Journal of Chemistry

Journal of Chemistry / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 9928543 | https://doi.org/10.1155/2021/9928543

Akbar Jahanbani, Rana Khoeilar, Hajar Shooshtari, "On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules", Journal of Chemistry, vol. 2021, Article ID 9928543, 22 pages, 2021. https://doi.org/10.1155/2021/9928543

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

Academic Editor: Marcelino Maneiro
Received02 Apr 2021
Accepted15 May 2021
Published01 Jun 2021

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 [13]. 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 [46].

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.

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 15 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.


Irregularity indices
SymbolThe function in equation (28)

IRDIF
IRL
IRLU
IRLF
IRLA
IRGA
IRA
IRB

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 15 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: