# On Computation and Analysis of Entropy Measures for Crystal Structures

Authors: Muhammad Kamran Siddiqui; Shazia Manzoor; Sarfraz Ahmad; Mohammed K. A. Kaabar
Journal: Mathematical Problems in Engineering
 (2021)

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## 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 cupriteCu2O and titanium difluoride TiF2.


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## Body

## 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 [3–5].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 [27–29].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:(1)ENTℑG=−∑r′s′∈EGℑr′s′∑rs∈EGℑrslogℑr′s′∑rs∈EGℑrs.(i)
The First Zagreb Entropy. If ℑrs=Θr+Θs, then equation (1) is reduced and is called the first Zagreb entropy:(2)ENTM1G=logM1G−1M1Glog∏rs∈EGΘr+ΘsΘr+Θs.(ii)
The Second Zagreb Entropy. If ℑrs=Θr×Θs, then equation (1) is reduced and is called the second Zagreb entropy:(3)ENTM2G=logM2G−1M2Glog∏rs∈EGΘr×ΘsΘr×Θs.(iii)
The Hyper Zagreb Entropy. If ℑrs=Θr+Θs2, then equation (1) is reduced and is called the hyper Zagreb entropy:(4)ENTHMG=logHMG−1HMGlog∏rs∈EGΘr+Θs2Θr+Θs2.(iv)
The Forgotten Entropy. If ℑrs=Θr2+Θs2, then equation (1) is reduced and is called the forgotten entropy:(5)ENTFG=logFG−1FGlog∏rs∈EGΘr2+Θs2Θr2+Θs2.(v)
The Augmented Zagreb Entropy. If ℑrs=ΘrΘs/Θr+Θs−23, then equation (1) is reduced and is called the augmented Zagreb entropy:(6)ENTAZIG=logAZIG−1AZIGlog∏rs∈EGΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23.(vi)
The Balaban Entropy. If ℑrs=q/q−p+2×1/ΘrΘs, then equation (1) is reduced and is called the Balaban entropy:(7)ENTJG=logJG−1JGlog∏rs∈EGqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs.For further details about these entropy measures, see [31–34].


## 2. Crystallographic Structure ofCu2Om,n,t

Among different progress metal oxides,Cu2O 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 Cu2O are particularly seen nearby mixture beams, sunlight-based cells, and catalysis [36]. The invention graph of crystallographic erection of Cu2O is depicted in Figures 1 and 2; see subtleties in [37].Figure 1
(a)Cu2O lattice. (b) Unit cell of Cu2O.
(a)(b)Figure 2
Crystallographic structure ofCu2O3,2,3.
(a)(b)The vertex partition and edge partition are depicted in Tables1 and 2, respectively.Table 1
Vertex partition ofCu2Om,n,t.
ΘrFrequencySet of vertices14t+4m+4n−8V122mt−4t+2mn−4n−4mnt+4m+2nt+6V24n+m+t+2nmt−mt−nm−nt−1V3Table 2
Edge partition ofCu2Om,n,t.
Θr,ΘsFrequencySet of edges1,24t+4n+4m−8E12,24mt+4nt+4nm−8m−8t−8n+12E22,44m+4n+8nmt−4nm−4nt−4mt+4t−4E3


