## 1. Introduction

Mathematical chemistry is a hypothetical science in which synthetic structures are observable by the use of scientific instruments. The synthetic diagram hypothesis is a part of this field where chart hypothesis devices are applied to scientifically demonstrate concoctions. This hypothesis noticeably contributes to the field of concoction science [

1].

A diagram G (V, E), with a vertex set V and an edge set E, is shown to be associated if an association between any pair of vertices in G exists. A system is essentially an associated diagram with no distinct edges and no circles. For diagram G, the level of vertex V is the quantity of edges with respect to V and is referred to as deg (v). A sub-atomic diagram is a synthetic structure in which vertices signify iotas and edges refer to bonds.

Atomic diagrams are typically described by various topological records regarding the relationship between the substance structures of a particle with natural, synthetic, or physical properties. Studies have determined a few applications of various topological files in quantitative structure–action relationships (QSARs), quantitative structure–property relationships (QSPRs), virtual screening, and computational medication planning.

A topological list is a numerical quantity, related to a chart, which describes the topology of a diagram and is invariant under diagram automorphism. A topological list (Top(

G)) of a diagram (G) is a number with the property that, for each chart

H isomorphic to

G, Top(

H) = Top(

G). The idea of a topological file originated from the work done by Wiener [

2] when he was attempting to determine the breaking point of paraffin. He named this value the way number. Later on, the way number was renamed in the Wiener file. The Wiener record is the first and most concentrated topological list, both from a hypothetical perspective and by its applications, and is characterized as the entirety of the separations between all the sets of vertices in

G (see [

3] for details).

One of the most seasoned topological records is the main Zagreb list, which was presented by Gutman and Trinajstic in 1972 and is based on the level of vertices of

G. In 2013, Shirdel, Reza Pour, and Sayadi [

4] presented a degree-based Zagreb list called the hyper-Zagreb record. It is as follows:

In 2012, Ghorbani and Azimi [

5] characterized two new forms of Zagreb record for diagram

G. The principal Zagreb records

PM_{1}(

G), the second numerous Zagreb list

PM_{2}(

G), and the following files are characterized below:

In Reference [

6], the Zagreb polynomials were defined as

Recently, there has been extensive research activity into the

HM(G),

vPM_{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

To process our results, we used the techniques of combinatorial figuring, the vertex segment strategy, the edge segment strategy, diagram hypothetical apparatuses, and the degree tallying technique. Further, we used MATLAB software for numerical calculations and checks. Additionally, we used Maple software to plot these numerical results. Furthermore, we processed the hyper-Zagreb file, the first various Zagreb file, the second numerous Zagreb file, and the Zagreb polynomials regarding the crystallographic structure of Cu_{2}O and titanium difluoride (TiF_{2}).

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

Among the different metal oxides, Cu

_{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].

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

Titanium difluoride is a water-insoluble titanium hotspot used in oxygen-sensitive applications such as metal creation. Fluoride mixes have various applications in current innovations and science, from oil refining and drawing to engineered natural science and the assembly of pharmaceuticals. The substance chart of the precious stone structure of titanium difluoride TiF

_{2}[

m,

n,

t] is depicted in

Figure 5. For more subtleties, see [

12,

20].

## 5. Comparisons and Discussion

The examination of the first and second various Zagreb lists and Zagreb polynomials of Cu_{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.

The correlation of the first and second various Zagreb files and Zagreb polynomials of TiF

_{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.

Since the first and second Zagreb records were found to occur for calculations of the absolute π electron vitality of the atoms, on account of Cu_{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

In this paper, we dealt with titanium difluoride (TiF_{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

H.Y. contributed in conceptualization, designing the experiments and funding. M.K.S. conceived and designed the experiments, and analyzed the data. M.H.M. and M.A.R. performed experiments and some computations. S.A. and M.N. contributed to methodology, software, and validation, and wrote the initial draft of the paper, which were validated and approved by M.K.S., and wrote the final draft. All authors read and approved the final version of the paper.

## Funding

This work was supported by the Soft Scientific Research of Sichuan Province under grant 2018ZR0265, Sichuan Military and Civilian Integration Strategy Research Center under grant JMRH-1818, and Sichuan Provincial Department of Education (Key Project) under grant 18ZA0118.

## Acknowledgments

The authors are grateful to the anonymous referees for their valuable comments and suggestions that improved this paper.

## Conflicts of Interest

The authors declare no conflicts of interest.

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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 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 2.**
Crystallographic structure of Cu_{2}O [3, 2, 3].

**Figure 2.**
Crystallographic structure of Cu_{2}O [3, 2, 3].

**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 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 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 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 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).

**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).

**Table 1.**
Edge partition of Cu_{2}O[rn, n, t] based on the degrees of the end vertices of each edge.

**Table 1.**
Edge partition of Cu_{2}O[rn, n, t] based on the degrees of the end vertices of each edge.

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} |

**Table 2.**
Edge partition of TiF_{2}[m, n, t] based on the degrees of end vertices of each edge.

**Table 2.**
Edge partition of TiF_{2}[m, n, t] based on the degrees of end vertices of each edge.

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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