Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2021 / Article
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Graph Invariants and Their Applications

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Research Article | Open Access

Volume 2021 |Article ID 9936949 | https://doi.org/10.1155/2021/9936949

Muhammad Kamran Siddiqui, Shazia Manzoor, Sarfraz Ahmad, Mohammed K. A. Kaabar, "On Computation and Analysis of Entropy Measures for Crystal Structures", Mathematical Problems in Engineering, vol. 2021, Article ID 9936949, 16 pages, 2021. https://doi.org/10.1155/2021/9936949

On Computation and Analysis of Entropy Measures for Crystal Structures

Academic Editor: Kamal Shah
Received28 Mar 2021
Accepted10 Jun 2021
Published28 Jun 2021

Abstract

In recent years, the study of topological indices associated to different molecular tubes and structures gained a lot of attention of the researchers—working in Chemistry and Mathematics. These descriptors play an important role in describing different properties associated to the objects of study. Moreover, Shannon’s entropy concept—a slightly different but more effective approach—provides structural information related to the molecular graphs. In this article, we have computed and analyzed different entropy measures associated to different crystallographic structures. In particular, we have worked on the Zagreb entropies, hyper and augmented Zagreb entropies, and forgotten and Balaban entropies for the crystallographic structures of the cuprite and titanium difluoride .

1. Introduction

The role of Graph Theory has been significantly improvised as applications in other areas of sciences, particularly in the direction of Chemical Graph Theory. Many researchers have been able to explore many new directions during last few years. However, there are plenty of gaps which need to be fixed in. The study of topological indices plays an important role in identifying many physical and chemical properties of the molecular structures of study. In recent time, another approach which is a bit different—but more effective—has been introduced in the literature, namely, using the concept of Shannon’s entropy [1, 2]. Concoction graph hypothesis is a part of numerical science wherein devices of graph hypothesis are applied to demonstrate the compound wonder scientifically. In addition, it has been identified with the insignificant uses of graph postulate for subatomic disputes. This hypothesis contributes a noticeable job in the field of compound sciences; for details, see [35].

The graph entropy gauges that partner likelihood disseminations with components (vertices, edges, and so forth) of a diagram can be delegated inherent and outward measures. There are a few distinct kinds of such chart entropy measures [6]. The degree powers are very critical invariants and concentrated broadly in chart hypothesis and system science, and they are utilized as the data functionals to investigate the systems [7, 8]. Dehmer presented chart entropies dependent on data functionals, which catch auxiliary data and contemplate their properties [9, 10]. For increasingly broad exploration, Estrada and Hatano recommended a truly solid entropy ration for systems/charts [11] and considered the walk-based diagram entropies [12].

The idea of entropy was presented first in Shannon’s celebrated paper [13] as “the entropy of a likelihood dissemination is known as a proportion of the unusualness of data content or a proportion of the liability of a framework.” Afterward, entropy was started to be applied to diagrams and substance systems. It was created for estimating the auxiliary data of diagrams and substance systems. In 1955, Rashevsky [14] presented the idea of graph entropy dependent on the orders of vertex circles. As of late, diagram entropies have been broadly applied in a wide range of fields, for example, science, biology, and humanism [15, 16].

The entropy measures for diagrams have been generally applied in art, chemical engineering, and basic science (see [17]). This issue is mind boggling as it is not sure about which diagram class the ration ought to be assessed. We guess that the presented degree-based entropy can be utilized to quantify organize heterogeneity. It is important to note that the mentioned applications have been identified by taking into account the hidden information investigation issue [18, 19]. Be that as it may, the purported auxiliary translation should be examined too. Comprehensively, the applications for entopic organize measures extend from assessable erection portrayal in basic science or programming innovation to investigate natural or synthetic properties of subatomic charts [20]. This calls to look at what sort of basic multifaceted nature does the measure identify. Comparable entropy measures which depend on vertex-degrees to identify arrange heterogeneity have been presented by Solé and Valverde [21] and Tan and Wu [22].

Shannon’s fundamental work [13] in the late nineteen-forties denotes the beginning stage of present day data hypothesis. Succeeding primary solicitations in semantics and electrical building, data hypothesis was applied broadly in science (see [23, 24]). Subsequently, this strategy has been utilized for investigating living frameworks, e.g., natural and concoction frameworks by methods for charts. These applications have been discussed by both Rashevsky [14] and Trucco [25]. Here, the fundamental oddity was thinking about a structure as a result of a subjective correspondence [26]. With the guide of this knowledge, Shannon’s entropy equations [13] were utilized to decide the basic data substance of a system [21]. In what follows, we survey in sequential request diagram entropy quantifies that have been utilized for considering organic and compound complexes [2729].

In 2014, Chen et al. [30] presented the meaning of the entropy of edge prejudiced graph. At that point, the entropy of edge slanted graph is given as follows:(i)  The First Zagreb Entropy. If , then equation (1) is reduced and is called the first Zagreb entropy:(ii)  The Second Zagreb Entropy. If , then equation (1) is reduced and is called the second Zagreb entropy:(iii)  The Hyper Zagreb Entropy. If , then equation (1) is reduced and is called the hyper Zagreb entropy:(iv)  The Forgotten Entropy. If , then equation (1) is reduced and is called the forgotten entropy:(v)  The Augmented Zagreb Entropy. If , then equation (1) is reduced and is called the augmented Zagreb entropy:(vi)  The Balaban Entropy. If , then equation (1) is reduced and is called the Balaban entropy:

For further details about these entropy measures, see [3134].

2. Crystallographic Structure of

Among different progress metal oxides, has pulled in huge consideration as of late attributable to its predictable assets and nonpoisonous nature and rudimentary establishment progression [35]. These days, the promising utilizations of are particularly seen nearby mixture beams, sunlight-based cells, and catalysis [36]. The invention graph of crystallographic erection of is depicted in Figures 1 and 2; see subtleties in [37].

The vertex partition and edge partition are depicted in Tables 1 and 2, respectively.


FrequencySet of vertices

1
2
4


FrequencySet of edges


2.1. Results for Crystallographic Structure of

(i)The First Zagreb Entropy. We computed the first Zagreb index and first Zagreb entropy as follows:(ii)  The Second Zagreb Entropy. We computed the second Zagreb index and second Zagreb entropy as follows:(iii)  The Hyper Zagreb Entropy of. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(iv)The Forgotten Entropy of. We computed the forgotten index and forgotten entropy as follows:(v)The Augmented Zagreb Entropy of . We computed the augmented Zagreb index and augmented Zagreb entropy as follows: .(vi)  The Balaban Entropy of. We computed the Balaban index and Balaban entropy as follows:

3. Crystallographic Structure of

Titanium difluoride is a water inexplicable titanium cradle for use in oxygen-sensitive solicitations, such as iron invention. Fluoride mixtures have assorted solicitations in existing machineries and science, from oil sanitizing and engraving to unreal animate chemistry and the fabrication of pharmaceuticals. The substance graph of mineral erection of titanium difluoride is designated in Figure 3; for more details, see [38].

The vertex partition and edge partition of are depicted in Tables 3 and 4.


FrequencySet of vertices

18
2
4
8


FrequencySet of edges

8

3.1. Results for Crystallographic Structure of

(i)The First Zagreb Entropy of. We calculated the first Zagreb index and first Zagreb entropy as follows:(ii)The Second Zagreb Entropy of. We computed the second Zagreb index and second Zagreb entropy as follows:(iii)  The Hyper Zagreb Entropy of. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(iv)The Forgotten Entropy of. We computed the forgotten index and forgotten entropy as follows: