# Degree-Based Topological Indices and QSPR Analysis of Antituberculosis Drugs

Authors: Mr. Adnan; Syed Ahtsham Ul Haq Bokhary; Ghulam Abbas; Tanveer Iqbal
Journal: Journal of Chemistry
 (2022)

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## Abstract

A topological index of graphG is a numerical quantity which describes its topology. If it is applied to molecular structure of a chemical compounds, then it reflects the theoretical properties of the chemical compounds. In this paper, well-known degree-based topological indices are applied on chemical structures of antituberculosis drugs. Chemical structure is considered as graph, where elements are taken as vertices and bounds between them are taken as edges. Furthermore, QSPR analysis of the said topological indices are discussed, and it is shown that these topological indices are highly correlated with the physical properties of antituberculosis drugs. This theocratical analysis may help the chemist and people working in pharmaceutical industry to predict properties of antituberculosis drugs without experimenting.


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## Body

## 1. Introduction

Before discovery/invention of antibiotics, lives of humans and animals were on great threat of being infected by some bacteria. In previous century, different bacterial infections were the most common causes of death until Alexander Flemings discovered Penicillin in 1929, the first antibiotic which was introduced to the world in 1940.Tuberculosis (TB) is a contagious infection caused by bacteria “Mycobacterium tuberculosis” that usually attacks lungs. It can also spread to other parts of body, like brain and spine.A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds between atoms.Cheminformatics is a new subject which is a combination of chemistry, mathematics, and information science. It studies quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of different chemical compounds. The molecular topological indices or simply the topological indices are used in chemistry. Wiener was the first who first showed that the Wiener index number is closely correlated with the boiling points of alkane molecules [18]. Later, work on quantitative structure-activity relationships showed that it is also correlated with other quantities including the parameters of its critical point [16], density, surface tension, viscosity of its liquid phase [14], and the van der Waals surface area of the molecule [10]. In the QSAR/QSPR study, physico-chemical properties and topological indices are used to predict bioactivity of the chemical compounds. In this work, it is shown that no single topological index exists that correlates with all the physical properties of chemical compounds.A graphGV,E with vertex set VG and edge set EG is connected if there exists a connection between any pair of vertices in G. The distance between two vertices u and v is denoted as du,v=dGu,v and is defined as the length of shortest path between u and v in graph G. The number of vertices of G adjacent to a given vertex v is the “degree” of this vertex and will be denoted by dvG or if misunderstanding is not possible simply by dv. The concept of degree is somewhat closely related to the concept of valence in chemistry.Some of the degree-based topological indices which we use in this work are defined as follows.Definition 1.
ABC index is proposed by Estrada et al. in [7], as(1)ABCG=∑e=uv∈EGdu+dv−2dudv.Definition 2.
The Randic´ index is proposed by Milan Randi c´ in [13], as(2)RG=∑e=uv∈EG1dudv.Definition 3.
The sum connectivity index is proposed by Zhou and Trinjstic in [19], as(3)SG=∑e=uv∈EG1du+dv.Definition 4.
TheGA index is proposed by Vukicevic et al. in [17], as(4)GAG=∑e=uv∈EG2dudvdu+dv.Definition 5.
The first and second Zagreb indices are proposed by Gutman and Trinajestic in [11], as(5)M1G=∑e=uv∈EGdu+dv,M2G=∑e=uv∈EGdudv.Definition 6.
The Harmonic index is proposed by Fajtlowicz et al. in [9], as(6)HG=∑e=uv∈EG2du+dv.Definition 7.
The hyper-Zagreb index is proposed by Shirdel et al. in [15], as(7)HMG=∑e=uv∈EGdu+dv2.Definition 8.
The third Zagreb index is proposed by Fath-Tabar et al. in [1], as(8)ZG3G=∑e=uv∈EGdu−dv.Definition 9.
The forgotten index is proposed by Furtula et al. in [8], as(9)FG=∑e=uv∈EGdu2+dv2.Definition 10.
The symmetric division index is proposed in [2], as(10)SSDG=∑e=uv∈EGPQ+QP,where P=mindu,dv and. Q=maxdu,dv.Some of the work in this area can be seen in [3–6, 12].


## 2. Results and Discussion

The above defined 11 topological indices are used for the modeling of six physical properties: boiling point (BP), enthalpy of vaporization (E), flash point (F), molar refractivity (MR), molar volume (MV), and polarizability (P) of 15 antituberculosis drugs: amikacin, bedaquiline, clofazimine, delamanid, ethambutol, ethionamide, imipenem-cilastatin, isoniazid, levofloxacin, linezolid, moxifloxacin, p-aminosalicylic acid, pyrazinamide, rifampin, and terizidone.


### 2.1. Regression Models

The following equation is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. We have used the following linear regression model:(11)P=A+bTI,where P is physical property of drug, A is constant, b is regression coefficient, and TI is topological index. Constant A and regression coefficient b is calculated from SPSS software for seven physical properties and eleven degree-based topological indices of molecular structure of fourteen drugs. Using equation (11), following are the linear regression model for the defined degree-based topological indices:(1)
Regression models for atom bond connectivity index:ABCGBoiling point=142.94+19.699ABCGEnthalpy=38.04+2.605ABCGFlash point=67.648+11.254ABCGMolar refraction=1.797+4.588ABCGMolar volume=8.698+12.686ABCGPolarizability=0.685+1.82ABCG(2)
Regression models for Randic´ index: RGBoiling point=124.752+33.178RGEnthalpy=35.127+4.424RGFlash point=59.012+18.814RGMolar refraction=−2.76+7.757RGMolar volume=−5.066+21.541RGPolarizability=−1.122+3.077RG(3)
. Regression models for sum-connectivity index.SGBoiling point=133.839+31.497SGEnthalpy=36.755+4.169SGFlash point=63.724+17.895SGMolar refraction=−0.81+7.373SGMolar volume=1.064+20.419SGPolarizability=−0.348+2.925SG(4)
Regression models for geometric-arithmetic index (GAG)Boiling point=197.225+14.102GAGEnthalpy=45.041+1.89GAGFlash point=97.24+8.114GAGMolar refraction=21.849+3.067GAGMolar volume=60.81+8.621GAGPolarizability=8.648+1.216GAG(5)
Regression models for first Zagreb index:M1GBoiling point=159.511+2.802M1GEnthalpy=40.359+0.37M1GFlash point=75.639+1.611M1GMolar refraction=4.118+0.664M1GMolar volume=15.714+1.3M1GPolarizability=1.606+0.264M1G(6)
Regression models for second Zagreb index:M2GBoiling point=176.377+2.294M2GEnthalpy=42.366+0.304M2GFlash point=85.303+1.32M2GMolar refraction=11.967+0.52M2GMolar volume=37.688+1.431M2GPolarizability=4.716+0.206M2G(7)
Regression models for harmonic index:HGBoiling point=124.25+34.754HGEnthalpy=35.279+4.616HGFlash point=59.249+19.664HGMolar refraction=−3.359+8.161HGMolar volume=−6.627+22.655HGPolarizability=−1.359+3.237HG(8)
Regression models for hyper-Zagreb index:HMGBoiling point=175.29+0.538HMGEnthalpy=42.25+0.071HMGFlash point=84.694+0.31HMGMolar refraction=10.754+0.123HMGMolar volume=34.659+0.34HMGPolarizability=4.237+0.049HMG(9)
Regression models for third Zagreb index:ZG3G.Boiling point=188.731+14.187ZG3GEnthalpy=45.595+1.819ZG3GFlash point=96.444+7.998ZG3GMolar refraction=18.618+3.077ZG3GMolar volume=49.304+8.748ZG3GPolarizability=7.358+1.221ZG3G(10)
Regression models for forgotten index:FGBoiling point=174.862+1.017FGEnthalpy=42.93+0.134FGFlash point=81.722+0.593FGMolar refraction=10.095+0.235FGMolar volume=32.425+0.648FGPolarizability=3.976+0.093FG(11)
Regression models for symmetric division index:SSDGBoiling point=147.016+5.962SSDGEnthalpy=38.141+0.795SSDGFlash point=68.953+3.421SSDGMolar refraction=4.877+1.371SSDGMolar volume=16.421+3.803SSDGPolarizability=1.908+0.544SSDG


