copper (I) Oxide. *Atoms* **2019**, *7*, 100.
https://doi.org/10.3390/atoms7040100

**AMA Style**

Yang H, Muhammad MH, Rashid MA, Ahmad S, Siddiqui MK, Naeem M. Topological Characterization of the Crystallographic Structure of Titanium Difluoride and Copper (I) Oxide. *Atoms*. 2019; 7(4):100.
https://doi.org/10.3390/atoms7040100

**Chicago/Turabian Style**

Yang, Hong, Mehwish Hussain Muhammad, Muhammad Aamer Rashid, Sarfraz Ahmad, Muhammad Kamran Siddiqui, and Muhammad Naeem. 2019. "Topological Characterization of the Crystallographic Structure of Titanium Difluoride and Copper (I) Oxide" *Atoms* 7, no. 4: 100.
https://doi.org/10.3390/atoms7040100