# Topological Indices of Novel Drugs Used in Diabetes Treatment and Their QSPR Modeling

Authors: Saima Parveen; Nadeem Ul Hassan Awan; Mogtaba Mohammed; Fozia Bashir Farooq; Nisha Iqbal
Journal: Journal of Mathematics
 (2022)

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## Abstract

A topological index is a real number obtained from the chemical graph structure. It is helpful to calculate the physicochemical and biological properties of numerous drugs. This is done through degree-based topological indices. In this paper, acarbose, tolazamide, miglitol, prandin, metformin, and so on used to treat diabetes are discussed, and the purpose of the QSPR study is to determine the mathematical relation between the properties under investigation (e.g., boiling point and flash point) and different descriptors related to the molecular structure of the drugs. In this study, it is observed that topological indices (TIs) applied to said drugs have a good correlation with physicochemical properties in this course.


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## Body

## 1. Introduction

Diabetes is a metabolic disease that develops when the pancreas cannot generate enough insulin, and its effective use is retarded. This chronic disease leads to high blood glucose and is named blood sugar. Glucose is obtained from food and becomes an energy source in our body. Pancreas generates a hormone named insulin. The hormone helps glucose to digest in cells and is used as energy. Glucagon is another hormone that associates with insulin and controls blood sugar. When our immune system is not functioning properly, it fights infections and kills insulin-producing agents. As a result, there is a chance that insulin does not work well and glucose becomes to stay in our blood and will not become part of our cells. With the passage of time, diabetes leads to severe damage to nerves and blood vessels, the amount of glucose becomes very high, and it will generate other health issues. Every year 422 million persons become sick with diabetes, and 1.5 million lead to death because of this disease. The risk of diabetes increases when you are 45 or above, and high blood pressure will also increase the chance of the said disease [1–4]. The disease has no proper treatment, but effective care can help you manage diabetes and live a healthier life. Drugs are used to cure this malignant disease, and many drugs tests are accompanied to fight the fatal disease. This needs timely diagnosis, screening, and medication that benefits patients to control the deadly disease in future. The ten essential drugs acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin are safe and the most efficient medicines that are required for well-being community. Figure 1 depicts the chemical structure of the said drugs.Figure 1
Molecular structure of drugs. (a) Acarbose. (b) Tolazamide. (c) Miglitol. (d) Prandin. (e) Metformin. (f) Glimepiride. (g) Linagliptin. (h) Pioglitazone. (i) Bromocriptine. (j) Alogliptin.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)Topological indices (TIs) are termed numeric descriptors that are obtained through a molecular graph in order to completely mention the chemical system and widely used in the investigation of physicochemical properties of many drugs. Since there are several kinds of polynomials and topological indices which are extensively calculated, represent the chemical structure, and have a vital position in chemical graph theory, among such families, degree-based topological indices are of great significance and play a vital part in chemical graph theory. The use of graph invariants (TIs) in QSPR and QSAR studies has taken key interest in recent years. Topological indices have applications in various areas of mathematics, bioinformatics, mathematics, informatics, biology, and so on, but their utmost significant use to date is in the nonempirical Quantitative Structure-Property Relationships (QSPR) [5, 6].ABC index, Wiener index, and Randic index are helpful to predict the bioactivity of drugs. The QSPR models assist in determining the optimal relationship between topological indices and psychochemical characteristics. These psychochemical qualities are being studied because they have a big impact on bioactivity and drug transit in the human body. In this paper, we have computed degree-based TIs related to diabetes drugs. Similarly, antidiabetes drugs represent a chemical compound on which the given topological indices are thoroughly defined and deliberate QSPR analysis. The corresponding characteristic estimated through this method is highly correlated with the characteristic of diabetes drugs with the help of linear regression. It is observed that a high correlation exists between the properties of drugs and TIs.