### 2.2. Computation of Topological Indices and Their Comparison with Correlation Coefficients of Some Physical Properties

Table1 shows the abovementioned physical properties of 15 drugs used for the treatment of tuberculosis.Table 1
Physical properties of drugs used for the treatment of tuberculosis.
Name of medicineBoiling pointMelting pointFlash pointEnthalpy of vaporizationMolar refractivityPolarizabilitySurface tensionMolar volumeAmikacin981.8203.5547.6162.2134.953.5103.3363.9Bedaquiline702.7176378.8108156.261.952.6420.1Clofazimine566.9210296.785.1136.25447.1366.1Delamanid653.7193349.196.3127.750.650368Ethambutol345.389113.768.358.623.238.1207Ethionamide247.9163103.746.54919.439.8142Imipenem-cilastatin530.2274.592.772.728.871183.9Isoniazid251.9717225136.914.657.8110.2Levofloxacin571.5224299.490.191.136.170.3244Linezolid585.5177307.987.58332.947.7259Moxifloxacin636270338.798.8101.840.460.6285p-Aminosalicylic aci380.8145184.166.339.315.683.4102.7Pyrazinamide173.3190119.154.131.912.660.787.7Rifampin937.4183561.3153.5213.184.548611.7Terizidone17576.130.262.5198.9Physical properties are taken from ChemSpider.In Table2, the 12 topological indices are computed of the graphs constructed from the molecular structures of the drugs.Table 2
Various drugs and topological indices values.
Name of medicineABC(G)R(G)S(G)GA(G)M1(G)M2(G)H(G)HM(G)ZG3(G)F(G)SSD(G)Amikacin30.26618.8519.2440.1819124717.83104638542101.33Bedaquiline28.4617.6118.4138.9519019717.089592950190Clofazimine27.0616.6817.4718.9718621616.169012946985.16Delamanid30.3818.2919.1540.4320423417.5310044253699.41Ethambutol10.68.197.6414.5360607.92521213234Ethionamide7.955.235.2210.57505452341012426Imipenem-cilastatin15.889.8710.0721.031081309.335501829053.33Isoniazid8.025.215.2110.5550554.966234815626.33Levofloxacin20.7512.3713.0527.9914618611.837542239467.66Linezolid19.4311.9512.426.0813015211.466404233662.66Moxifloxacin24.114.415.2916.9717221513.8668942446477.33p-Aminosalicylic acid8.135.15.110.3852584.762521213628Pyrazinamide6.544.34.34.8455434.1318269621Rifampin44.8727.3228.1459.1930937425.78159355855151.5Terizidone18.6611.5411.9911.7412414411.136042031650.33In Table3, the correlation coefficients of the 6 physical properties with respect to each topological index are computed. The graph of the correlation coefficient of all the physical properties such as boiling point, enthalpy, flash point, molar refraction, molar volume, and polarizability on different topological indices are shown in Figure 1. The graphical representation of topological indices of the different medicine is shown in Figure 2.Table 3
Correlation coefficients.
Topological indexCorrelation coefficients of boiling pointCorrelation coefficients of enthalpyCorrelation coefficients of flash pointCorrelation coefficients of molar refractionCorrelation coefficients of molar volumeCorrelation coefficients of polarizabilityABC(G)0.9210.8660.9030.9800.9730.980R(G)0.9210.8720.8960.9840.9810.984S(G)0.9190.8640.8970.9830.9780.984GA(G)0.8940.8650.8830.9090.9170.908M1(G)0.8990.8400.8870.9740.9650.974M2(G)0.9210.8690.9100.9530.9430.954H(G)0.9180.8650.8910.9850.9820.985HM(G)0.9250.8720.9140.9710.9590.971ZG3(G)0.8770.7960.8490.8720.8900.872F(G)0.9170.8680.9170.9700.9590.970SSD(G)0.9250.8790.9120.9780.9740.978Figure 1
Physical properties and topological indices. (a) Boiling Point on TI. (b) Enthalpy on TI. (c) Flash point on TI. (d) Molar Refraction on TI. (e) Molar Volume on TI. (f) Polarizability on TI.
(a)(b)(c)(d)(e)(f)Figure 2
Medicine with topological indices.


### 2.3. Computation of Statistical Parameters

In this section, regression parameters have been computed.N is the sample size, A is constant or Y-intercept, b is slope, r is correlation coefficient, and r2 is the percentage of the dependent variable variation that a linear model explains. The p value for each term tests the null hypothesis that the coefficient is equal to zero (no effect), whereas a larger (insignificant) p value suggests that changes in the predictor are not associated with changes in the response. Suppose we are doing a test in which the null hypothesis is that all of the regression coefficients are zero. The result of this kind of a test gives us a value called F value. In this case, the model has no predictive capability. With the help of this test, one can compare their model with zero predictor variables and decides whether their added coefficients have improved the model. In Tables 4–13, these statistical parameters are computed for linear QSPR models for different topological indices. All of these analyses show that the p value in each of the model is zero that indicates the significance of the results.Table 4
Statistical parameters for the linear QSPR model forABCG.
Physical propertyNAbrr2FpIndicatorBoiling point14142.94019.6990.9210.84867.034≤0.0001SignificantEnthalpy1338.0402.6050.8660.75033.027≤0.0001SignificantFlash point1467.64811.2540.9030.81653.137≤0.0001SignificantMolar refraction141.7974.5880.9800.961320.196≤0.0001SignificantMolar volume158.69812.6860.9730.947233.580≤0.0001SignificantPolarizability150.6851.8200.9800.961322.025≤0.0001SignificantTable 5
Statistical parameters for the linear QSPR model forRG.
Physical propertyNAbrr2FpIndicatorBoiling point14124.75233.1780.9210.84866.986≤0.0001SignificantEnthalpy1335.1274.4240.8720.76034.889≤0.0001SignificantFlash point1459.01218.8140.8960.80449.123≤0.0001SignificantMolar refraction14−2.7607.7570.9840.968398.969≤0.0001SignificantMolar volume15−5.06621.5410.9810.963338.102≤0.0001SignificantPolarizability15−1.1223.0770.9840.969401.220≤0.0001SignificantTable 6
Statistical parameters for the linear QSPR model forSG.
Physical propertyNAbrr2FpIndicatorBoiling point14133.83931.4970.9190.84565.492≤0.0001SignificantEnthalpy1336.7554.1690.8640.74732.410≤0.0001SignificantFlash point1463.72417.8950.8970.80449.215≤0.0001SignificantMolar refraction14−0.8107.3730.9830.967382.171≤0.0001SignificantMolar volume151.06420.4190.9780.956285.344≤0.0001SignificantPolarizability15−0.3482.9250.9840.967384.473≤0.0001SignificantTable 7
Statistical parameters for the linear QSPR model forGAG.
Physical propertyNAbrr2FpIndicatorBoiling point14197.22514.1020.8940.79947.624≤0.0001SignificantEnthalpy1345.0411.8900.8650.74832.691≤0.0001SignificantFlash point1497.2408.1140.8830.77942.398≤0.0001SignificantMolar refraction1421.8493.0670.9090.82661.559≤0.0001SignificantMolar volume1560.8108.6210.9170.84168.909≤0.0001SignificantPolarizability158.6481.2160.9080.82561.422≤0.0001SignificantTable 8
Statistical parameters for the linear QSPR model forM1G.
Physical propertyNAbrr2FpIndicatorBoiling point14159.5112.8020.8990.80850.411≤0.0001SignificantEnthalpy1340.3590.3700.8400.70626.417≤0.0001SignificantFlash point1475.6391.6110.8870.78744.458≤0.0001SignificantMolar refraction144.1180.6640.9740.949241.578≤0.0001SignificantMolar volume1515.7141.8330.9650.931175.131≤0.0001SignificantPolarizability151.6060.2640.9740.949242.643≤0.0001SignificantTable 9
Statistical parameters for the linear QSPR model forM2G.
Physical propertyNAbrr2FpIndicatorBoiling point14176.3772.2940.9210.84867.069≤0.0001SignificantEnthalpy1342.3660.3040.8690.75533.832≤0.0001SignificantFlash point1485.3031.3200.9100.82757.458≤0.0001SignificantMolar refraction1411.9670.5200.9530.909130.015≤0.0001SignificantMolar volume1537.6881.4310.9430.889104.388≤0.0001SignificantPolarizability154.7160.2060.9540.909130.608≤0.0001SignificantTable 10
Statistical parameters for the linear QSPR model forHG.
Physical propertyNAbrr2FpIndicatorBoiling point14124.25034.7540.9180.8420.842≤0.0001SignificantEnthalpy1335.2794.6160.8650.74832.682≤0.0001SignificantFlash point1459.24919.6640.8910.79546.457≤0.0001SignificantMolar refraction14−3.3598.1610.9850.970422.174≤0.0001SignificantMolar volume15−6.62722.6550.9820.964347.348≤0.0001SignificantPolarizability15−1.3593.2370.9850.970424.568≤0.0001SignificantTable 11
Statistical parameters for the linear QSPR model forHMG.
Physical propertyNAbrr2FpIndicatorBoiling point14175.2900.5380.9250.85771.653≤0.0001SignificantEnthalpy1342.2500.0710.8720.76134.961≤0.0001SignificantFlash point1484.6940.3100.9140.83560.823≤0.0001SignificantMolar refraction1410.7540.1230.9710.943213.272≤0.0001SignificantMolar volume1534.6590.3400.9590.919148.407≤0.0001SignificantPolarizability154.2370.0490.9710.943214.463≤0.0001SignificantTable 12
Statistical parameters for the linear QSPR model forZG3G.
Physical propertyNAbrr2FpIndicatorBoiling point14188.73114.1870.8770.77040.068≤0.0001SignificantEnthalpy1345.5951.8190.7960.63419.044≤0.0012SignificantFlash point1496.4447.9980.8490.72130.987≤0.0001SignificantMolar refraction1418.6183.0770.8720.76141.284≤0.0001SignificantMolar volume1549.3048.7480.8900.79349.736≤0.0001SignificantPolarizability157.3581.2210.8720.76141.320≤0.0001SignificantTable 13
Statistical parameters for the linear QSPR model forFG.
Physical propertyNAbrr2FpIndicatorBoiling point14174.8621.0170.9170.84163.583≤0.0001SignificantEnthalpy1342.9300.1340.8680.75333.540≤0.0001SignificantFlash point1481.7220.5930.9170.84263.799≤0.0001SignificantMolar refraction1410.0950.2350.9700.940204.155≤0.0001SignificantMolar volume1532.4250.6480.9590.920150.473≤0.0001SignificantPolarizability153.9760.0930.9700.940205.151≤0.0001Significant


