# On Curvilinear Regression Analysis via Newly Proposed Entropies for Some Benzene Models

Authors: Guangwu Liu; Muhammad Kamran Siddiqui; Shazia Manzoor; Muhammad Naeem; Douhadji Abalo
Journal: Complexity
 (2022)

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## Abstract

To avoid exorbitant and extensive laboratory experiments, QSPR analysis, based on topological descriptors, is a very constructive statistical approach for analyzing the numerous physical and chemical properties of compounds. Therefore, we presented some new entropy measures which are based on the sum of the neighborhood degree of the vertices. Firstly, we made the partition of the edges of benzene derivatives which are based on the degree sum of neighboring vertices and then computed the neighborhood version of entropies. Secondly, we made use of the software SPSS for developing a correlation between newly introduced entropies and the physicochemical properties of benzene derivatives. Our obtained results demonstrated that the critical temperatureCT, critical pressure CP, and critical volume CV can be predicted through fifth geometric arithmetic entropy, second SK entropy, and fifth ND entropy, respectively. Other remaining physical characteristics include Gibb’s energy qℰ, logP, molar refractivity ℳℛ, and Henry’s law ℋℒ that can be predicted by using sixth ND entropy.


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## Body

## 1. Introduction

In chemistry, models are classified into two types. The first type of modeling is based on the quantum-chemical method and theoretically identical models derived from statistical mechanics. The second type consists of chemical thinking, which compares related systems. According to the core principle of chemistry, all the features of matter correlate with its molecular structure, and thus, molecules with similar structures have similar properties. This has resulted in several empirical approaches such as SAR, SPC, QSPR, and QSAR [1, 2].The identical factor of all these methods is the correlation between physicochemical properties and molecular descriptors. There are numerous descriptors to utilize [3], but one family of descriptors has been demonstrated exceptionally basic and valuable in foreseeing multiple molecular properties. These are known as topological indices TℐS. Topological indices TℐS are numerical parameters that are correlated with a graph and help identify its topology. There are three types of TℐS: degree-based TℐS, spectrum-based TℐS, and distance-based TℐS [4, 5]. Degree-based TℐS, which are defined in terms of the degrees of the vertices of a graph, are one of the most studied TℐS used in mathematical chemistry [6]. Imran et al. [7] discuss the topological properties of symmetric chemical structures. Zou et al. [8] computed topological indices for polyphenylene. Recently, some neighborhood versions of TℐS have also been introduced [9, 10]. Zhang et al. [11–13] discuss the topological indices of generalized bridge molecular graphs, carbon nanotubes, and the product of chemical graphs.The idea of entropy first appeared in thermodynamics in the nineteenth century, when it was closely linked to the heat flow and a key component of the second law of thermodynamics. Subsequently, statistical mechanics used the notion to illuminate thermodynamics physically. Leading physicists such as Boltzmann and Gibbs [14, 15], who employed entropy as a measure of the disorder of the massive dynamical system underlying molecule collections, were responsible for this. Fisher employed similar notions in the establishment of the foundations of theoretical statistics in 1920 [16]. In the 1950s, Kullback and Leibler [17] developed this concept further. Zhang et al. [18–20] provided the physical analysis of heat for the formation and entropy of ceria oxide.Claude Shannon, a mathematician and electrical engineer who worked at Bell Labs in New Jersey in the midtwentieth century, identified the link between entropy and information content [21]. The concept formed a key element of the emerging field of information theory at that time. Afterward, in the 1950s, Jaynes explained the explicit association between Shannon’s entropy and that of statistical mechanics in a series of excellent works [22, 23]. Since then, information theory has found applications in a variety of fields, including networking analysis, mathematical statistics, complexity theory, and financial mathematics.We made a bibliometric analysis grounded on the Scopus databasehttps://www.scopus.com. This analysis is based upon 986 research articles with entropy and graph entropy [24–26] as key factors. The percentage of publications in different subjects is shown in Figure 1.Figure 1