### 2.1. Results for Crystallographic Structure ofCu2Om,n,t

(i)
The First Zagreb Entropy. We computed the first Zagreb index and first Zagreb entropy as follows:(8)M1G=48mnt−8mn+mt+nt+4m+4n+4t,ENTM1Cu2O=logM1−1M1log∏rs∈E1GΘr+ΘsΘr+Θs×∏rs∈E2GΘr+ΘsΘr+Θs×∏rs∈E3GΘr+ΘsΘr+Θs=logM1−1M1log4n+4m+4t−8×27×4nt+4nm−8m+4mt−8t−8n+12×256×8nmt−4nm+4n+4m−4nt−4mt+4t−4×46656,ENTM1Cu2O=log48mnt−8mn−8mt−8nt+4m+4n+4t−log4n+4m+4t−8×2748mnt−8mn−8mt−8nt+4m+4n+4t−log4nt+4nm−8m+4mt−8t−8n+12×25648mnt−8mn−8mt−8nt+4m+4n+4t−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×4665648mnt−8mn−8mt−8nt+4m+4n+4t.(ii)
The Second Zagreb Entropy. We computed the second Zagreb index and second Zagreb entropy as follows:(9)M2G=64mnt−16mn−16mt−16nt+8m+8n+8t,ENTM2Cu2O=logM2−1M2log∏rs∈E1GΘr×ΘsΘr×Θs×∏rs∈E2GΘr×ΘsΘr×Θs×∏rs∈E3GΘr×ΘsΘr×Θs=logM2−1M2log4n+4m+4t−8×4×4nt+4nm−8m+4mt−8t−8n+12×256×8nmt−4nm+4n+4m−4nt−4mt+4t−4×16777216,ENTM2Cu2O=log64mnt+8m−16mn−16nt+8n−16mt+8t−log4n+4m+4t−8×464mnt+8m−16mn−16nt+8n−16mt+8t−log4nt+4nm−8m+4mt−8t−8n+12×25664mnt+8m−16mn−16nt+8n−16mt+8t−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1677721664mnt+8m−16mn−16nt+8n−16mt+8t.(iii)
The Hyper Zagreb Entropy ofCu2Om,n,t. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(10)HMG=8mnt−80mn−80mt−80nt+52m+52n+52t−24,ENTHMCu2O=logHM−1HMlog∏rs∈E1GΘr+Θs2Θr+Θs2×∏rs∈E2GΘr+Θs2Θr+Θs2×∏rs∈E3GΘr+Θs2Θr+Θs2=logHM−1HMlog4n+4m+4t−8×9×4nt+4nm−8m+4mt−8t−8n+12×16×8nmt−4nm+4n+4m−4nt−4mt+4t−4×36,ENTHMCu2O=log8mnt−80mn−80mt−80nt+52m+52n+52t−24−log4n+4m+4t−8×98mnt−80mn−80mt−80nt+52m+52n+52t−24−log4nt+4nm−8m+4mt−8t−8n+12×168mnt−80mn−80mt−80nt+52m+52n+52t−24−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×368mnt−80mn−80mt−80nt+52m+52n+52t−24.(iv)
The Forgotten Entropy ofCu2Om,n,t. We computed the forgotten index and forgotten entropy as follows:(11)FG=36n+36m+36t−24−48nm−48nt−48mt+160nmt,ENTFCu2O=logFG−1FGlog∏rs∈E1GΘr2+Θs2Θr2+Θs2×∏rs∈E2GΘr2+Θs2Θr2+Θs2×∏rs∈E3GΘr2+Θs2Θr2+Θs2=logFG−1FGlog4n+4m+4t−8×3125×4nt+4nm−8m+4mt−8t−8n+12×16777216×8nmt−4nm+4n+4m−4nt−4mt+4t−4×1.048576×1026,ENTFCu2O=log36n+36m+36t−24−48nm−48nt−48mt+160nmt−log4n+4m+4t−8×312536n+36m+36t−24−48nm−48nt−48mt+160nmt−log4nt+4nm−8m+4mt−8t−8n+12×1677721636n+36m+36t−24−48nm−48nt−48mt+160nmt−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1.048576×102636n+36m+36t−24−48nm−48nt−48mt+160nmt.(v)
The Augmented Zagreb Entropy ofCu2Om,n,t. We computed the augmented Zagreb index and augmented Zagreb entropy as follows: AZIG=64nmt.(12)ENTAZICu2O=logAZIG−1AZIGlog∏rs∈E1GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E2GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E3GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23=logAZIG−1AZIGlog4n+4m+4t−8×16777216×4nt+4nm−8m+4mt−8t−8n+12×16777216×8nmt−4nm+4n+4m−4nt−4mt+4t−4×16777216,ENTAZICu2O=log64nmt−log4n+4m+4t−8×1677721664nmt−log4nt+4nm−8m+4mt−8t−8n+12×1677721664nmt−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1677721664nmt.(vi)
The Balaban Entropy ofCu2Om,n,t. We computed the Balaban index and Balaban entropy as follows:(13)JG=8mnt2mnt−mt−nt−t+1−mn−m−n×124n+4m+4t−8+124nt+4nm−8m+4mt−8t−8n+12+8mnt2mnt−mt−nt−t+1−mn−m−n×188nmt−4nm+4n+4m−4nt−4mt+4t−4=8mnt2mnt−mt−nt−t+1−mn−m−n×224n+4m+4t−8+8mnt2mnt−mt−nt−t+1−mn−m−n×2mn+2nt+2mt−4n−4m−4t+6+8mnt2mnt−mt−nt−t+1−mn−m−n×222nmt−nm−nt−mt+n+m+t−1,ENTJCu2O=logJG−1JGlog∏rs∈E1Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs×∏rs∈E2Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs×∏rs∈E3Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs,ENTJCu2O=logJG−1JGlog4n+4m+4t−8×q2q−p+2q/2q−p+2×4nt+4nm−8m+4mt−8t−8n+12×q2q−p+2q/2q−p+2×8nmt−4nm+4n+4m−4nt−4mt+4t−4×q22q−p+2q/22q−p+2,ENTJCu2O=logJG−log4n+4m+4t−8×q/2q−p+2q/2q−p+2JG−log4nt+4nm−8m+4mt−8t−8n+12×q/2q−p+2q/2q−p+2JG−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×q/22q−p+2q/22q−p+2JG.