### 2.4. Standard Error of Estimate

The Standard error of estimate is the measure of variation of an observation made around the computed regression line. Table14 shows the standard error of estimate for six physical properties corresponding to each topological index.Table 14
Standard error of estimate.
Topological indexStd. error of the estimate for boiling pointStd. error of the estimate for enthalpyStd. error of the estimate for molar refractionStd. error of the estimate for flash pointStd. error of the estimate for molar volumeStd. error of the estimate for polarizabilityABC(G)98.8084217.7158710.5370163.4010134.109374.16784R(G)98.8384017.352749.4762165.4495128.584853.74830S(G)99.7863617.841279.6755365.4005431.009443.82645GA(G)113.7597717.7838522.2750469.3776259.181708.84300M1(G)111.1907319.2171512.0547168.1000039.050164.77125M2(G)98.7860817.5561716.0833261.3972649.435716.36590H(G)100.6852217.785759.2201066.9257528.215763.64692HM(G)96.0416917.3390912.7865059.9617442.159125.05818ZG3(G)121.7345221.4458826.1055278.0439367.6230510.35071F(G)76.3245611.7577617.4156793.3923673.685256.90532SSD(G)96.1847916.8842011.1984160.7350833.752944.43398


### 2.5. Correlation Determination

The correlation determination describes the percentage of relation which gives you more information about the relationship between variables. When you square the correlation coefficient, you end up with the correlation of determinationr2.


### 2.6. Comparison

In this section, the comparison for known values and computed values from our regression models is drawn. In Tables15–20, the comparison of each physical property is shown.Table 15
Comparison of actual and computed values for boiling point from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 16
Comparison of actual values computed values for enthalpy from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 17
Comparison of actual values computed values for Flash point from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 18
Comparison of actual values computed values for molar refraction from regression models.
Name of drugMolar refraction of drugMolar refraction computed from regression model for ABC indexMolar refraction computed from regression model for R(G) indexMolar refraction computed from regression model for S(G) indexMolar refraction computed from regression model for GA(G) indexMolar refraction computed from regression model for M1(G) indexMolar refraction computed from regression model for M2(G) indexMolar refraction computed from regression model H(G) indexMolar refraction computed from regression of HM(G) indexMolar refraction computed from regression model for ZG3(G)Molar refraction computed from regression model of F(G) indexMolar refraction computed from regression model of SSD(G)Amikacin134.9140.67143.46141.05145.08131.02140.32142.16139.93135.54137.52143.79Bedaquiline156.2132.38133.84134.93141.30130.35114.34136.04129.19107.85127.88128.26Clofazimine136.2125.96126.63128.0080.03127.70124.21128.53122.02107.85120.36121.63Delamanid127.7141.19139.12140.38145.84139.65133.57139.71134.74147.85136.11141.16Ethambutol58.650.4360.7755.5266.4143.9843.1561.1241.8755.5441.1351.49Ethionamide4938.2737.8137.6854.2737.3440.0337.4539.6549.3939.2540.52Imipenem-cilastatin72.774.6673.8073.4486.3575.8779.5272.7978.6874.0078.2777.99Isoniazid36.938.6037.6537.6054.2037.3440.5537.1739.6543.2346.7740.97Levofloxacin91.197.0193.2095.41107.69101.12108.6293.19103.8786.31102.7297.63Linezolid8390.9589.9490.62101.8390.4990.9690.1789.79147.8589.0990.78Moxifloxacin101.8112.38108.94111.9273.89118.39123.69109.81121.1692.46119.18110.89p-Aminosalicylic acid39.339.1036.8036.7953.6838.6742.1135.4941.8755.5442.0743.26Pyrazinamide31.931.8030.6030.8936.6940.6634.3130.3533.2337.0832.6633.67Rifampin213.1207.68209.17206.67203.38209.42206.32207.04207.48187.85211.10212.57Table 19
Comparison of actual values computed values for molar volume from regression models.
Name of drugMolar volume of drugMolar volume computed from regression model for ABC indexMolar volume computed from regression model for R(G) indexMolar volume computed from regression model for S(G) indexMolar volume computed from regression model for GA(G) indexMolar volume computed from regression model for M1(G) indexMolar volume computed from regression model for M2(G) indexMolar volume computed from regression model H(G) indexMolar volume computed from regression of HM(G)IndexMolar volume computed from regression model for ZG3(G)Molar volume computed from regression model of F(G) indexMolar volume computed from regression model of SSD(G)Amikacin363.9392.65400.98393.92407.20365.72391.20397.31389.95381.74383.54401.74Bedaquiline420.1369.74374.27376.97396.60363.89319.64380.32360.40303.01356.98358.65Clofazimine366.1351.98354.23357.78224.35356.56346.84359.48340.70303.01336.25340.25Delamanid368394.10388.92392.08409.36389.54372.60390.51375.69416.73379.65394.44Ethambutol207143.17171.35157.06186.07125.66123.56172.35120.26154.28117.94145.71Ethionamide142109.55107.59107.65151.93107.34114.97106.65114.14136.79112.75115.29Imipenem-cilastatin183.9210.15207.54206.68242.11213.62223.75204.74221.48206.77220.29219.21Isoniazid110.2110.44107.16107.44151.76107.34116.41105.88114.14119.29133.48116.54Levofloxacin244271.93261.39267.53302.11283.26303.90261.38290.77241.77287.66273.70Linezolid259255.19252.35254.25285.65253.94255.24253.00252.05416.73250.09254.69Moxifloxacin285314.43305.12313.26207.11330.90345.40307.51338.32259.26333.01310.48p-Aminosalicylic acid102.7111.84104.79105.20150.30111.00120.70101.21120.26154.28120.53122.89Pyrazinamide87.791.6687.5688.86102.54116.5099.2386.9496.48101.7994.6196.28Rifampin611.7577.92583.43575.64571.09581.96572.97577.42575.75530.46586.30592.51Table 20
Comparison of actual values computed values for polarizability from regression models.
Name of drugMolar volume of drugMolar volume computed from regression model for ABC indexMolar volume computed from regression model forR(G) indexMolar volume computed from regression model forS(G) indexMolar volume computed from regression model forGA(G) indexMolar volume computed from regression model forM1(G) indexMolar volume computed from regression model for M2(G) indexMolar volume computed from regression model H(G) indexMolar volume computed from regression of HM(G) indexMolar volume computed from regression model for ZG3(G)Molar volume computed from regression model of F(G) indexMolar volume computed from regression model of SSD(G)Amikacin363.9392.65400.98393.92407.20365.72391.20397.31389.95381.74383.54401.74Bedaquiline420.1369.74374.27376.97396.60363.89319.64380.32360.40303.01356.98358.65Clofazimine366.1351.98354.23357.78224.35356.56346.84359.48340.70303.01336.25340.25Delamanid368394.10388.92392.08409.36389.54372.60390.51375.69416.73379.65394.44Ethambutol207143.17171.35157.06186.07125.66123.56172.35120.26154.28117.94145.71Ethionamide142109.55107.59107.65151.93107.34114.97106.65114.14136.79112.75115.29Imipenem-cilastatin183.9210.15207.54206.68242.11213.62223.75204.74221.48206.77220.29219.21Isoniazid110.2110.44107.16107.44151.76107.34116.41105.88114.14119.29133.48116.54Levofloxacin244271.93261.39267.53302.11283.26303.90261.38290.77241.77287.66273.70Linezolid259255.19252.35254.25285.65253.94255.24253.00252.05416.73250.09254.69Moxifloxacin285314.43305.12313.26207.11330.90345.40307.51338.32259.26333.01310.48p-Aminosalicylic acid102.7111.84104.79105.20150.30111.00120.70101.21120.26154.28120.53122.89Pyrazinamide87.791.6687.5688.86102.54116.5099.2386.9496.48101.7994.6196.28Rifampin611.7577.92583.43575.64571.09581.96572.97577.42575.75530.46586.30592.51