Bibliometric analysis of the publications of entropy in various disciplines.We made a bibliometric analysis of the research conducted in different countries on the concept of entropy in Figure2.Figure 2
Bibliometric analysis of countrywide research based on the topic of entropy. The size of the circles denotes the frequency of articles, and distinct colors are used to signify distinct clusters.In recent times, another approach which is a bit different in the literature, namely, using the concept of Shannon’s entropy in terms of topological indices was introduced by Manzoor et al. in [27]. Continuing their work, they also introduced eccentricity-based graph entropies [28, 29] and bond additive graph entropies [30]. In this paper, the present authors formulated some new graph entropies, namely, neighborhood versions of graph entropies. The graph entropy is represented in the following formula:(1)EIB=−∑u′v′∈EBIu′v′∑uv∈EBIuvlog2Iu′v′∑uv∈EBIuv,where VB is the vertex set, EB is the edge set, and Iuv is the edge weight of the edge uv.(i)
Neighborhood version of forgotten entropyIfIuv=Au2+Av2, then(2)∑uv∈EBIuv=∑uv∈EBAu2+Av2=FN∗.So, equation (1) is called the neighborhood version of forgotten entropy.(3)EFN∗B=log2ℍN∗−1ℍN∗∑i=1q∑uv∈EiBAu2+Av2log2Au2+Av2.(ii)
Neighborhood version of second Zagreb entropyIfIuv=Au×Av, then(4)∑uv∈EBIuv=∑uv∈EBAu×Av=M2∗.So, equation (2) is called the neighborhood version of second Zagreb entropy.(5)EM2∗B=log2M2∗−1M2∗∑i=1q∑uv∈EiBAu×Avlog2Au×Av.(iii)
Neighborhood version of hyper-Zagreb entropyIfIuv=Au+Av2, then(6)∑uv∈EBIuv=∑uv∈EBAu+Av2=ℍMN.So, equation (3) is called the neighborhood version of hyper-Zagreb entropy.(7)EℍMNB=log2ℍMN−1ℍMN∑i=1q∑uv∈EiBAu+Av2log2Au+Av2.(iv)
FirstND entropyIfIuv=Au×Av, then(8)∑uv∈EBIuv=∑uv∈EBAu×Av=ℕD1.So, equation (4) is called the first ND entropy.(9)EℕD1B=log2ℕD1−1ℕD1∑i=1q∑uv∈EiBAu×Avog2Au×Av1.(v)
SecondND entropyIfIuv=1/Au+Av, then(10)∑uv∈EBIuv=∑uv∈EB1Au+Av=ℕD2.So, equation (5) is called the second ND entropy.(11)EℕD2B=log2ℕD2−1ℕD2∑i=1q∑uv∈EiB1Au+Avlog21Au+Av.(vi)
ThirdND entropy.IfIuv=Au×AvAu+Av, then(12)∑uv∈EBIuv=∑uv∈EBAu×AvAu+Av=ℕD3.So, equation (6) is called the third ND entropy.(13)EℕD3B=log2ℕD3−1ℕD3∑i=1q∑uv∈EiBAu×AvAu+Avlog2Au×AvAu+Av.(vii)
FourthND entropy.IfIuv=1/Au×Av, then(14)∑uv∈EBIuv=∑uv∈EB1Au×Av=ℕD4.So, equation (7) is called the fourth ND entropy.(15)EℕD4B=log2ℕD4−1ℕD4∑i=1q∑uv∈EiB1Au×Avlog21Au×Av.(viii)
FifthND entropy.IfIuv=Au/Av+Av/Au, then(16)∑uv∈EBIuv=∑uv∈EBAuAv+AvAu=ℕD5.So, equation (8) is called the fifth ND entropy.(17)EℕD5B=log2ℕD5−1ℕD5∑i=1q∑uv∈EiBAuAv+AvAulog2AuAv+AvAu.(ix)
SixthND entropyIfIuv=drAu+dsAv, then(18)∑uv∈EBIuv=∑uv∈EBdrAu+dsAv=ℕD6.So, equation (9) is called the sixth ND entropy.(19)EℕD6B=log2ℕD6−1ℕD6∑i=1q∑uv∈EiBdrAu+dsAvlog2drAu+dsAv.(x)
Neighborhood version ofSK entropyIfIuv=Au+Av/2, then(20)∑uv∈EBIuv=∑uv∈EBAu+Av2=SKN.So, equation (10) is called the neighborhood version of SK entropy.(21)ESKNB=log2SKN−1SKN∑i=1q∑uv∈EiBAu+Av2log2Au+Av2.(xi)
Neighborhood version of firstSK entropy.IfIuv=Au×Av/2, then(22)∑uv∈EBIuv=∑uv∈EBAu×Av2=SK1N.So, equation (11) is called the neighborhood version of SK entropy.(23)ESK1NB=log2SK1N−1SK1N∑i=1q∑uv∈EiBAu×Av2log2Au×Av2.(xii)
Neighborhood version of secondSK entropy.IfIuv=Au+Av/22, then(24)∑uv∈EBIuv=∑uv∈EBAu+Av22=SK1N.So, equation (12) is called the neighborhood version of SK entropy.(25)ESK2NB=log2SK2N−1SK2N∑i=1q∑uv∈EiBAu+Av22log2Au+Av22.(xiii)
Neighborhood version of modified Randić entropy.IfIuv=1/maxAu,Av, then(26)∑uv∈EBIuv=∑uv∈EB1maxAu,Av=mℝN.So, equation (13) is called the neighborhood version of modified Randi ć entropy.(27)EmℝNB=log2mℝN−1mℝN∑i=1q∑uv∈EiB1maxAu,Avlog21maxAu,Av.(xiv)
Neighborhood version of redefined first Zagreb entropy.IfIuv=Au+Av/Au×Av, then(28)∑uv∈EBIuv=∑uv∈EBAu+AvAu×Av=ℝez1N.So, equation (14) is called the neighborhood version of redefined first Zagreb entropy.(29)Eℝez1NB=log2ℝez1N−1ℝez1N∑i=1q∑uv∈EiBAu+AvAu×Avlog2Au+AvAu×Av.(xv)
Neighborhood version of redefined second Zagreb entropy.IfIuv=Au×Av/Au+Av, then(30)∑uv∈EBIuv=∑uv∈EBAu×AvAu+Av=ℝez2N.So, equation (15) is called the neighborhood version of redefined second Zagreb entropy.(31)Eℝez2NB=log2ℝez2N−1ℝez2N∑i=1q∑uv∈EiBAu×AvAu+Avlog2Au×AvAu+Av.(xvi)
Neighborhood version of atom bond connectivity entropy.IfIuv=Au+Av−2/Au×Av, then(32)∑uv∈EBIuv=∑uv∈EBAu+Av−2Au×Av=ABCN.So, equation (16) is called the neighborhood version of atom bond connectivity entropy [31].(33)EABCNB=log2ABCN−1ABCN∑i=1q∑uv∈EiBAu+Av−2Au×Avlog2Au+Av−2Au×Av.(xvii)
Neighborhood version of geometric arithmetic entropy.IfIuv=2×Au×Av/Au+Av, then(34)∑uv∈EBIuv=∑uv∈EB2×Au×AvAu+Av=GAN.So, equation (17) is called the neighborhood version of geometric arithmetic entropy [31].(35)EGANB=log2GAN−1GAN∑i=1q∑uv∈EiB2×Au×AvAu+Avlog22×Au×AvAu+Av.