## 2.1. Results for Crystallographic Structure ofCu2Om,n,t

(i)
The First Zagreb Entropy. We computed the first Zagreb index and first Zagreb entropy as follows:(8)M1G=48mnt−8mn+mt+nt+4m+4n+4t,ENTM1Cu2O=logM1−1M1log∏rs∈E1GΘr+ΘsΘr+Θs×∏rs∈E2GΘr+ΘsΘr+Θs×∏rs∈E3GΘr+ΘsΘr+Θs=logM1−1M1log4n+4m+4t−8×27×4nt+4nm−8m+4mt−8t−8n+12×256×8nmt−4nm+4n+4m−4nt−4mt+4t−4×46656,ENTM1Cu2O=log48mnt−8mn−8mt−8nt+4m+4n+4t−log4n+4m+4t−8×2748mnt−8mn−8mt−8nt+4m+4n+4t−log4nt+4nm−8m+4mt−8t−8n+12×25648mnt−8mn−8mt−8nt+4m+4n+4t−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×4665648mnt−8mn−8mt−8nt+4m+4n+4t.(ii)
The Second Zagreb Entropy. We computed the second Zagreb index and second Zagreb entropy as follows:(9)M2G=64mnt−16mn−16mt−16nt+8m+8n+8t,ENTM2Cu2O=logM2−1M2log∏rs∈E1GΘr×ΘsΘr×Θs×∏rs∈E2GΘr×ΘsΘr×Θs×∏rs∈E3GΘr×ΘsΘr×Θs=logM2−1M2log4n+4m+4t−8×4×4nt+4nm−8m+4mt−8t−8n+12×256×8nmt−4nm+4n+4m−4nt−4mt+4t−4×16777216,ENTM2Cu2O=log64mnt+8m−16mn−16nt+8n−16mt+8t−log4n+4m+4t−8×464mnt+8m−16mn−16nt+8n−16mt+8t−log4nt+4nm−8m+4mt−8t−8n+12×25664mnt+8m−16mn−16nt+8n−16mt+8t−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1677721664mnt+8m−16mn−16nt+8n−16mt+8t.(iii)
The Hyper Zagreb Entropy ofCu2Om,n,t. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(10)HMG=8mnt−80mn−80mt−80nt+52m+52n+52t−24,ENTHMCu2O=logHM−1HMlog∏rs∈E1GΘr+Θs2Θr+Θs2×∏rs∈E2GΘr+Θs2Θr+Θs2×∏rs∈E3GΘr+Θs2Θr+Θs2=logHM−1HMlog4n+4m+4t−8×9×4nt+4nm−8m+4mt−8t−8n+12×16×8nmt−4nm+4n+4m−4nt−4mt+4t−4×36,ENTHMCu2O=log8mnt−80mn−80mt−80nt+52m+52n+52t−24−log4n+4m+4t−8×98mnt−80mn−80mt−80nt+52m+52n+52t−24−log4nt+4nm−8m+4mt−8t−8n+12×168mnt−80mn−80mt−80nt+52m+52n+52t−24−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×368mnt−80mn−80mt−80nt+52m+52n+52t−24.(iv)