## 2.1. Regression Models

The following equation is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. We have used the following linear regression model:(11)P=A+bTI,where P is physical property of drug, A is constant, b is regression coefficient, and TI is topological index. Constant A and regression coefficient b is calculated from SPSS software for seven physical properties and eleven degree-based topological indices of molecular structure of fourteen drugs. Using equation (11), following are the linear regression model for the defined degree-based topological indices:(1)
Regression models for atom bond connectivity index:ABCGBoiling point=142.94+19.699ABCGEnthalpy=38.04+2.605ABCGFlash point=67.648+11.254ABCGMolar refraction=1.797+4.588ABCGMolar volume=8.698+12.686ABCGPolarizability=0.685+1.82ABCG(2)
Regression models for Randic´ index: RGBoiling point=124.752+33.178RGEnthalpy=35.127+4.424RGFlash point=59.012+18.814RGMolar refraction=−2.76+7.757RGMolar volume=−5.066+21.541RGPolarizability=−1.122+3.077RG(3)
. Regression models for sum-connectivity index.SGBoiling point=133.839+31.497SGEnthalpy=36.755+4.169SGFlash point=63.724+17.895SGMolar refraction=−0.81+7.373SGMolar volume=1.064+20.419SGPolarizability=−0.348+2.925SG(4)
Regression models for geometric-arithmetic index (GAG)Boiling point=197.225+14.102GAGEnthalpy=45.041+1.89GAGFlash point=97.24+8.114GAGMolar refraction=21.849+3.067GAGMolar volume=60.81+8.621GAGPolarizability=8.648+1.216GAG(5)
Regression models for first Zagreb index:M1GBoiling point=159.511+2.802M1GEnthalpy=40.359+0.37M1GFlash point=75.639+1.611M1GMolar refraction=4.118+0.664M1GMolar volume=15.714+1.3M1GPolarizability=1.606+0.264M1G(6)
Regression models for second Zagreb index:M2GBoiling point=176.377+2.294M2GEnthalpy=42.366+0.304M2GFlash point=85.303+1.32M2GMolar refraction=11.967+0.52M2GMolar volume=37.688+1.431M2GPolarizability=4.716+0.206M2G(7)
Regression models for harmonic index:HGBoiling point=124.25+34.754HGEnthalpy=35.279+4.616HGFlash point=59.249+19.664HGMolar refraction=−3.359+8.161HGMolar volume=−6.627+22.655HGPolarizability=−1.359+3.237HG(8)
Regression models for hyper-Zagreb index:HMGBoiling point=175.29+0.538HMGEnthalpy=42.25+0.071HMGFlash point=84.694+0.31HMGMolar refraction=10.754+0.123HMGMolar volume=34.659+0.34HMGPolarizability=4.237+0.049HMG(9)
Regression models for third Zagreb index:ZG3G.Boiling point=188.731+14.187ZG3GEnthalpy=45.595+1.819ZG3GFlash point=96.444+7.998ZG3GMolar refraction=18.618+3.077ZG3GMolar volume=49.304+8.748ZG3GPolarizability=7.358+1.221ZG3G(10)
Regression models for forgotten index:FGBoiling point=174.862+1.017FGEnthalpy=42.93+0.134FGFlash point=81.722+0.593FGMolar refraction=10.095+0.235FGMolar volume=32.425+0.648FGPolarizability=3.976+0.093FG(11)
Regression models for symmetric division index:SSDGBoiling point=147.016+5.962SSDGEnthalpy=38.141+0.795SSDGFlash point=68.953+3.421SSDGMolar refraction=4.877+1.371SSDGMolar volume=16.421+3.803SSDGPolarizability=1.908+0.544SSDG


## 2.2. Computation of Topological Indices and Their Comparison with Correlation Coefficients of Some Physical Properties