## 2. Curvilinear Regression Analysis of Proposed Entropies

In this section, we analyze the entropies given above with the following physical characteristics of the benzene derivatives [32, 33]: critical pressure CP, critical temperature CT, critical volume CV, Gibb’s energy qℰ, logP, molar refractivity ℳℛ, and Henry’s law ℋℒ. The experimental values of physical characteristics of benzene derivatives have been referred to from [34] and presented in Table 1. We have presented the values of the proposed indices for the benzene derivatives in Tables 2 and 3. Figures 3 and 4 illustrate the structure of benzene derivatives.Table 1
The physical characteristics of benzene derivatives.
Chemical structuresCPHLGECTlogPCVMRBenzeneS147.690.66121.68323.792.03263.525.28NaphthaleneS238.971.67252.38484.953.03409.542.45PhenanthreneS332.432.68383.08586.114.03555.559.62AnthraceneS432.432.68383.08586.114.03555.559.62ChryseneS527.413.69513.78650.85.03701.576.79Benz[a]anthreneS627.413.69513.78650.85.03701.576.79TriphenyleneS727.413.69513.78650.85.03701.576.79TetraceneS827.413.69513.78650.85.9701.576.79Benzo[a]pyreneS926.084.48621.88689.175.34765.585.53Benzo[e]pyreneS1026.084.48621.88689.175.34765.585.53PeryleneS1126.084.48621.88689.175.34765.585.53Benzo[ghi]peryleneS1224.855.27729.98728.065.66829.594.28Dibenzo[a, c]anthracene S1323.474.7644.48714.536.02847.593.96Dibenzo[a, h]anthracene S1423.474.7644.48714.536.02847.593.96Dibenzo[a, j]anthracene S1523.474.7644.48714.536.02847.593.96PiceneS1623.474.7644.48714.536.02847.593.96CoroneneS1723.76.06838.08767.685.98893.5103.02PyreneS1830.733.47491.18625.655.08619.568.36Table 2
Numerical values of neighborhood version of entropies.
CompoundEFN∗EM2∗EℍMNEℕD1EℕD2EℕD3EℕD4EℕD5EℕD6S12.58502.58502.58502.58492.58492.58492.58492.58492.5849S23.35653.35993.35833.43443.45323.24163.43593.45913.3767S33.84663.86453.86473.96513.99113.71263.90323.99913.8986S43.90623.90483.90563.97393.99283.80953.97103.99973.8981S54.26014.25754.25914.35683.34134.11343.74534.39124.2926S64.27124.28324.27204.36284.38424.16994.35994.39184.3107S74.21584.21034.21354.34294.37944.03334.34264.39084.3401S84.31404.31204.31314.36944.38564.23734.36424.39214.3302S94.45044.44334.44854.34804.57564.29594.54774.58434.4903S104.42594.35874.36334.54034.28384.47254.53954.58394.4729S114.42864.42064.42494.54104.57384.25504.54084.58404.4736S124.61334.60554.60974.71574.74515.22904.71594.75424.6580S134.55614.55064.55384.66004.69844.39974.65804.69354.5958S144.59254.59014.59154.67044.69224.47704.66714.69984.6181S154.59254.59014.59154.67044.69224.47704.66714.69984.6181S164.57534.57114.57364.66594.69124.43664.66404.69914.6118S174.78054.77354.77714.87294.89884.62524.87524.90644.8235S184.11114.10794.10964.21314.23964.10954.21604.24764.1527Table 3
Numerical values of neighborhood version of entropies.
CompoundESKNESK1NESK2NEmℝNEℝez1NEℝez2NEABCNEGANS12.58492.58492.58492.58492.58492.58492.58492.5849S23.43383.35993.36243.42443.43673.43503.45613.4594S33.28303.61263.86473.95333.90963.94793.99503.9999S43.92653.90483.90563.86413.96033.97383.99593.9990S54.35674.25754.14294.34484.35694.34913.38694.3922S64.36284.28324.28414.31824.36034.36274.38754.3923S74.34314.21034.29854.33344.34364.34263.38474.3922S84.36954.31204.31314.36094.36424.36913.38844.3923S94.54884.44624.44854.55634.54804.54824.57914.5849S104.54094.41834.42244.53604.53974.53954.78984.5849S114.54174.42064.42494.29404.52134.54014.57774.5849S124.71654.63534.60974.71494.71584.71484.74834.7548S134.66034.55064.78474.64954.65884.62084.69404.7004S144.65354.59014.59154.65834.66754.67024.67024.6955S154.65354.59014.59154.65834.66754.67024.67024.6955S164.66604.57114.57364.65374.66474.66564.69494.7004S174.87364.77354.77714.87074.87504.88194.90124.9069S184.21324.10794.10964.21114.21624.21294.24264.2479Figure 3
An illustration of benzene derivatives for the first type.Figure 4
An illustration of benzene derivatives for the second type.We analyze the topological indices vis a vis the physical characteristics using the following regression models:(36)P=a˜+b˜X,where P is the physical property, X is the entropy, and a˜ and b˜ represent the coefficient and constant, respectively. For the seven physicochemical properties, we found the correlation between the properties and the seventeen entropies proposed by us. We now present the analysis of the linear model based on the R2 value. Based on the recommendations of the International Academy of Mathematical Chemistry (IAMC), we have only considered regression models with R2≥0.8.