The Forgotten Entropy ofCu2Om,n,t. We computed the forgotten index and forgotten entropy as follows:(11)FG=36n+36m+36t−24−48nm−48nt−48mt+160nmt,ENTFCu2O=logFG−1FGlog∏rs∈E1GΘr2+Θs2Θr2+Θs2×∏rs∈E2GΘr2+Θs2Θr2+Θs2×∏rs∈E3GΘr2+Θs2Θr2+Θs2=logFG−1FGlog4n+4m+4t−8×3125×4nt+4nm−8m+4mt−8t−8n+12×16777216×8nmt−4nm+4n+4m−4nt−4mt+4t−4×1.048576×1026,ENTFCu2O=log36n+36m+36t−24−48nm−48nt−48mt+160nmt−log4n+4m+4t−8×312536n+36m+36t−24−48nm−48nt−48mt+160nmt−log4nt+4nm−8m+4mt−8t−8n+12×1677721636n+36m+36t−24−48nm−48nt−48mt+160nmt−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1.048576×102636n+36m+36t−24−48nm−48nt−48mt+160nmt.(v)
The Augmented Zagreb Entropy ofCu2Om,n,t. We computed the augmented Zagreb index and augmented Zagreb entropy as follows: AZIG=64nmt.(12)ENTAZICu2O=logAZIG−1AZIGlog∏rs∈E1GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E2GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E3GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23=logAZIG−1AZIGlog4n+4m+4t−8×16777216×4nt+4nm−8m+4mt−8t−8n+12×16777216×8nmt−4nm+4n+4m−4nt−4mt+4t−4×16777216,ENTAZICu2O=log64nmt−log4n+4m+4t−8×1677721664nmt−log4nt+4nm−8m+4mt−8t−8n+12×1677721664nmt−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×1677721664nmt.(vi)
The Balaban Entropy ofCu2Om,n,t. We computed the Balaban index and Balaban entropy as follows:(13)JG=8mnt2mnt−mt−nt−t+1−mn−m−n×124n+4m+4t−8+124nt+4nm−8m+4mt−8t−8n+12+8mnt2mnt−mt−nt−t+1−mn−m−n×188nmt−4nm+4n+4m−4nt−4mt+4t−4=8mnt2mnt−mt−nt−t+1−mn−m−n×224n+4m+4t−8+8mnt2mnt−mt−nt−t+1−mn−m−n×2mn+2nt+2mt−4n−4m−4t+6+8mnt2mnt−mt−nt−t+1−mn−m−n×222nmt−nm−nt−mt+n+m+t−1,ENTJCu2O=logJG−1JGlog∏rs∈E1Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs×∏rs∈E2Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs×∏rs∈E3Gqq−p+2×1ΘrΘsq/q−p+2×1/ΘrΘs,ENTJCu2O=logJG−1JGlog4n+4m+4t−8×q2q−p+2q/2q−p+2×4nt+4nm−8m+4mt−8t−8n+12×q2q−p+2q/2q−p+2×8nmt−4nm+4n+4m−4nt−4mt+4t−4×q22q−p+2q/22q−p+2,ENTJCu2O=logJG−log4n+4m+4t−8×q/2q−p+2q/2q−p+2JG−log4nt+4nm−8m+4mt−8t−8n+12×q/2q−p+2q/2q−p+2JG−log8nmt−4nm+4n+4m−4nt−4mt+4t−4×q/22q−p+2q/22q−p+2JG.


