Abstract
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 and crystallographic structure of titanium difluoride of . These theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.
1. 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 [1–3]. 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 structure-activity relationships (QSARs). There are three kinds of topological indices:(1)Degree-based.(2)Distance-based.(3)Spectral-based.
In theoretical chemistry and biology, topological indices have been used for working out the information on molecules in the form of numerical coding. This relates to characterizing physicochemical, biological, toxicologic, pharmacologic, and other properties of chemical compounds. Thousands of molecular structure descriptors have been suggested in order to characterize the physical and chemical properties of molecules [4–6].
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 structure-property relationship (QSPR) studies of molecular graphs [9].
Let be the chemical graph of with unit cells in the plane and layers. Copper(I) oxide or cuprous oxide is the inorganic compound with the formula . Figure 1 describes the graph of the molecule [10]. Note that copper atoms are shown in Figure 1 by red dots and oxygen atoms are shown by blue dots. In the lattice graph, each copper atom is attached to two oxygen atoms, and every oxygen atom is attached to four copper atoms.
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It is one of the principal oxides of copper, the other being or cupric oxide. Nowadays, the crystallographic structure of the molecule has attracted attention due to its interesting properties, low-cost, abundance, nontoxic nature [11]. This is the main reason to choose 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 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 and and computing and comparing the irregularities of some relevant chemical graphs.
Throughout this article, all graphs are finite, undirected, and simple. Let be such a graph with vertex set and edge set . The order of is the cardinality of its vertex set, and size is the cardinality of its edge set. The vertices of correspond to atoms, and an edge between two vertices is related to the chemical bond between these vertices. The degree of a vertex of a graph is symbolized by and is defined as the number of edges incident with . 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 , Albertson [22] defines the imbalance of an edge as and the irregularity of as
For more information about the , you can see [23]. Abdo et al. [24] introduced the total irregularity index, as follows:
Gutman et al. [25] introduced the irregularity index of the graph as follows:
Simplified ways of expressing the irregularities are irregularity indices. These irregularity indices have been studied recently in a novel way [26, 27]. The 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 is an appropriately selected function.
2. 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 contains vertices and edges.
We begin with obtaining formulas for irregularity measures of , , and .
Theorem 1. Let be the crystallographic structure. Then, the irregularity indices of are
Proof. Consider the graph crystallographic structure . From the graph of crystallographic structure, we can see that there are three partitions, , , and . The edge set of can be partitioned as follows:From the molecular graph of , we can observe in Figures 1–5 that , and . Thus, by the definition of Albertson index for , we havewhich is the required formula result (5).
By the definition of total irregularity measure of , we havewhich is the required formula result (7).
By the definition of irregularity index for , we havewhich is the required formula result (8).
Note that throughout the paper, colors red, blue, and green represent , and , respectively.
The next example represents some values of the calculated irregularity indices of , , and for some test values of .
Example 1. Let be crystallographic structure described in Figure 2. Then,
Theorem 2. Let be the crystallographic structure. Then, the irregularity indices of are
Proof. By definition of irregularity index of , we haveBy applying the definition of for , we haveFinally, by applying the definition of for , we haveNote that throughout the paper, colors violet, yellow, and Peru represent , and , respectively.
Theorem 3. Let be the crystallographic structure. Then, the irregularity indices of are
Proof. By applying the definition of for , we haveBy applying the definition of for , we haveFinally, by applying the definition of for , we haveNote that throughout the paper, colors aquamarine, black, and blue violet represent , and , respectively.
Theorem 4. Let be the crystallographic structure. Then, the irregularity indices of are
Proof. By definition of irregularity index of , we haveBy applying the definition of for , we haveBy applying the definition of for , we haveNote that throughout the paper, colors dark orange, fuchsia, and gray represent , and , respectively.
Next example represents some values of the calculated irregularity indices of , , and for some test values of , and .
Example 2. Let be crystallographic structure described in Figure 3.
Then,
3. Results for the Crystallographic Structure of Titanium Difluoride
In this section, we compute irregularity measures of crystal structure of titanium difluoride of .
Titanium difluoride is a water-insoluble titanium hotspot for use in oxygen-delicate applications, for example, metal generation. The concoction chart of the crystallographic structure of is depicted in Figure 1 graph unit cell of (see [29]). In Figures 4 and 5, red dots are for atoms and green dots are for atoms.
From [18], we know that the graph contains vertices and edges.
Here, we obtain some formulas for irregularity measures of , , and .
Theorem 5. Let be the crystallographic structure of titanium difluoride. Then, the irregularity indices of are
Proof. Consider the graph crystallographic structure of titanium difluoride . From the graph of crystallographic structure of titanium difluoride, we can see that there are four partitions, , , , and . The edge set of can be partitioned as follows:From the molecular graph of , we can observe in Figures 4 and 5 that , , and . By applying the definition of Albertson index for , we havewhich is the required result (26). By applying the definition of total irregularity measure for , we havewhich is the required result (27).
By applying the definition of for , we havewhich is the required result (28).
The next example represents some values of the calculated irregularity indices of , , and for some test values of .
Example 3. Let be the crystallographic structure of titanium difluoride described in Figure 4. Then,
Theorem 6. Let be the crystallographic structure of titanium difluoride. Then, the irregularity indices of are
Proof. By applying the definition of for , we haveBy applying the definition of for , we haveFinally, by applying the definition of for , we have
Theorem 7. Let be the crystallographic structure of titanium difluoride. Then, the irregularity indices of are
Proof. By applying the definition of for , we haveBy applying the definition of for , we haveBy applying the definition of for , we have
Theorem 8. Let be the crystallographic structure of titanium difluoride. Then, the irregularity indices of are
Proof. By applying the definition of for , we havewhich is the required result (42).By applying the definition of for , we havewhich is the required result (43).
By applying the definition of for , we havewhich is the required formula result (44).
Finally, the next example represents some values of the calculated irregularity indices of , , , and for some test values of , and .
Example 4. Let be the crystallographic structure of titanium difluoride in Figure 5. Then,
4. Comparisons and Discussion
The 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.
Through the means of a graph structural analysis and derivation, we compute some irregularity measures of crystallographic structure of molecules and the crystallographic structure of titanium difluoride of . Similar works have been done in [12, 13, 17]. From Tables 2–9, we can easily see that all indices are in increasing order as the values of , and . On the other hand, indices showed higher values for and . The graphical representations of irregularity indices of and are depicted in Figures 6–13 for certain values of , and . Moreover, we presented the comparison of all irregularity indices in Figures 14 and 15 for and , respectively.
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5. 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 and crystallographic structure of titanium difluoride of .
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.