Table1 shows the abovementioned physical properties of 15 drugs used for the treatment of tuberculosis.Table 1
Physical properties of drugs used for the treatment of tuberculosis.
Name of medicineBoiling pointMelting pointFlash pointEnthalpy of vaporizationMolar refractivityPolarizabilitySurface tensionMolar volumeAmikacin981.8203.5547.6162.2134.953.5103.3363.9Bedaquiline702.7176378.8108156.261.952.6420.1Clofazimine566.9210296.785.1136.25447.1366.1Delamanid653.7193349.196.3127.750.650368Ethambutol345.389113.768.358.623.238.1207Ethionamide247.9163103.746.54919.439.8142Imipenem-cilastatin530.2274.592.772.728.871183.9Isoniazid251.9717225136.914.657.8110.2Levofloxacin571.5224299.490.191.136.170.3244Linezolid585.5177307.987.58332.947.7259Moxifloxacin636270338.798.8101.840.460.6285p-Aminosalicylic aci380.8145184.166.339.315.683.4102.7Pyrazinamide173.3190119.154.131.912.660.787.7Rifampin937.4183561.3153.5213.184.548611.7Terizidone17576.130.262.5198.9Physical properties are taken from ChemSpider.In Table2, the 12 topological indices are computed of the graphs constructed from the molecular structures of the drugs.Table 2
Various drugs and topological indices values.
Name of medicineABC(G)R(G)S(G)GA(G)M1(G)M2(G)H(G)HM(G)ZG3(G)F(G)SSD(G)Amikacin30.26618.8519.2440.1819124717.83104638542101.33Bedaquiline28.4617.6118.4138.9519019717.089592950190Clofazimine27.0616.6817.4718.9718621616.169012946985.16Delamanid30.3818.2919.1540.4320423417.5310044253699.41Ethambutol10.68.197.6414.5360607.92521213234Ethionamide7.955.235.2210.57505452341012426Imipenem-cilastatin15.889.8710.0721.031081309.335501829053.33Isoniazid8.025.215.2110.5550554.966234815626.33Levofloxacin20.7512.3713.0527.9914618611.837542239467.66Linezolid19.4311.9512.426.0813015211.466404233662.66Moxifloxacin24.114.415.2916.9717221513.8668942446477.33p-Aminosalicylic acid8.135.15.110.3852584.762521213628Pyrazinamide6.544.34.34.8455434.1318269621Rifampin44.8727.3228.1459.1930937425.78159355855151.5Terizidone18.6611.5411.9911.7412414411.136042031650.33In Table3, the correlation coefficients of the 6 physical properties with respect to each topological index are computed. The graph of the correlation coefficient of all the physical properties such as boiling point, enthalpy, flash point, molar refraction, molar volume, and polarizability on different topological indices are shown in Figure 1. The graphical representation of topological indices of the different medicine is shown in Figure 2.Table 3
Correlation coefficients.
Topological indexCorrelation coefficients of boiling pointCorrelation coefficients of enthalpyCorrelation coefficients of flash pointCorrelation coefficients of molar refractionCorrelation coefficients of molar volumeCorrelation coefficients of polarizabilityABC(G)0.9210.8660.9030.9800.9730.980R(G)0.9210.8720.8960.9840.9810.984S(G)0.9190.8640.8970.9830.9780.984GA(G)0.8940.8650.8830.9090.9170.908M1(G)0.8990.8400.8870.9740.9650.974M2(G)0.9210.8690.9100.9530.9430.954H(G)0.9180.8650.8910.9850.9820.985HM(G)0.9250.8720.9140.9710.9590.971ZG3(G)0.8770.7960.8490.8720.8900.872F(G)0.9170.8680.9170.9700.9590.970SSD(G)0.9250.8790.9120.9780.9740.978Figure 1
Physical properties and topological indices. (a) Boiling Point on TI. (b) Enthalpy on TI. (c) Flash point on TI. (d) Molar Refraction on TI. (e) Molar Volume on TI. (f) Polarizability on TI.
(a)(b)(c)(d)(e)(f)Figure 2
Medicine with topological indices.


## 2.3. Computation of Statistical Parameters

In this section, regression parameters have been computed.N is the sample size, A is constant or Y-intercept, b is slope, r is correlation coefficient, and r2 is the percentage of the dependent variable variation that a linear model explains. The p value for each term tests the null hypothesis that the coefficient is equal to zero (no effect), whereas a larger (insignificant) p value suggests that changes in the predictor are not associated with changes in the response. Suppose we are doing a test in which the null hypothesis is that all of the regression coefficients are zero. The result of this kind of a test gives us a value called F value. In this case, the model has no predictive capability. With the help of this test, one can compare their model with zero predictor variables and decides whether their added coefficients have improved the model. In Tables 4–13, these statistical parameters are computed for linear QSPR models for different topological indices. All of these analyses show that the p value in each of the model is zero that indicates the significance of the results.Table 4
Statistical parameters for the linear QSPR model forABCG.
Physical propertyNAbrr2FpIndicatorBoiling point14142.94019.6990.9210.84867.034≤0.0001SignificantEnthalpy1338.0402.6050.8660.75033.027≤0.0001SignificantFlash point1467.64811.2540.9030.81653.137≤0.0001SignificantMolar refraction141.7974.5880.9800.961320.196≤0.0001SignificantMolar volume158.69812.6860.9730.947233.580≤0.0001SignificantPolarizability150.6851.8200.9800.961322.025≤0.0001SignificantTable 5
Statistical parameters for the linear QSPR model forRG.
Physical propertyNAbrr2FpIndicatorBoiling point14124.75233.1780.9210.84866.986≤0.0001SignificantEnthalpy1335.1274.4240.8720.76034.889≤0.0001SignificantFlash point1459.01218.8140.8960.80449.123≤0.0001SignificantMolar refraction14−2.7607.7570.9840.968398.969≤0.0001SignificantMolar volume15−5.06621.5410.9810.963338.102≤0.0001SignificantPolarizability15−1.1223.0770.9840.969401.220≤0.0001SignificantTable 6
Statistical parameters for the linear QSPR model forSG.
Physical propertyNAbrr2FpIndicatorBoiling point14133.83931.4970.9190.84565.492≤0.0001SignificantEnthalpy1336.7554.1690.8640.74732.410≤0.0001SignificantFlash point1463.72417.8950.8970.80449.215≤0.0001SignificantMolar refraction14−0.8107.3730.9830.967382.171≤0.0001SignificantMolar volume151.06420.4190.9780.956285.344≤0.0001SignificantPolarizability15−0.3482.9250.9840.967384.473≤0.0001SignificantTable 7
Statistical parameters for the linear QSPR model forGAG.
Physical propertyNAbrr2FpIndicatorBoiling point14197.22514.1020.8940.79947.624≤0.0001SignificantEnthalpy1345.0411.8900.8650.74832.691≤0.0001SignificantFlash point1497.2408.1140.8830.77942.398≤0.0001SignificantMolar refraction1421.8493.0670.9090.82661.559≤0.0001SignificantMolar volume1560.8108.6210.9170.84168.909≤0.0001SignificantPolarizability158.6481.2160.9080.82561.422≤0.0001SignificantTable 8
Statistical parameters for the linear QSPR model forM1G.
Physical propertyNAbrr2FpIndicatorBoiling point14159.5112.8020.8990.80850.411≤0.0001SignificantEnthalpy1340.3590.3700.8400.70626.417≤0.0001SignificantFlash point1475.6391.6110.8870.78744.458≤0.0001SignificantMolar refraction144.1180.6640.9740.949241.578≤0.0001SignificantMolar volume1515.7141.8330.9650.931175.131≤0.0001SignificantPolarizability151.6060.2640.9740.949242.643≤0.0001SignificantTable 9
Statistical parameters for the linear QSPR model forM2G.
Physical propertyNAbrr2FpIndicatorBoiling point14176.3772.2940.9210.84867.069≤0.0001SignificantEnthalpy1342.3660.3040.8690.75533.832≤0.0001SignificantFlash point1485.3031.3200.9100.82757.458≤0.0001SignificantMolar refraction1411.9670.5200.9530.909130.015≤0.0001SignificantMolar volume1537.6881.4310.9430.889104.388≤0.0001SignificantPolarizability154.7160.2060.9540.909130.608≤0.0001SignificantTable 10
Statistical parameters for the linear QSPR model forHG.
Physical propertyNAbrr2FpIndicatorBoiling point14124.25034.7540.9180.8420.842≤0.0001SignificantEnthalpy1335.2794.6160.8650.74832.682≤0.0001SignificantFlash point1459.24919.6640.8910.79546.457≤0.0001SignificantMolar refraction14−3.3598.1610.9850.970422.174≤0.0001SignificantMolar volume15−6.62722.6550.9820.964347.348≤0.0001SignificantPolarizability15−1.3593.2370.9850.970424.568≤0.0001SignificantTable 11
Statistical parameters for the linear QSPR model forHMG.
Physical propertyNAbrr2FpIndicatorBoiling point14175.2900.5380.9250.85771.653≤0.0001SignificantEnthalpy1342.2500.0710.8720.76134.961≤0.0001SignificantFlash point1484.6940.3100.9140.83560.823≤0.0001SignificantMolar refraction1410.7540.1230.9710.943213.272≤0.0001SignificantMolar volume1534.6590.3400.9590.919148.407≤0.0001SignificantPolarizability154.2370.0490.9710.943214.463≤0.0001SignificantTable 12
Statistical parameters for the linear QSPR model forZG3G.
Physical propertyNAbrr2FpIndicatorBoiling point14188.73114.1870.8770.77040.068≤0.0001SignificantEnthalpy1345.5951.8190.7960.63419.044≤0.0012SignificantFlash point1496.4447.9980.8490.72130.987≤0.0001SignificantMolar refraction1418.6183.0770.8720.76141.284≤0.0001SignificantMolar volume1549.3048.7480.8900.79349.736≤0.0001SignificantPolarizability157.3581.2210.8720.76141.320≤0.0001SignificantTable 13
Statistical parameters for the linear QSPR model forFG.
Physical propertyNAbrr2FpIndicatorBoiling point14174.8621.0170.9170.84163.583≤0.0001SignificantEnthalpy1342.9300.1340.8680.75333.540≤0.0001SignificantFlash point1481.7220.5930.9170.84263.799≤0.0001SignificantMolar refraction1410.0950.2350.9700.940204.155≤0.0001SignificantMolar volume1532.4250.6480.9590.920150.473≤0.0001SignificantPolarizability153.9760.0930.9700.940205.151≤0.0001Significant