## 3. Crystallographic Structure ofTiF2m,n,t

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 difluorideTiF2m,n,t is designated in Figure 3; for more details, see [38].Figure 3
Crystal structure titanium difluorideTiF2m,n,t: (a) unit cell of of TiF2m,n,t; (b) crystal structure of TiF24,1,2.
(a)(b)The vertex partition and edge partition ofTiF2m,n,t are depicted in Tables 3 and 4.Table 3
Vertex partition ofTiF2m,n,t.
ΘrFrequencySet of vertices18V124m+4n+4t−12V244mn+4mt−4m−4t+6+8mnt+4nt−4nV38n+t−2mn+mt+nt−1+4mnt+mV4Table 4
Edge partition ofTiF2m,n,t.
Θr,ΘsFrequencySet of edges1,48E12,48m+8n+8t−24E24,416mn+16mt−16n−16m+16nt−16t+24E34,88m+8n+16mn−16nt+8t−8+32mnt−16mtE4


### 3.1. Results for Crystallographic Structure ofTiF2m,n,t

(i)
The First Zagreb Entropy ofTiF2m,n,t. We calculated the first Zagreb index and first Zagreb entropy as follows:(14)M1G=384mnt−64mn−64mt−64nt+16m+16n+16t−8,ENTM1TiF2=logM1−1M1log∏rs∈E1GΘr+ΘsΘr+Θs×∏rs∈E2GΘr+ΘsΘr+Θs×∏rs∈E3GΘr+ΘsΘr+Θs×∏rs∈E4GΘr+ΘsΘr+Θs=logM1−1M1log8×3125×8m+8t−24+8n×46656×16mn+16mt+16nt−16m−16n−16t+24×16777216×32mnt−16mt−16mn−16nt+8m+8n+8t−8×8916100448456,ENTM1TiF2=log384mnt−64mn−64mt−64nt+16m+16n+16t−8−log8×3125384mnt−64mn−64mt−64nt+16m+16n+16t−8−log8m+8n+8t−24×46656×256384mnt−64mn−64mt−64nt+16m+16n+16t−8−log16mn+16nt−16n−16t+24+16mt−16m×16777216384mnt−64mn−64mt−64nt+16m+16n+16t−8−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×8916100448456384mnt−64mn−64mt−64nt+16m+16n+16t−8.(ii)
The Second Zagreb Entropy ofTiF2m,n,t. We computed the second Zagreb index and second Zagreb entropy as follows:(15)M2G=1024mnt−256mn−256mt−256nt+64m+64n+64t−32,ENTM2TiF2=logM2−1M2log∏rs∈E1GΘr×ΘsΘr×Θs×∏rs∈E2GΘr×ΘsΘr×Θs×∏rs∈E3GΘr×ΘsΘr×Θs×∏rs∈E4GΘr×ΘsΘr×Θs=logM2−1M2log8×256×8m+8n+8t−24×16777216×16mn+16nt−16n−16t+24+16mt−16m×1616×32mnt−16mt−16mn−16nt+8m+8n+8t−8×3232,(16)ENTM2TiF2=log1024mnt−256mn−256mt−256nt+64m+64n+64t−32−log8×2561024mnt−256mn−256mt−256nt+64m+64n+64t−32−log8m+8n+8t−24×167772161024mnt−256mn−256mt−256nt+64m+64n+64t−32−log16mn+16nt−16m+16mt−16n−16t+24×16161024mnt−256mn−256mt−256nt+64m+64n+64t−32−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×32321024mnt−256mn−256mt−256nt+64m+64n+64t−32.(iii)