### 2.1. Linear Regression Models

Using equation (36), we obtained the linear regression models (LRM) for the seven physicochemical properties via each of the proposed indices, and the results are presented in Tables 4–9.Table 4
The significant LRM: critical temperature of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9980.9960.9966.66225Neighborhood version of second Zagreb entropy0.9980.9960.9966.87851FirstND entropy0.9980.9970.9966.59836SecondND entropy0.9990.9970.9975.94872ThirdND entropy0.9990.9970.9966.18795FourthND entropy0.9990.9980.9966.41912FifthND entropy0.9990.9980.9976.12268SixthND entropy0.9990.9980.9966.43054SK entropy0.9610.9230.91829.78877FirstSK entropy0.9970.9930.9929.06386SecondSK entropy0.9970.9940.9938.62473Neighborhood version of modified Randic entropy0.9970.9950.9938.55077Neighborhood version of redefined first Zagreb entropy0.9990.9980.9975.48691Fourth atom bond connectivity entropy0.9990.9980.9975.71456Neighborhood version of hyper-Zagreb entropy0.9980.9950.9957.23697Neighborhood version of redefined second Zagreb entropy0.9990.9980.9975.4923Fifth geometric arithmetic entropy0.999.9980.9975.45456Table 5
The significant LRM: critical pressureCP of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sNeighborhood version of second Zagreb entropy0.9930.9850.9830.81336FirstND entropy0.9930.9860.9830.8299SecondND entropy0.9930.9860.9810.85676ThirdND entropy0.9940.9880.9830.812FourthND entropy0.9940.9880.9820.8429FifthND entropy0.9940.9890.980.8797SixthND entropy0.9950.9890.9790.90475SK entropy0.9580.9170.9121.87022FirstSK entropy0.9910.9820.980.88984SecondSKentropy0.9950.9890.9870.72501Neighborhood version of modified Randic entropy0.9950.9890.9860.72720Neighborhood version of redefined first Zagreb entropy0.9950.9890.9850.77312Fourth atom bond connectivity entropy0.9950.9900.9840.7966Neighborhood version of hyper-Zagreb entropy0.9920.9850.9840.79918Neighborhood version of redefined second Zagreb entropy0.9920.9850.9830.825Fifth geometric arithmetic entropy0.9940.9880.9860.75756Table 6
The significant LRM: critical volume of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9810.9630.96133.08152Neighborhood version of second Zagreb entropy0.9820.9640.95933.90248FirstND entropy0.9820.9650.95734.7795SecondND entropy0.9820.9650.95436.03779ThirdND entropy0.9820.9650.9537.50893FourthND entropy0.9820.9650.94639.09405FifthNDentropy0.9910.9830.96730.46046SixthND entropy0.9870.9740.95635.1254SK entropy0.9610.9230.91847.88923FirstSK entropy0.9820.9650.96133.26689SecondSK entropy0.9840.9680.96133.15523Neighborhood version of modified Randic entropy0.9840.9680.95834.26668Neighborhood version of redefined first Zagreb entropy0.9850.9710.95934.09436Fourth atom bond connectivity entropy0.9860.9710.95635.26197Neighborhood version of hyper-Zagreb entropy0.9810.9620.95933.85292Neighborhood version of redefined second Zagreb entropy0.9820.9640.95933.95369Fifth geometric arithmetic entropy0.9820.9640.95635.14404Table 7
The significant LRM: Gibb’s energyGE of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9570.9170.91151.08458Neighborhood version of second Zagreb entropy0.9600.9220.91250.92216First ND entropy0.9620.9250.90851.96242Second ND entropy0.9630.9270.90453.0685Third ND entropy0.9670.9350.90951.91691Fourth ND entropy0.9670.9360.90154.00602Fifth ND entropy0.9730.9460.90852.04307SixthNDentropy0.9780.9560.91749.48688SK entropy0.9430.8890.88258.99765First SK entropy0.9590.920.9151.58078Second SK entropy0.9600.9210.90453.17509Neighborhood version of modified Randic entropy0.9600.9220.89854.93049Neighborhood version of redefined first Zagreb entropy0.9600.9220.88957.16546Fourth atom bond connectivity entropy0.9600.9220.87959.6619Neighborhood version of hyper-Zagreb entropy0.9550.9130.90752.24284Neighborhood version of redefined second Zagreb entropy0.9560.9130.90253.85496Fifth geometric arithmetic entropy0.9590.920.90353.56609Table 8
The significant LRM:logP of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9690.940.9360.28258Neighborhood version of second Zagreb entropy0.9700.940.9320.29159FirstND entropy0.9700.940.9280.30098SecondND entropy0.9710.9430.9260.30508ThirdND entropy0.9710.9430.920.31671FourthND entropy0.9720.9440.9140.32831FifthND entropy0.9810.9630.9360.28207SixthNDentropy0.9850.9710.9450.26278SK entropy0.9510.9050.8990.35614FirstSK entropy0.9710.9430.9350.28406SecondSK entropy0.9720.9450.9330.28894Neighborhood version of modified Randic entropy0.9720.9450.9280.29977Neighborhood version of redefined first Zagreb entropy0.9770.9550.9360.28365Fourth atom bond connectivity entropy0.9770.9550.930.29626Neighborhood version of hyper-Zagreb entropy0.9690.940.9360.2827Neighborhood version of redefined second Zagreb entropy0.9750.9510.9440.2637Fifth geometric arithmetic entropy0.9750.9510.940.27288Table 9
The significant LRM: molar refractivityMR of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9810.9620.964.02512Neighborhood version of second Zagreb entropy0.9820.9640.9594.09119FirstND entropy0.9820.9640.9574.1829SecondND entropy0.9820.9640.9544.34063ThirdND entropy0.9820.9650.954.49344FourthND entropy0.9820.9650.9464.67932FifthND entropy0.9870.9750.9574.18104SixthNDentropy0.9920.9830.9693.56484SK entropy0.9610.9240.925.70614FirstSK entropy0.9820.9650.964.03334SecondSK entropy0.9830.9660.9584.10153Neighborhood version of modified Randic entropy0.9830.9660.9554.25288Neighborhood version of redefined first Zagreb entropy0.9840.9680.9544.31552Fourth atom bond connectivity entropy0.9840.9680.9514.47577Neighborhood version of hyper-Zagreb entropy0.9800.960.9584.14024Neighborhood version of redefined second Zagreb entropy0.9800.9610.9564.21432Fifth geometric arithmetic entropy0.9810.9620.9534.34898Table4 shows the relation between entropy and critical temperature. In Table 4, we can easily see that all the entropies show the highest positive correlation with critical temperature. The most significant regression models are shown as follows:(37)CT=−159.668+185.780EGANCT=−168.156+191.581EℕD6CT=−111.034+177.475EℕD4CT=−161.636+187.887Eℝez1NCT=−148.444+182.909EABCNCT=−165.387+664Eℝez2N.Figure5 shows that all the points fall near the fitted line. From all entropies, the fifth geometric arithmetic entropy shows the best relation with critical temperature CT. So, the critical temperature can be predicted by using the fifth geometric arithmetic entropy.Figure 5