## 2.4. Standard Error of Estimate

The Standard error of estimate is the measure of variation of an observation made around the computed regression line. Table14 shows the standard error of estimate for six physical properties corresponding to each topological index.Table 14
Standard error of estimate.
Topological indexStd. error of the estimate for boiling pointStd. error of the estimate for enthalpyStd. error of the estimate for molar refractionStd. error of the estimate for flash pointStd. error of the estimate for molar volumeStd. error of the estimate for polarizabilityABC(G)98.8084217.7158710.5370163.4010134.109374.16784R(G)98.8384017.352749.4762165.4495128.584853.74830S(G)99.7863617.841279.6755365.4005431.009443.82645GA(G)113.7597717.7838522.2750469.3776259.181708.84300M1(G)111.1907319.2171512.0547168.1000039.050164.77125M2(G)98.7860817.5561716.0833261.3972649.435716.36590H(G)100.6852217.785759.2201066.9257528.215763.64692HM(G)96.0416917.3390912.7865059.9617442.159125.05818ZG3(G)121.7345221.4458826.1055278.0439367.6230510.35071F(G)76.3245611.7577617.4156793.3923673.685256.90532SSD(G)96.1847916.8842011.1984160.7350833.752944.43398


## 2.5. Correlation Determination

The correlation determination describes the percentage of relation which gives you more information about the relationship between variables. When you square the correlation coefficient, you end up with the correlation of determinationr2.


## 2.6. Comparison

In this section, the comparison for known values and computed values from our regression models is drawn. In Tables15–20, the comparison of each physical property is shown.Table 15
Comparison of actual and computed values for boiling point from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 16
Comparison of actual values computed values for enthalpy from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 17
Comparison of actual values computed values for Flash point from regression models.
Name of drugBoiling point of drugBoiling point computed from regression model for ABC indexBoiling point computed from regression model for R(G) indexBoiling point computed from regression model for S(G) indexBoiling point computed from regression model for GA(G) indexBoiling point computed from regression model for M1(G) indexBoiling point computed from regression model for M2(G) indexBoiling point computed from regression model H(G) indexBoiling point computed from regression of HM(G) indexBoiling point computed from regression model for ZG3(G) indexBoiling point computed from regression model of F(G) indexBoiling point computed from regression model of SSD(G) indexAmikacin981.8739.16750.16739.84763.82694.65743.07743.91738.32727.82726.12751.10Bedaquiline702.7703.58709.02713.70746.48691.85628.36717.84691.49600.14684.42683.56Clofazimine566.9676.00678.16684.09464.73680.64671.95685.87660.27600.14651.87654.70Delamanid653.7741.41731.58737.01767.35731.08713.25733.48715.72784.57720.01739.65Ethambutol345.3351.75396.48374.48402.12327.62314.04398.80310.94358.97309.12349.71Ethionamide247.9299.55298.27298.25346.28299.60300.27298.02301.25330.60300.98302.02Imipenem-cilastatin530.2455.77452.22451.02493.78462.10474.64448.50471.34444.09469.81464.95Isoniazid251.97300.93297.61297.94346.00299.60302.56296.84301.25302.22333.53303.98Levofloxacin571.5551.70535.16544.88591.93568.57603.12535.39581.15500.84575.59550.37Linezolid585.5525.70521.23524.40564.99523.74525.11522.53519.79784.57516.60520.57Moxifloxacin636617.69602.52615.43436.53641.42669.66606.14656.51529.21646.78608.02p-Aminosalicylic acid380.8303.10293.96294.47343.60305.20309.45289.68310.94358.97313.18313.94Pyrazinamide173.3271.77267.42269.28265.48313.61275.03267.78273.26273.85272.50272.21Table 18
Comparison of actual values computed values for molar refraction from regression models.
Name of drugMolar refraction of drugMolar refraction computed from regression model for ABC indexMolar refraction computed from regression model for R(G) indexMolar refraction computed from regression model for S(G) indexMolar refraction computed from regression model for GA(G) indexMolar refraction computed from regression model for M1(G) indexMolar refraction computed from regression model for M2(G) indexMolar refraction computed from regression model H(G) indexMolar refraction computed from regression of HM(G) indexMolar refraction computed from regression model for ZG3(G)Molar refraction computed from regression model of F(G) indexMolar refraction computed from regression model of SSD(G)Amikacin134.9140.67143.46141.05145.08131.02140.32142.16139.93135.54137.52143.79Bedaquiline156.2132.38133.84134.93141.30130.35114.34136.04129.19107.85127.88128.26Clofazimine136.2125.96126.63128.0080.03127.70124.21128.53122.02107.85120.36121.63Delamanid127.7141.19139.12140.38145.84139.65133.57139.71134.74147.85136.11141.16Ethambutol58.650.4360.7755.5266.4143.9843.1561.1241.8755.5441.1351.49Ethionamide4938.2737.8137.6854.2737.3440.0337.4539.6549.3939.2540.52Imipenem-cilastatin72.774.6673.8073.4486.3575.8779.5272.7978.6874.0078.2777.99Isoniazid36.938.6037.6537.6054.2037.3440.5537.1739.6543.2346.7740.97Levofloxacin91.197.0193.2095.41107.69101.12108.6293.19103.8786.31102.7297.63Linezolid8390.9589.9490.62101.8390.4990.9690.1789.79147.8589.0990.78Moxifloxacin101.8112.38108.94111.9273.89118.39123.69109.81121.1692.46119.18110.89p-Aminosalicylic acid39.339.1036.8036.7953.6838.6742.1135.4941.8755.5442.0743.26Pyrazinamide31.931.8030.6030.8936.6940.6634.3130.3533.2337.0832.6633.67Rifampin213.1207.68209.17206.67203.38209.42206.32207.04207.48187.85211.10212.57Table 19
Comparison of actual values computed values for molar volume from regression models.
Name of drugMolar volume of drugMolar volume computed from regression model for ABC indexMolar volume computed from regression model for R(G) indexMolar volume computed from regression model for S(G) indexMolar volume computed from regression model for GA(G) indexMolar volume computed from regression model for M1(G) indexMolar volume computed from regression model for M2(G) indexMolar volume computed from regression model H(G) indexMolar volume computed from regression of HM(G)IndexMolar volume computed from regression model for ZG3(G)Molar volume computed from regression model of F(G) indexMolar volume computed from regression model of SSD(G)Amikacin363.9392.65400.98393.92407.20365.72391.20397.31389.95381.74383.54401.74Bedaquiline420.1369.74374.27376.97396.60363.89319.64380.32360.40303.01356.98358.65Clofazimine366.1351.98354.23357.78224.35356.56346.84359.48340.70303.01336.25340.25Delamanid368394.10388.92392.08409.36389.54372.60390.51375.69416.73379.65394.44Ethambutol207143.17171.35157.06186.07125.66123.56172.35120.26154.28117.94145.71Ethionamide142109.55107.59107.65151.93107.34114.97106.65114.14136.79112.75115.29Imipenem-cilastatin183.9210.15207.54206.68242.11213.62223.75204.74221.48206.77220.29219.21Isoniazid110.2110.44107.16107.44151.76107.34116.41105.88114.14119.29133.48116.54Levofloxacin244271.93261.39267.53302.11283.26303.90261.38290.77241.77287.66273.70Linezolid259255.19252.35254.25285.65253.94255.24253.00252.05416.73250.09254.69Moxifloxacin285314.43305.12313.26207.11330.90345.40307.51338.32259.26333.01310.48p-Aminosalicylic acid102.7111.84104.79105.20150.30111.00120.70101.21120.26154.28120.53122.89Pyrazinamide87.791.6687.5688.86102.54116.5099.2386.9496.48101.7994.6196.28Rifampin611.7577.92583.43575.64571.09581.96572.97577.42575.75530.46586.30592.51Table 20
Comparison of actual values computed values for polarizability from regression models.
Name of drugMolar volume of drugMolar volume computed from regression model for ABC indexMolar volume computed from regression model forR(G) indexMolar volume computed from regression model forS(G) indexMolar volume computed from regression model forGA(G) indexMolar volume computed from regression model forM1(G) indexMolar volume computed from regression model for M2(G) indexMolar volume computed from regression model H(G) indexMolar volume computed from regression of HM(G) indexMolar volume computed from regression model for ZG3(G)Molar volume computed from regression model of F(G) indexMolar volume computed from regression model of SSD(G)Amikacin363.9392.65400.98393.92407.20365.72391.20397.31389.95381.74383.54401.74Bedaquiline420.1369.74374.27376.97396.60363.89319.64380.32360.40303.01356.98358.65Clofazimine366.1351.98354.23357.78224.35356.56346.84359.48340.70303.01336.25340.25Delamanid368394.10388.92392.08409.36389.54372.60390.51375.69416.73379.65394.44Ethambutol207143.17171.35157.06186.07125.66123.56172.35120.26154.28117.94145.71Ethionamide142109.55107.59107.65151.93107.34114.97106.65114.14136.79112.75115.29Imipenem-cilastatin183.9210.15207.54206.68242.11213.62223.75204.74221.48206.77220.29219.21Isoniazid110.2110.44107.16107.44151.76107.34116.41105.88114.14119.29133.48116.54Levofloxacin244271.93261.39267.53302.11283.26303.90261.38290.77241.77287.66273.70Linezolid259255.19252.35254.25285.65253.94255.24253.00252.05416.73250.09254.69Moxifloxacin285314.43305.12313.26207.11330.90345.40307.51338.32259.26333.01310.48p-Aminosalicylic acid102.7111.84104.79105.20150.30111.00120.70101.21120.26154.28120.53122.89Pyrazinamide87.791.6687.5688.86102.54116.5099.2386.9496.48101.7994.6196.28Rifampin611.7577.92583.43575.64571.09581.96572.97577.42575.75530.46586.30592.51