The Hyper Zagreb Entropy ofTiF2m,n,t. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(17)HMG=4608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280,ENTHMTiF2=logHM−1HMlog∏rs∈E1GΘr+Θs2Θr+Θs2×∏rs∈E2GΘr+Θs2Θr+Θs2×∏rs∈E3GΘr+Θs2Θr+Θs2×∏rs∈E4GΘr+Θs2Θr+Θs2,ENTHMTiF2=logHM−1HMlog8×2525×8m+8n+8t−24×3636×16mn+16nt−16m+16mt−16n−16t+24×6464×32mnt−16mt−16mn−16nt+8m+8n+8t−8×144144,ENTHMTiF2=log4608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log8×25254608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log8m+8n+8t−24×36364608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log16mn+16nt−16m+16mt−16n−16t+24×64644608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×1441444608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280.(iv)
The Forgotten Entropy ofTiF2m,n,t. We computed the forgotten index and forgotten entropy as follows:(18)FG=2560mnt−216+288m+288n+288t−768mn−768mt−768nt,ENTFTiF2=logFG−1FGlog∏rs∈E1GΘr2+Θs2Θr2+Θs2×∏rs∈E2GΘr2+Θs2Θr2+Θs2×∏rs∈E3GΘr2+Θs2Θr2+Θs2×∏rs∈E4GΘr2+Θs2Θr2+Θs2=logFG−1FGlog8×1717×8m+8n+8t−24×2020×16mn+16nt−16n−16t+24+16mt−16m×3232×32mnt−16mt−16mn−16nt+8m+8n+8t−8×8080,ENTFTiF2=log2560mnt−216+288m+288n+288t−768mn−768mt−768nt−log8×17172560mnt−216+288m+288n+288t−768mn−768mt−768nt−log8m+8n+8t−24×20202560mnt−216+288m+288n+288t−768mn−768mt−768nt−log16mn+16nt−16n−16t+24+16mt−16m×32322560mnt−216+288m+288n+288t−768mn−768mt−768nt−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×80802560mnt−216+288m+288n+288t−768mn−768mt−768nt.(v)
The Augmented Zagreb Entropy ofTiF2m,n,t. We computed the augmented Zagreb index and augmented Zagreb entropy as follows:(19)AZITiF2=672643375+767363375m+767363375n+767363375t−7454723375mn−7454723375mt−7454723375nt+131072125mnt,ENTAZITiF2=logAZIG−1AZIGlog∏rs∈E1GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E2GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E3GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E4GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23=logAZIG−1AZIGlog8×642764/27×8m+8n+8t−24×88×16mn+16nt−16n−16t+24+16mt−16m×51227512/27×32mnt−16mt−16mn−16nt+8m+8n+8t−8×40961254096/125,ENTAZITiF2=logAZITiF2−log8×64/2764/27AZITiF2−log8m+8n+8t−24×88AZITiF2−log16mn+16nt−16n−16t+24+16mt−16m×512/27512/27AZITiF2−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×4096/1254096/125AZITiF2.


