On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules

Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? In this paper, we are interested in formulating closed forms of irregularity measures of some of the crystallographic structures of Cu2O[p, q, r] and crystallographic structure of titanium difluoride of TiF2[p, q, r]. &ese theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.


Introduction
In the medicines mathematical model, the structure of medication is taken as an undirected graph, where vertices and edges are taken as atoms and chemical bonds. Mathematical chemistry gives tools, for example, polynomials and numbers, to obtain properties of chemical compounds without utilizing quantum mechanics [1][2][3]. A topological index is a numerical parameter of a graph and describes its topology. It depicts the molecular structure numerically and is utilized in the advancement of qualitative structureactivity relationships (QSARs).
ere are three kinds of topological indices: (1) Degree-based.
In theoretical chemistry and biology, topological indices have been used for working out the information on molecules in the form of numerical coding.
is relates to characterizing physicochemical, biological, toxicologic, pharmacologic, and other properties of chemical compounds. ousands of molecular structure descriptors have been suggested in order to characterize the physical and chemical properties of molecules [4][5][6].
Degree-based indices can be further classified in the class of irregularity indices that measure the irregularity of the given graph. Recently, Réti et al. [7,8] showed that the graph irregularity indices are efficient in quantitative structureproperty relationship (QSPR) studies of molecular graphs [9].
Let Cu 2 O[p, q, r] be the chemical graph of Cu 2 O with p × q unit cells in the plane and r layers. Copper(I) oxide or cuprous oxide is the inorganic compound with the formula Cu 2 O. Figure 1 describes the graph of the molecule Cu 2 O [10]. Note that copper atoms are shown in Figure 1 by red dots and oxygen atoms are shown by blue dots. In the Cu 2 O lattice graph, each copper atom is attached to two oxygen atoms, and every oxygen atom is attached to four copper atoms.
It is one of the principal oxides of copper, the other being CuO or cupric oxide. Nowadays, the crystallographic structure of the molecule Cu 2 O has attracted attention due to its interesting properties, low-cost, abundance, nontoxic nature [11]. is is the main reason to choose Cu 2 O and compute irregularity indices for it. Figures 1-5 represent molecular graphs of these two systems.
Zahid et al. [12] computed the irregularity indices of a nanotube, Gao et al. [13] recently computed irregularity measures of some dendrimer structures, in [14], they had discussed irregularity molecular descriptors of hourglass, jagged-rectangle, and triangular benzenoid systems, Iqbal et al. [15] computed the irregularity indices of nanosheets, Zheng et al. [16] discussed irregularity measures of subdivision vertex-edge, Abdo et al. [17] computed irregularity of some molecular structures, Iqbal [12] studied irregularity measures of some nanotubes, and Gao et al. [18] obtained M-polynomials of the crystallographic structure of molecules. In [19], the authors investigated the total irregularity of trees with bounded maximal degree Δ and state integer linear programming problem which gives standard information about extremal trees. In many applications and problems, it is of importance to know how much a given graph deviates from being regular, i.e., how great its irregularity is [20]. Since then, irregular graphs and the degree of irregularity have become one of the core open problems of graph theory. In the current article, we are interested in finding the irregularity of the crystallographic structure of molecules Cu 2 O and T i F 2 [p, q, r] and computing and comparing the irregularities of some relevant chemical graphs. roughout this article, all graphs are finite, undirected, and simple. Let G � (V(G), E(G)) be such a graph with vertex set V(G) and edge set E(G). e order of G is the cardinality of its vertex set, and size is the cardinality of its edge set. e vertices of G correspond to atoms, and an edge between two vertices is related to the chemical bond between these vertices. e degree of a vertex u of a graph G is symbolized by d G (u) � d(u) and is defined as the number of edges incident with u. A graph is said to be regular if all its vertices have the same degree; otherwise, it is irregular. For details on the bases of graph theory, we refer to the book [21].
For a graph G, Albertson [22] defines the imbalance of an edge e � uv ∈ E(G) as |dG(u) − dG(v)| and the irregularity of G as For more information about the IRR(G), you can see [23]. Abdo et al. [24] introduced the total irregularity index, as follows: Gutman et al. [25] introduced the IRF(G) irregularity index of the graph G as follows: Simplified ways of expressing the irregularities are irregularity indices. ese irregularity indices have been  Journal of Chemistry 3 studied recently in a novel way [26,27]. e first such irregularity index was introduced in [28]. Table 1 shows the rest of the irregularity indices used in this paper. Most of the well-known degree-based irregularity indices can be obtained from the following general setting: where f(d(u), d(v)) is an appropriately selected function.