## 3. Conclusion and Future Study

### 3.1. Conclusion

Tuberculosis (TB) is a disease and its infection is caused by bacteria “Mycobacterium tuberculosis” that usually attacks lungs. It can also spread to other parts of body, like brain and spine and cause death.Amolecular descriptor is actually a mathematical formula that can be applied to any graph which models some molecular structure. More precisely, a single number represents a chemical structure. In graph theory, this number is called topological descriptor. When a topological descriptor correlates with a molecular property, it is called molecular index or topological index TI. Thus, a topological graph index is also called a molecular descriptor. Mathematically, a topological index is a numeric quantity associated with a graph which characterizes the topology of a graph. By computing these topological indices, it is possible to analyze mathematical values and further investigate some physio-chemical properties of a molecule. Actually topological indices are designed on the ground of transformation of molecular graph into a number which characterizes the topology of graph.Molecular topological indices play a significant role in mathematical chemistry, especially in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) investigations. (11) is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. Therefore, it is an efficient way to avoid expensive and time-consuming laboratory experiments. The purpose of computing these 11 topological indices is that no single topological index is found yet that can be efficient (given in various tables) for all physical properties of these drugs.Tables1 and 2 gives the values of physical properties and topological indices of medicines (Figure 3), respectively. Table 3 and graphs (Figure 1) shows how degree-based topological indices and physical properties of medicine correlate. Upon examining correlation coefficients horizontally for physical properties under consideration, we see that HG index gives highest correlation coefficient for molar refraction r=0.985, molar volume r=0.982, and polarizability r=0.985. HMG index gives highest correlation coefficient for boiling point r=0.925. FG has highest correlation coefficient for flash point r=0.925. SSD gives highest correlation coefficient for boiling point and enthalpy. When we look vertically, boiling point has also good correlation with ABCG and RG, i.e., r=.921 for both. Enthalpy has also good correlation with RG and HMG, i.e., r=0.872. Flash point has also good correlation with SSDG, i.e., r=0.812. Molar refraction has also good correlation with RG, i.e., r=0.984. Molar volume has also good correlation with RG, i.e., r=0.981. Polarizability has also good correlation with RG and SG, i.e., r=.984.Figure 3
Molecular structure of drugs. (a) Amikacin. (b) Bedaquiline. (c) Clofazimine. (d) Delamanid. (e) Ethambutol. (f) Ethionamide. (g) Imipenem-cilastatin. (h) Isoniazid. (i) Levofloxacin. (j) Linezolid. (k) Moxifloxacin. (l)p-aminosalicylic acid. (m) Pyrazinamide. (n) Rifampin. (o) Terizidone.While examining correlation determination (Table21) horizontally for physical properties under consideration, we see that HG index gives highest correlation determination for molar refraction r2=0.970, molar volume r2=0.964, and polarizability r2=0.970. HMG index gives highest correlation determination for boiling point r2=0.857. FG has highest correlation determination for flash point r2=0.842. SSD gives highest correlation determination for boiling point and enthalpy r2=0.773. When we look vertically, boiling point has also good correlation with ABCG and RG, i.e., r2=0.848 for both. Enthalpy has also good correlation with RG and HMG, i.e., r≥0.760. Flash point has also good correlation with SSDG and HMG, i.e., r2=.0.831. Molar refraction has also good correlation with RG, i.e., r2=0.968. Molar volume has also good correlation with RG, i.e., r2=0.963. Polarizability has also good correlation with RG and SG, i.e., r2=0.969. One can easily gather from the table of statistical parameters for the linear QSPR models for different degree-based topological indices that are as follows:Harmonic indexHG has positive and highly significant correlation coefficient for molar refraction r=0.985, molar volume r=0.982, and polarizability r=0.985.Sum-connectivitySG index and hyper-Zagreb index HMG has highly significant correlation coefficient for boiling point, i.e., r=0.925.Sum-connectivitySG index also has highly significant correlation coefficient for enthalpy.Forgotten topological indexFG has highly significant correlation coefficient for flash point r=0.917.From Table21, we can also see high percentage correlation between degree-based topological index and physical properties of medicines.Harmonic indexHG has positive and high percentage of correlation for molar refraction r2=0.970, molar volume r2=0.964, and polarizability r2=0.970.Hyper-Zagreb indexHMG has high percentage of correlation correlation for boiling point, i.e., r2=0.857.Sum-connectivitySG index also has high percentage of correlation for enthalpy (r2=0.773 ).Forgotten topological indexFG has high percentage of correlation for flash point r2=0.842.Table 21
Correlation determination.
Topological indexCoefficient of determination of boiling pointCoefficient of determination of enthalpyCoefficient of determination of flash pointCoefficient of determination of molar refractionCoefficient of determination of molar volumeCoefficient of determination of polarizabilityABC(G)0.8480.7500.8160.9610.9470.961R(G)0.8480.7600.8040.9680.9630.969S(G)0.8450.7470.8040.9670.9560.967GA(G)0.7990.7480.7790.8260.8410.825M1(G)0.8080.7060.7870.9490.9310.949M2(G)0.8480.7550.8270.9090.8890.909H(G)0.8420.7480.7950.9700.9640.970HM(G)0.8570.7610.8350.9430.9190.943ZG3(G)0.7700.6340.7210.7610.7930.761F(G)0.8410.7530.8420.9400.9200.940SSD(G)0.8440.7730.8310.9560.9480.956Tables4–13 and 22 show different statistical parameters of correlation between values of eleven degree based topological indices and six physical properties of medicine. We could not find any correlation between degree-based topological index and melting point of antituberculosis drugs.Table 22
Statistical parameters for the linear QSPR model forSSDG.
Physical propertyNAbrr2FpIndicatorBoiling point14147.0165.9620.9250.84471.404≤0.0001SignificantEnthalpy1338.1410.7950.8790.77337.471≤0.0001SignificantFlash point1468.9533.4210.9120.83158.981≤0.0001SignificantMolar refraction144.8771.3710.9780.956282.000≤0.0001SignificantMolar volume1516.4213.8030.9740.948238.815≤0.0001SignificantPolarizability151.9080.5440.9780.956283.014≤0.0001SignificantThis work indicated that this theocratical analysis may help the chemist and people working in pharmaceutical industry to predict properties of antituberculosis drugs without experimenting. It is also possible that different composition of these drugs may be used for different diseases; of course it depends on the range of the topological indices which are computed in this work. In this work, we have found the correlation coefficient for different topological indices; this will help the chemist to design new drugs based on the combination of positively high correlated drugs.