## 3.1. Results for Crystallographic Structure ofTiF2m,n,t

(i)
The First Zagreb Entropy ofTiF2m,n,t. We calculated the first Zagreb index and first Zagreb entropy as follows:(14)M1G=384mnt−64mn−64mt−64nt+16m+16n+16t−8,ENTM1TiF2=logM1−1M1log∏rs∈E1GΘr+ΘsΘr+Θs×∏rs∈E2GΘr+ΘsΘr+Θs×∏rs∈E3GΘr+ΘsΘr+Θs×∏rs∈E4GΘr+ΘsΘr+Θs=logM1−1M1log8×3125×8m+8t−24+8n×46656×16mn+16mt+16nt−16m−16n−16t+24×16777216×32mnt−16mt−16mn−16nt+8m+8n+8t−8×8916100448456,ENTM1TiF2=log384mnt−64mn−64mt−64nt+16m+16n+16t−8−log8×3125384mnt−64mn−64mt−64nt+16m+16n+16t−8−log8m+8n+8t−24×46656×256384mnt−64mn−64mt−64nt+16m+16n+16t−8−log16mn+16nt−16n−16t+24+16mt−16m×16777216384mnt−64mn−64mt−64nt+16m+16n+16t−8−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×8916100448456384mnt−64mn−64mt−64nt+16m+16n+16t−8.(ii)
The Second Zagreb Entropy ofTiF2m,n,t. We computed the second Zagreb index and second Zagreb entropy as follows:(15)M2G=1024mnt−256mn−256mt−256nt+64m+64n+64t−32,ENTM2TiF2=logM2−1M2log∏rs∈E1GΘr×ΘsΘr×Θs×∏rs∈E2GΘr×ΘsΘr×Θs×∏rs∈E3GΘr×ΘsΘr×Θs×∏rs∈E4GΘr×ΘsΘr×Θs=logM2−1M2log8×256×8m+8n+8t−24×16777216×16mn+16nt−16n−16t+24+16mt−16m×1616×32mnt−16mt−16mn−16nt+8m+8n+8t−8×3232,(16)ENTM2TiF2=log1024mnt−256mn−256mt−256nt+64m+64n+64t−32−log8×2561024mnt−256mn−256mt−256nt+64m+64n+64t−32−log8m+8n+8t−24×167772161024mnt−256mn−256mt−256nt+64m+64n+64t−32−log16mn+16nt−16m+16mt−16n−16t+24×16161024mnt−256mn−256mt−256nt+64m+64n+64t−32−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×32321024mnt−256mn−256mt−256nt+64m+64n+64t−32.(iii)
The Hyper Zagreb Entropy ofTiF2m,n,t. We computed the hyper Zagreb index and hyper Zagreb entropy as follows:(17)HMG=4608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280,ENTHMTiF2=logHM−1HMlog∏rs∈E1GΘr+Θs2Θr+Θs2×∏rs∈E2GΘr+Θs2Θr+Θs2×∏rs∈E3GΘr+Θs2Θr+Θs2×∏rs∈E4GΘr+Θs2Θr+Θs2,ENTHMTiF2=logHM−1HMlog8×2525×8m+8n+8t−24×3636×16mn+16nt−16m+16mt−16n−16t+24×6464×32mnt−16mt−16mn−16nt+8m+8n+8t−8×144144,ENTHMTiF2=log4608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log8×25254608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log8m+8n+8t−24×36364608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log16mn+16nt−16m+16mt−16n−16t+24×64644608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×1441444608mnt−1280mn−1280mt−1280nt+416m+416n+416t−280.(iv)
The Forgotten Entropy ofTiF2m,n,t. We computed the forgotten index and forgotten entropy as follows:(18)FG=2560mnt−216+288m+288n+288t−768mn−768mt−768nt,ENTFTiF2=logFG−1FGlog∏rs∈E1GΘr2+Θs2Θr2+Θs2×∏rs∈E2GΘr2+Θs2Θr2+Θs2×∏rs∈E3GΘr2+Θs2Θr2+Θs2×∏rs∈E4GΘr2+Θs2Θr2+Θs2=logFG−1FGlog8×1717×8m+8n+8t−24×2020×16mn+16nt−16n−16t+24+16mt−16m×3232×32mnt−16mt−16mn−16nt+8m+8n+8t−8×8080,ENTFTiF2=log2560mnt−216+288m+288n+288t−768mn−768mt−768nt−log8×17172560mnt−216+288m+288n+288t−768mn−768mt−768nt−log8m+8n+8t−24×20202560mnt−216+288m+288n+288t−768mn−768mt−768nt−log16mn+16nt−16n−16t+24+16mt−16m×32322560mnt−216+288m+288n+288t−768mn−768mt−768nt−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×80802560mnt−216+288m+288n+288t−768mn−768mt−768nt.(v)
The Augmented Zagreb Entropy ofTiF2m,n,t. We computed the augmented Zagreb index and augmented Zagreb entropy as follows:(19)AZITiF2=672643375+767363375m+767363375n+767363375t−7454723375mn−7454723375mt−7454723375nt+131072125mnt,ENTAZITiF2=logAZIG−1AZIGlog∏rs∈E1GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E2GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E3GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23×∏rs∈E4GΘrΘsΘr+Θs−23ΘrΘs/Θr+Θs−23=logAZIG−1AZIGlog8×642764/27×8m+8n+8t−24×88×16mn+16nt−16n−16t+24+16mt−16m×51227512/27×32mnt−16mt−16mn−16nt+8m+8n+8t−8×40961254096/125,ENTAZITiF2=logAZITiF2−log8×64/2764/27AZITiF2−log8m+8n+8t−24×88AZITiF2−log16mn+16nt−16n−16t+24+16mt−16m×512/27512/27AZITiF2−log32mnt−16mt−16mn−16nt+8m+8n+8t−8×4096/1254096/125AZITiF2.