Graphical representation of LRM between critical temperature and the fifth geometric arithmetic entropy.Table5 shows the relation between entropy and critical pressure CP, and all the entropies show the highest positive correlation with critical pressure. The most significant regression models are shown as follows:(38)CP=77.137−11.477ESK2NCP=96.285−11.005EℕD4CP=77.014−11.192EℕD5CP=77.547−11.545EℕD6CP=76.958−11.329EmℝNCP=77.121−11.314Eℝez1N.From all the entropies, the secondSK entropy shows the best relation with critical pressure CP, and all the points fall near the fitted line shown in Figure 6. So, the critical pressure CP can be predicted by using the second SK entropy.Figure 6
Graphical representation of LRM between critical pressureCP and the second SK entropy.Table6 shows the relation between entropy and critical volume. From Table 6, we can easily see that all the entropies show the highest positive correlation with critical volume. The most significant regression models are shown as follows:(39)CV=−569.955+293.079EℕD5CV=−589.308+303.572EℕD6CV=−576.671+297.184Eℝez1NCV=−551.360+288.287EABCNCV=−577.343+65.676ESK2NCV=−573.522+297.843EmℝN.From all the entropies, the fifthND entropy shows the best relation with critical volume CV, and all the points fall near the fitted line as can be seen in Figure 7. So, the critical temperature can be predicted by using the fifth ND entropy.Figure 7
Graphical representation of LRM between critical volume and the fifth ND entropy.Table7 shows the relation between entropy and Gibb’s energy qℰ of benzene derivatives. From Table 7, we can easily see that all the entropies show the highest positive correlation with Gibb’s energy qℰ.(40)qℰ=−751.66+303.591EℕD6qℰ=−732.155+293.063EℕD5qℰ=−674.923+284.469EℕD4qℰ=−639.259+283.942EℕD3qℰ=−725.293+293.202ESK2Nqℰ=−739.593+297.729EℕD1.Of all the entropies, the fifth ND entropy shows the best relation with Gibb’s energyqℰ, and all the points fall near the fitted line as seen in Figure 8. So, Gibb’s energy can be predicted by using the sixth ND entropy.Figure 8
Graphical representation of LRM between Gibb’s energyGE and the sixth ND entropy.Table8 shows the relation between entropy and logP. From Table 8, we can easily see that all the entropies show the highest positive correlation with critical volume. The most significant regression models are shown as follows:(41)logP=−3.443+1.998EℕD6logP=−3.304+1.927EℕD5logP=−3.159+1.890EℕD4logP=−3.357+1.956Eℝez1NlogP=−3.300+1.925EGANlogP=−3.371+1.958Eℝez2N.From all the entropies, the sixthND entropy shows the best relation with logP, and all the points fall near the fitted line as can be seen in Figure 9. So, the logP can be predicted by using the sixth ND entropy.Figure 9
Graphical representation of LRM betweenlogP and the sixth ND entropy.Table9 shows the relation between entropy and molar refractivity ℳℛ of benzene derivatives. From Table 9, we can easily see that all the entropies show the highest positive correlation with molar refractivity ℳℛ.(42)ℳℛ=−77.645+36.463EℕD6ℳℛ=−75.316+35.202EℕD5ℳℛ=−76.136+35.698Eℝez1Nℳℛ=−73.151+34.642EABCNℳℛ=−75.860+36.134ESK2Nℳℛ=−75.730+35.771E.mℝNOf all the entropies, the sixthND entropy shows the best relation with molar refractivity ℳℛ, and all the points fall near the fitted line as seen in Figure 10. So, the molar refractivity can be predicted by using the sixth ND entropy.Figure 10
Graphical representation of LRM between molar refractivity and the sixthND entropy.Table10 shows the relation between entropy and Henry’s law. From Table 10, we can easily see that all the entropies show the highest positive correlation with Henry’s law. The most significant regression models are shown as follows:(43)ℋℒ=−5.991+2.318EℕD6ℋℒ=−5.842+2.237EℕD5ℋℒ=−5.395+2.169EℕD4ℋℒ=−5.102+2.160EℕD3ℋℒ=−5.793+2.239EℕD2ℋℒ=−5.902+2.074EℕD1.Table 10
The significant LRM: Henry’s law of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9620.9260.9210.36535Neighborhood version of second Zagreb entropy0.9650.9310.9220.36445FirstND entropy0.9660.9330.9190.37174SecondND entropy0.9670.9350.9150.38046ThirdND entropy0.9700.9420.9180.37439FourthND entropy0.9710.9420.9110.38944FifthND entropy0.9760.9520.9190.37152SixthNDentropy0.9810.9620.9290.3485SK entropy0.9470.8970.890.43156FirstSK entropy0.9640.930.920.36835SecondSK entropy0.9640.930.9150.38044Neighborhood version of modified Randic entropy0.9650.930.9090.39337Neighborhood version of redefined first Zagreb entropy0.9650.930.9010.40943Fourth atom bond connectivity entropy0.9650.9310.8930.42717Neighborhood version of hyper-Zagreb entropy0.9600.9220.9180.37422Neighborhood version of redefined second Zagreb entropy0.9610.9230.9120.38616Fifth geometric arithmetic entropy0.9630.9280.9130.38561From all the entropies, the sixthND entropy shows the best relation with Henry’s law ℋℒ, and all the points fall near the fitted line as can be seen in Figure 11. So, Henry’s law ℋℒ can be predicted by using the sixth ND entropy.Figure 11
Graphical representation of LRM between Henry’s law and the sixthND entropy.