### 3.2. Future Study

In a similar pattern, relation between physical properties for different drugs/medicine for the treatment/preventive measures of a particular disease and Topological indices can be established to estimate physical properties of newly discovered medicine or candidate drug(s) of particular disease.It will be very important to find a topological index that can correlate with all the physical properties.


## 3.1. Conclusion

Tuberculosis (TB) is a disease and its infection is caused by bacteria “Mycobacterium tuberculosis” that usually attacks lungs. It can also spread to other parts of body, like brain and spine and cause death.Amolecular descriptor is actually a mathematical formula that can be applied to any graph which models some molecular structure. More precisely, a single number represents a chemical structure. In graph theory, this number is called topological descriptor. When a topological descriptor correlates with a molecular property, it is called molecular index or topological index TI. Thus, a topological graph index is also called a molecular descriptor. Mathematically, a topological index is a numeric quantity associated with a graph which characterizes the topology of a graph. By computing these topological indices, it is possible to analyze mathematical values and further investigate some physio-chemical properties of a molecule. Actually topological indices are designed on the ground of transformation of molecular graph into a number which characterizes the topology of graph.Molecular topological indices play a significant role in mathematical chemistry, especially in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) investigations. (11) is used to correlate the various physical properties of various drugs used for the treatment of tuberculosis with some topological indices. Therefore, it is an efficient way to avoid expensive and time-consuming laboratory experiments. The purpose of computing these 11 topological indices is that no single topological index is found yet that can be efficient (given in various tables) for all physical properties of these drugs.Tables1 and 2 gives the values of physical properties and topological indices of medicines (Figure 3), respectively. Table 3 and graphs (Figure 1) shows how degree-based topological indices and physical properties of medicine correlate. Upon examining correlation coefficients horizontally for physical properties under consideration, we see that HG index gives highest correlation coefficient for molar refraction r=0.985, molar volume r=0.982, and polarizability r=0.985. HMG index gives highest correlation coefficient for boiling point r=0.925. FG has highest correlation coefficient for flash point r=0.925. SSD gives highest correlation coefficient for boiling point and enthalpy. When we look vertically, boiling point has also good correlation with ABCG and RG, i.e., r=.921 for both. Enthalpy has also good correlation with RG and HMG, i.e., r=0.872. Flash point has also good correlation with SSDG, i.e., r=0.812. Molar refraction has also good correlation with RG, i.e., r=0.984. Molar volume has also good correlation with RG, i.e., r=0.981. Polarizability has also good correlation with RG and SG, i.e., r=.984.Figure 3
Molecular structure of drugs. (a) Amikacin. (b) Bedaquiline. (c) Clofazimine. (d) Delamanid. (e) Ethambutol. (f) Ethionamide. (g) Imipenem-cilastatin. (h) Isoniazid. (i) Levofloxacin. (j) Linezolid. (k) Moxifloxacin. (l)p-aminosalicylic acid. (m) Pyrazinamide. (n) Rifampin. (o) Terizidone.While examining correlation determination (Table21) horizontally for physical properties under consideration, we see that HG index gives highest correlation determination for molar refraction r2=0.970, molar volume r2=0.964, and polarizability r2=0.970. HMG index gives highest correlation determination for boiling point r2=0.857. FG has highest correlation determination for flash point r2=0.842. SSD gives highest correlation determination for boiling point and enthalpy r2=0.773. When we look vertically, boiling point has also good correlation with ABCG and RG, i.e., r2=0.848 for both. Enthalpy has also good correlation with RG and HMG, i.e., r≥0.760. Flash point has also good correlation with SSDG and HMG, i.e., r2=.0.831. Molar refraction has also good correlation with RG, i.e., r2=0.968. Molar volume has also good correlation with RG, i.e., r2=0.963. Polarizability has also good correlation with RG and SG, i.e., r2=0.969. One can easily gather from the table of statistical parameters for the linear QSPR models for different degree-based topological indices that are as follows:Harmonic indexHG has positive and highly significant correlation coefficient for molar refraction r=0.985, molar volume r=0.982, and polarizability r=0.985.Sum-connectivitySG index and hyper-Zagreb index HMG has highly significant correlation coefficient for boiling point, i.e., r=0.925.Sum-connectivitySG index also has highly significant correlation coefficient for enthalpy.Forgotten topological indexFG has highly significant correlation coefficient for flash point r=0.917.From Table21, we can also see high percentage correlation between degree-based topological index and physical properties of medicines.Harmonic indexHG has positive and high percentage of correlation for molar refraction r2=0.970, molar volume r2=0.964, and polarizability r2=0.970.Hyper-Zagreb indexHMG has high percentage of correlation correlation for boiling point, i.e., r2=0.857.Sum-connectivitySG index also has high percentage of correlation for enthalpy (r2=0.773 ).Forgotten topological indexFG has high percentage of correlation for flash point r2=0.842.Table 21
Correlation determination.
Topological indexCoefficient of determination of boiling pointCoefficient of determination of enthalpyCoefficient of determination of flash pointCoefficient of determination of molar refractionCoefficient of determination of molar volumeCoefficient of determination of polarizabilityABC(G)0.8480.7500.8160.9610.9470.961R(G)0.8480.7600.8040.9680.9630.969S(G)0.8450.7470.8040.9670.9560.967GA(G)0.7990.7480.7790.8260.8410.825M1(G)0.8080.7060.7870.9490.9310.949M2(G)0.8480.7550.8270.9090.8890.909H(G)0.8420.7480.7950.9700.9640.970HM(G)0.8570.7610.8350.9430.9190.943ZG3(G)0.7700.6340.7210.7610.7930.761F(G)0.8410.7530.8420.9400.9200.940SSD(G)0.8440.7730.8310.9560.9480.956Tables4–13 and 22 show different statistical parameters of correlation between values of eleven degree based topological indices and six physical properties of medicine. We could not find any correlation between degree-based topological index and melting point of antituberculosis drugs.Table 22
Statistical parameters for the linear QSPR model forSSDG.
Physical propertyNAbrr2FpIndicatorBoiling point14147.0165.9620.9250.84471.404≤0.0001SignificantEnthalpy1338.1410.7950.8790.77337.471≤0.0001SignificantFlash point1468.9533.4210.9120.83158.981≤0.0001SignificantMolar refraction144.8771.3710.9780.956282.000≤0.0001SignificantMolar volume1516.4213.8030.9740.948238.815≤0.0001SignificantPolarizability151.9080.5440.9780.956283.014≤0.0001SignificantThis work indicated that this theocratical analysis may help the chemist and people working in pharmaceutical industry to predict properties of antituberculosis drugs without experimenting. It is also possible that different composition of these drugs may be used for different diseases; of course it depends on the range of the topological indices which are computed in this work. In this work, we have found the correlation coefficient for different topological indices; this will help the chemist to design new drugs based on the combination of positively high correlated drugs.


## 3.2. Future Study

In a similar pattern, relation between physical properties for different drugs/medicine for the treatment/preventive measures of a particular disease and Topological indices can be established to estimate physical properties of newly discovered medicine or candidate drug(s) of particular disease.It will be very important to find a topological index that can correlate with all the physical properties.



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