Results for the Crystallographic Structure
In this section, we will compute the irregularity indices of some popular crystallographic structure of molecules From [18], we know that the graph Cu 2 O[p, q, r] contains 6pqr + pq + qr + pr + q + p + r + 1 vertices and 8pqr edges. We Proof.   which is the required formula result (5).
By the definition of total irregularity measure of which is the required formula result (7 which is the required formula result (8). Note that throughout the paper, colors red, blue, and green represent IRR, IRR t , and IRF, respectively. e next example represents some values of the calculated irregularity indices of IRR ( [8].

Irregularity indices
Symbol Proof. By definition of IRL irregularity index of By applying the definition of IRD 1 for Cu 2 O[p, q, r], we have Finally, by applying the definition of IRGA for Cu 2 O[p, q, r], we have 6 Journal of Chemistry Note that throughout the paper, colors violet, yellow, and Peru represent IRL, IRD 1 , and IRGA, respectively. □ Proof. By applying the definition of IRA for Cu 2 O[p, q, r], we have Finally, by applying the definition of IRDIF for Note that throughout the paper, colors aquamarine, black, and blue violet represent IRA, IRB, and IRDIF, respectively.
By applying the definition of IRLA for Note that throughout the paper, colors dark orange, fuchsia, and gray represent IRLU, IRLA, and IRLF, respectively.

Results for the Crystallographic Structure of Titanium Difluoride
In this section, we compute irregularity measures of crystal structure of titanium difluoride of T i F 2 [p, q, r].
Titanium difluoride is a water-insoluble titanium hotspot for use in oxygen-delicate applications, for example, metal generation. e concoction chart of the crystallographic structure of T i F 2 [p, q, r] is depicted in Figure 1 graph unit cell of Cu 2 O (see [29]). In Figures 4 and 5, red dots are for F atoms and green dots are for T i atoms.
Here, we obtain some formulas for irregularity measures of Proof. Consider the graph crystallographic structure of titanium difluoride T i F 2 [p, q, r]. From the graph of T i F 2 [p, q, r] crystallographic structure of titanium difluoride, we can see that there are four partitions, . e edge set of T i F 2 [p, q, r] can be partitioned as follows: From the molecular graph of T i F 2 [p, q, r], we can observe in Figures 4 and 5 which is the required result (26). By applying the definition of total irregularity measure for T i F 2 [p, q, r], we have which is the required result (27). By applying the definition of IRF for T i F 2 [p, q, r], we have which is the required result (28). e next example represents some values of the calculated irregularity indices of IRAL( 1, 1] be the crystallographic structure of titanium difluoride described in Figure 4. en, � 22.1807097779168pqr − 11.0903548889584qr − 11.0903548889584pq − 11.0903548889584pr + 11.0903548889584p + 11.0903548889584q + 11.0903548889584r − 11.0903548889576.

(35)
By applying the definition of IRD 1 for T i F 2 [p, q, r], we have By applying the definition of IRB for T i F 2 [p, q, r], we have √ ) 2 +(32pqr − 16(pq + pr + qr) By applying the definition of IRDIF for T i F 2 [p, q, r], we have 14 Journal of Chemistry Proof. By applying the definition of IRLU for T i F 2 [p, q, r], we have (2,4) +(32pqr − 16(pq + pr +(p + qqr) which is the required result (42).By applying the definition of IRLA for T i F 2 [p, q, r], we have which is the required result (43). By applying the definition of IRLF for T i F 2 [p, q, r], we have

Comparisons and Discussion
e main motivation comes from the fact that graphs of the irregularity indices show close accurate results about properties like entropy, standard enthalpy, vaporization, and acentric factors of octane isomers [8]. Irregularity indices may help to measure the chemical, biological, and nano properties which are widely popular in developing areas.
rough the means of a graph structural analysis and derivation, we compute some irregularity measures of crystallographic structure of molecules Cu 2 O[p, q, r] and the crystallographic structure of titanium difluoride of T i F 2 [p, q, r]. Similar works have been done in [12,13,17]

Conclusion
In this paper, we studied popular crystallographic structure of molecules and also applied analytical methods to compute the irregularity measures for crystallographic structure of molecules Cu 2 O[p, q, r] and crystallographic structure of titanium difluoride of T i F 2 [p, q, r].

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Journal of Chemistry 21