## 2.1. Linear Regression Models

Using equation (36), we obtained the linear regression models (LRM) for the seven physicochemical properties via each of the proposed indices, and the results are presented in Tables 4–9.Table 4
The significant LRM: critical temperature of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9980.9960.9966.66225Neighborhood version of second Zagreb entropy0.9980.9960.9966.87851FirstND entropy0.9980.9970.9966.59836SecondND entropy0.9990.9970.9975.94872ThirdND entropy0.9990.9970.9966.18795FourthND entropy0.9990.9980.9966.41912FifthND entropy0.9990.9980.9976.12268SixthND entropy0.9990.9980.9966.43054SK entropy0.9610.9230.91829.78877FirstSK entropy0.9970.9930.9929.06386SecondSK entropy0.9970.9940.9938.62473Neighborhood version of modified Randic entropy0.9970.9950.9938.55077Neighborhood version of redefined first Zagreb entropy0.9990.9980.9975.48691Fourth atom bond connectivity entropy0.9990.9980.9975.71456Neighborhood version of hyper-Zagreb entropy0.9980.9950.9957.23697Neighborhood version of redefined second Zagreb entropy0.9990.9980.9975.4923Fifth geometric arithmetic entropy0.999.9980.9975.45456Table 5
The significant LRM: critical pressureCP of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sNeighborhood version of second Zagreb entropy0.9930.9850.9830.81336FirstND entropy0.9930.9860.9830.8299SecondND entropy0.9930.9860.9810.85676ThirdND entropy0.9940.9880.9830.812FourthND entropy0.9940.9880.9820.8429FifthND entropy0.9940.9890.980.8797SixthND entropy0.9950.9890.9790.90475SK entropy0.9580.9170.9121.87022FirstSK entropy0.9910.9820.980.88984SecondSKentropy0.9950.9890.9870.72501Neighborhood version of modified Randic entropy0.9950.9890.9860.72720Neighborhood version of redefined first Zagreb entropy0.9950.9890.9850.77312Fourth atom bond connectivity entropy0.9950.9900.9840.7966Neighborhood version of hyper-Zagreb entropy0.9920.9850.9840.79918Neighborhood version of redefined second Zagreb entropy0.9920.9850.9830.825Fifth geometric arithmetic entropy0.9940.9880.9860.75756Table 6
The significant LRM: critical volume of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9810.9630.96133.08152Neighborhood version of second Zagreb entropy0.9820.9640.95933.90248FirstND entropy0.9820.9650.95734.7795SecondND entropy0.9820.9650.95436.03779ThirdND entropy0.9820.9650.9537.50893FourthND entropy0.9820.9650.94639.09405FifthNDentropy0.9910.9830.96730.46046SixthND entropy0.9870.9740.95635.1254SK entropy0.9610.9230.91847.88923FirstSK entropy0.9820.9650.96133.26689SecondSK entropy0.9840.9680.96133.15523Neighborhood version of modified Randic entropy0.9840.9680.95834.26668Neighborhood version of redefined first Zagreb entropy0.9850.9710.95934.09436Fourth atom bond connectivity entropy0.9860.9710.95635.26197Neighborhood version of hyper-Zagreb entropy0.9810.9620.95933.85292Neighborhood version of redefined second Zagreb entropy0.9820.9640.95933.95369Fifth geometric arithmetic entropy0.9820.9640.95635.14404Table 7
The significant LRM: Gibb’s energyGE of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9570.9170.91151.08458Neighborhood version of second Zagreb entropy0.9600.9220.91250.92216First ND entropy0.9620.9250.90851.96242Second ND entropy0.9630.9270.90453.0685Third ND entropy0.9670.9350.90951.91691Fourth ND entropy0.9670.9360.90154.00602Fifth ND entropy0.9730.9460.90852.04307SixthNDentropy0.9780.9560.91749.48688SK entropy0.9430.8890.88258.99765First SK entropy0.9590.920.9151.58078Second SK entropy0.9600.9210.90453.17509Neighborhood version of modified Randic entropy0.9600.9220.89854.93049Neighborhood version of redefined first Zagreb entropy0.9600.9220.88957.16546Fourth atom bond connectivity entropy0.9600.9220.87959.6619Neighborhood version of hyper-Zagreb entropy0.9550.9130.90752.24284Neighborhood version of redefined second Zagreb entropy0.9560.9130.90253.85496Fifth geometric arithmetic entropy0.9590.920.90353.56609Table 8