## 4. Comparisons and Discussion forCu2Om,n,t

We develop Tables5 and 6 for tiny estimations of m,n,t for the structure of Cu2Om,n,t. The graphical portrayals of registered outcomes area unit are described in Figures 4–6 for specific estimations of m,n,t.Table 5
Comparison ofENTM1 and ENTM2 for Cu2Om,n,t.
m,n,tENTM1ENTM21,1,11.121.212,2,21.421.613,3,31.821.914,4,42.122.515,5,53.413.32Table 6
Comparison ofENTHM, ENTF, ENTAZI, and ENTJ entropies for Cu2Om,n,t.
m,n,tENTHMENTFENTAZIENTJ1,1,12.122.412.222.422,2,23.653.523.523.413,3,34.544.654.424.814,4,45.255.745.325.515,5,56.436.686.216.72Figure 4
(a) The first Zagreb entropy. (b) The second Zagreb entropy.
(a)(b)Figure 5
(a) The hyper Zagreb entropy. (b) The forgotten entropy.
(a)(b)Figure 6
(a) The augmented Zagreb entropy. (b) The Balaban entropy.
(a)(b)


## 5. Comparisons and Discussion forTiF2m,n,t

Presently, from Tables7 and 8, we are able to notice that while there is not much of a stretch, we see that each one of the estimations of entropy is in increasing request because the estimations of m,n,t are increments. The graphical portrayals of registered outcomes are diagrammatic in Figures 7–9 for specific estimations of m,n,t.Table 7
Comparison ofENTM1 and ENTM2 for TiF2m,n,t.
m,n,tENTM1ENTM21,1,11.221.312,2,21.621.813,3,31.922.314,4,42.322.915,5,53.613.42Table 8
Comparison ofENTHM, ENTF, ENTAZI, and ENTJ entropies for TiF2m,n,t.
m,n,tENTHMENTFENTAZIENTJ1,1,12.022.212.122.322,2,23.453.323.423.513,3,34.344.454.524.714,4,45.555.545.625.415,5,56.636.886.316.22Figure 7
(a) The first Zagreb entropy. (b) The second Zagreb entropy.
(a)(b)Figure 8
(a) The hyper Zagreb entropy. (b) The forgotten entropy.
(a)(b)Figure 9
(a) The augmented Zagreb entropy. (b) The Balaban entropy.
(a)(b)


## 6. Conclusion

In this paper, in lightweight of applied scientist,s entropy, we have a tendency to study the graph entropies known with another information work. We have a tendency to present an association between the gradation established topological indices with gradation established entropies. We are particularly interested in forming the gradation that is based on entropies for crystallographic erection of oxide Cu2Om,n,t and metal difluoride TiF2m,n,t. In addition, the arithmetic estimations of these entropies have been registered in tables that provide the correlation between the gradation focused topological lists and gradation established entropies that drives the United States of America to differentiate the physio-substance possessions of those crystallographic erection of Cu2Om,n,t and TiF2m,n,t.



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*Source: 9936949-2021-06-28.xml*