The significant LRM:logP of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9690.940.9360.28258Neighborhood version of second Zagreb entropy0.9700.940.9320.29159FirstND entropy0.9700.940.9280.30098SecondND entropy0.9710.9430.9260.30508ThirdND entropy0.9710.9430.920.31671FourthND entropy0.9720.9440.9140.32831FifthND entropy0.9810.9630.9360.28207SixthNDentropy0.9850.9710.9450.26278SK entropy0.9510.9050.8990.35614FirstSK entropy0.9710.9430.9350.28406SecondSK entropy0.9720.9450.9330.28894Neighborhood version of modified Randic entropy0.9720.9450.9280.29977Neighborhood version of redefined first Zagreb entropy0.9770.9550.9360.28365Fourth atom bond connectivity entropy0.9770.9550.930.29626Neighborhood version of hyper-Zagreb entropy0.9690.940.9360.2827Neighborhood version of redefined second Zagreb entropy0.9750.9510.9440.2637Fifth geometric arithmetic entropy0.9750.9510.940.27288Table 9
The significant LRM: molar refractivityMR of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9810.9620.964.02512Neighborhood version of second Zagreb entropy0.9820.9640.9594.09119FirstND entropy0.9820.9640.9574.1829SecondND entropy0.9820.9640.9544.34063ThirdND entropy0.9820.9650.954.49344FourthND entropy0.9820.9650.9464.67932FifthND entropy0.9870.9750.9574.18104SixthNDentropy0.9920.9830.9693.56484SK entropy0.9610.9240.925.70614FirstSK entropy0.9820.9650.964.03334SecondSK entropy0.9830.9660.9584.10153Neighborhood version of modified Randic entropy0.9830.9660.9554.25288Neighborhood version of redefined first Zagreb entropy0.9840.9680.9544.31552Fourth atom bond connectivity entropy0.9840.9680.9514.47577Neighborhood version of hyper-Zagreb entropy0.9800.960.9584.14024Neighborhood version of redefined second Zagreb entropy0.9800.9610.9564.21432Fifth geometric arithmetic entropy0.9810.9620.9534.34898Table4 shows the relation between entropy and critical temperature. In Table 4, we can easily see that all the entropies show the highest positive correlation with critical temperature. The most significant regression models are shown as follows:(37)CT=−159.668+185.780EGANCT=−168.156+191.581EℕD6CT=−111.034+177.475EℕD4CT=−161.636+187.887Eℝez1NCT=−148.444+182.909EABCNCT=−165.387+664Eℝez2N.Figure5 shows that all the points fall near the fitted line. From all entropies, the fifth geometric arithmetic entropy shows the best relation with critical temperature CT. So, the critical temperature can be predicted by using the fifth geometric arithmetic entropy.Figure 5
Graphical representation of LRM between critical temperature and the fifth geometric arithmetic entropy.Table5 shows the relation between entropy and critical pressure CP, and all the entropies show the highest positive correlation with critical pressure. The most significant regression models are shown as follows:(38)CP=77.137−11.477ESK2NCP=96.285−11.005EℕD4CP=77.014−11.192EℕD5CP=77.547−11.545EℕD6CP=76.958−11.329EmℝNCP=77.121−11.314Eℝez1N.From all the entropies, the secondSK entropy shows the best relation with critical pressure CP, and all the points fall near the fitted line shown in Figure 6. So, the critical pressure CP can be predicted by using the second SK entropy.Figure 6
Graphical representation of LRM between critical pressureCP and the second SK entropy.Table6 shows the relation between entropy and critical volume. From Table 6, we can easily see that all the entropies show the highest positive correlation with critical volume. The most significant regression models are shown as follows:(39)CV=−569.955+293.079EℕD5CV=−589.308+303.572EℕD6CV=−576.671+297.184Eℝez1NCV=−551.360+288.287EABCNCV=−577.343+65.676ESK2NCV=−573.522+297.843EmℝN.From all the entropies, the fifthND entropy shows the best relation with critical volume CV, and all the points fall near the fitted line as can be seen in Figure 7. So, the critical temperature can be predicted by using the fifth ND entropy.Figure 7
Graphical representation of LRM between critical volume and the fifth ND entropy.Table7 shows the relation between entropy and Gibb’s energy qℰ of benzene derivatives. From Table 7, we can easily see that all the entropies show the highest positive correlation with Gibb’s energy qℰ.(40)qℰ=−751.66+303.591EℕD6qℰ=−732.155+293.063EℕD5qℰ=−674.923+284.469EℕD4qℰ=−639.259+283.942EℕD3qℰ=−725.293+293.202ESK2Nqℰ=−739.593+297.729EℕD1.Of all the entropies, the fifth ND entropy shows the best relation with Gibb’s energyqℰ, and all the points fall near the fitted line as seen in Figure 8. So, Gibb’s energy can be predicted by using the sixth ND entropy.Figure 8
Graphical representation of LRM between Gibb’s energyGE and the sixth ND entropy.Table8 shows the relation between entropy and logP. From Table 8, we can easily see that all the entropies show the highest positive correlation with critical volume. The most significant regression models are shown as follows:(41)logP=−3.443+1.998EℕD6logP=−3.304+1.927EℕD5logP=−3.159+1.890EℕD4logP=−3.357+1.956Eℝez1NlogP=−3.300+1.925EGANlogP=−3.371+1.958Eℝez2N.From all the entropies, the sixthND entropy shows the best relation with logP, and all the points fall near the fitted line as can be seen in Figure 9. So, the logP can be predicted by using the sixth ND entropy.Figure 9
Graphical representation of LRM betweenlogP and the sixth ND entropy.Table9 shows the relation between entropy and molar refractivity ℳℛ of benzene derivatives. From Table 9, we can easily see that all the entropies show the highest positive correlation with molar refractivity ℳℛ.(42)ℳℛ=−77.645+36.463EℕD6ℳℛ=−75.316+35.202EℕD5ℳℛ=−76.136+35.698Eℝez1Nℳℛ=−73.151+34.642EABCNℳℛ=−75.860+36.134ESK2Nℳℛ=−75.730+35.771E.mℝNOf all the entropies, the sixthND entropy shows the best relation with molar refractivity ℳℛ, and all the points fall near the fitted line as seen in Figure 10. So, the molar refractivity can be predicted by using the sixth ND entropy.Figure 10
Graphical representation of LRM between molar refractivity and the sixthND entropy.Table10 shows the relation between entropy and Henry’s law. From Table 10, we can easily see that all the entropies show the highest positive correlation with Henry’s law. The most significant regression models are shown as follows:(43)ℋℒ=−5.991+2.318EℕD6ℋℒ=−5.842+2.237EℕD5ℋℒ=−5.395+2.169EℕD4ℋℒ=−5.102+2.160EℕD3ℋℒ=−5.793+2.239EℕD2ℋℒ=−5.902+2.074EℕD1.Table 10
The significant LRM: Henry’s law of benzene derivatives on indices.
Entropiesℜℜ-squareAdjℜ2sModified neighborhood version of forgotten entropy0.9620.9260.9210.36535Neighborhood version of second Zagreb entropy0.9650.9310.9220.36445FirstND entropy0.9660.9330.9190.37174SecondND entropy0.9670.9350.9150.38046ThirdND entropy0.9700.9420.9180.37439FourthND entropy0.9710.9420.9110.38944FifthND entropy0.9760.9520.9190.37152SixthNDentropy0.9810.9620.9290.3485SK entropy0.9470.8970.890.43156FirstSK entropy0.9640.930.920.36835SecondSK entropy0.9640.930.9150.38044Neighborhood version of modified Randic entropy0.9650.930.9090.39337Neighborhood version of redefined first Zagreb entropy0.9650.930.9010.40943Fourth atom bond connectivity entropy0.9650.9310.8930.42717Neighborhood version of hyper-Zagreb entropy0.9600.9220.9180.37422Neighborhood version of redefined second Zagreb entropy0.9610.9230.9120.38616Fifth geometric arithmetic entropy0.9630.9280.9130.38561From all the entropies, the sixthND entropy shows the best relation with Henry’s law ℋℒ, and all the points fall near the fitted line as can be seen in Figure 11. So, Henry’s law ℋℒ can be predicted by using the sixth ND entropy.Figure 11
Graphical representation of LRM between Henry’s law and the sixthND entropy.


## 3. Concluding Remarks

Quantitative structure-property relationships (QSPRs) mathematically link physical or chemical properties with the structure of a molecule. Entropies are defined on molecular structures for a better understanding of the physical features [35] and chemical reactivity. In this paper, we developed the QSPR between ND entropy and the physical characteristics of benzene derivatives. Based on the linear regression model, we analyzed that critical temperature CT, critical pressure CP, and critical volume CV can be predicted through the fifth geometric arithmetic entropy, second SK entropy, and fifth ND entropy, respectively. Other remaining physical characteristics such as Gibb’s energy qℰ, logP, molar refractivity ℳℛ, and Henry’s law ℋℒ can be predicted by using the sixth ND entropy.



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*Source: 4416647-2022-09-23.xml*