## 2. Material and Method

In drug structures, atoms denote vertices, and the corresponding bonds connecting the atoms are termed edges. Graph G (V, E) is considered as simple, finite, and connected, whereas V and E represented in the chemical graph are termed as vertex and edge set, respectively. The degree of a vertex in a graph G is the number of vertices adjacent to it and is denoted bydu. Valence of a compound in chemistry and the degree of a vertex in a graph are meticulously related concepts [7–10]. Degree-based topological indices used are given as follows.Definition 1.
The ABC index [10] of a molecular graph G is defined as(1)ABCG=∑uv∈EGdu+dv−2dudv.Definition 2.
The first degree-based topological index is the Randic indexRAG introduced by Milan Randic in 1975 [11]. Randic index is defined as(2)RAG=∑uv∈EG1dudv.Definition 3.
The sum connectivity index [12] of a molecular graph G is defined as(3)SG=∑uv∈EG1du+dv.Definition 4.
The GA index [13] of a molecular graph G is defined as(4)GAG=∑uv∈EG2dudvdu+dv.Definition 5.
The first and second Zagreb indices [14] of a molecular graph G are defined as(5)M1G=∑uv∈EGdu+dv,M2G=∑uv∈EGdudv.Definition 6.
The harmonic index [15] of a molecular graph G is defined as(6)HG=∑uv∈EG2du+dv.Definition 7.
The hyper-Zagreb index [16] of a molecular graph G is defined as(7)HMG=∑uv∈EGdu+dv2.Definition 8.
The forgotten index [17] of a molecular graph G is defined as(8)FG=∑uv∈EGdu2+dv2.
Theπ−electron energy of molecule was found with the help of the first and second Zagreb indices in [18]. Alkane heat of formation is best predicted by applying the augmented Zagreb index in [19]. The values of physical properties are taken from ChemSpider. It is observed from data in Table 1 and is found that these data values are normally distributed. Therefore, the linear regression model is the most adequate to test and adopt for the said analysis. For more insight on degree-based topological indices, we offer the reader to visit the following articles [5, 7, 20–24].Table 1
The TIs values of candidate drugs.
Name of drugABC(G)RA(G)S(G)GA(G)M1(G)M2(G)F(G)H(G)HM(G)Acarbose65.4438.1838.5781.165066821730309434.29Tolazamide32.7318.2318.4338.61242310806142615.98Miglitol23.0713.6713.5527.99174230596105612.16Prandin/repaglinide53.6130.2330.6264.243965091300231826.73Metformin14.758.758.5717.00961042904987.72Glimepiride53.2729.4629.8662.773985151342237025.75Linagliptin50.2327.9228.8160.923724721196214024.99Pioglitazone35.2220.0220.5243.02256322808145217.96Bromocriptine66.5736.3337.5780.685186991762316032.12Alogliptin35.8820.3820.7743.86266344858154618.21


## 3. Results and Discussions

In this section, degree-based TIs are executed on diabetes drugs. The relation between QSPR analysis and topological indices portrays that the properties are vastly correlated in terms of physicochemical properties for the said disease. The ten medicines acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin are used in the analysis for diabetes. The drug structures are displayed in Figure1. We consider the molecular structure as a graph, the drug elements denote vertices, and bonds among atoms are their edges. We use regression analysis calculation for drugs study.


### 3.1. Regression Model

In this paper, drug computable structure analysis about ten topological indices is done for QSPR modeling tenacity. The six physical properties, flash point (FP), polarity, boiling point (BP), molar volume (MV), and refractivity (R) for ten medicine arranged in Table2 are used in diabetes treatment. We execute the regression analysis for the drugs, and the linear regression model is tested with the help of equation(9)P=A+bTI.Table 2
Correlation coefficients.
Topological indexCorrelation coefficient of boiling pointCorrelation coefficient of refractive indexCorrelation coefficient of flash pointCorrelation coefficient of polarityCorrelation coefficient of molar volumeCorrelation coefficient of complexityABC(G)0.9100.9610.9290.9930.9600.946RA(G)0.8980.9470.9390.9850.9470.931S(G)0.9040.9530.9380.9890.9510.938GA(G)0.9090.9560.9380.9910.9520.943M1(G)0.9120.9530.9400.9900.9490.947M2(G)0.9120.9400.9500.9830.9350.943HM(G)0.8960.9440.9420.9830.9420.928F(G)0.9070.9370.9470.9820.9350.941H(G)0.9090.9390.9490.9830.9350.942Here, P represents the physicochemical property of given drugs. TI is the topological index, A is constant, andb represents the regression coefficient. The IBM SPSS Statistics version-24 software is helpful to find out the results. Nine TIs of diabetes drugs and physiochemical properties are analyzed with a linear QSPR model. Equation (9) is suitable for the said calculation purpose.Theorem 1.
LetG1 be the graph metformin. Various topological indices of G1 are given as follows:(i)
ABCG1=14.75.(ii)
RAG1=8.75.(iii)
SG1=8.57.(iv)
GAG1=17.00.(v)
M1G1=96.(vi)
M2G1=104.(vii)
FG1=290.(viii)
HG1=498.(ix)
HMG1=7.72.Proof.
LetG1 be the graph of metformin with edge set E. Let Em,n represent the class of edges of G1 joining vertices of degrees m and n. With E1,3=5, E1,4=6,E2,3=3,E2,4=1,E3,3=3, and E3,4=1, one has the following: (i)
By using Definition1 and the above given edge partitions Em,n, we get(10)ABG1=51+3−21×3+61+4−21×4+32+3−22×3+12+4−22×4+33+3−23×3+13+4−23×4=14.75.(ii)
By using Definition2 and the above given edge partitions Em,n, we get(11)RAGG1=511×3+611×4+312×3+112×4+313×3+113×4=8.75.(iii)
By using Definition3 and the above given edge partitions Em,n, we get(12)SG1=511+3+611+4+312+3+112+4+313+3+113+4=8.57.(iv)
By using Definition4 and the above given edge partitions Em,n, we get(13)GAG1=51×31+3+61×41+4+32×32+3+12×42+4+33×33+3+13×43+4=17.00.(v)
By using Definition5 and the above given edge partitions Em,n, we get(14)M1G1=51+3+61+4+32+3+12+4+33+3+13+4=96.(vi)
By using Definition5 and the above given edge partitions Em,n, we get(15)M2G1=51×3+61×4+32×3+12×4+33×3+13×4=104.(vii)
By using Definition6 and the above given edge partitions Em,n, we get(16)HG1=511+3+611+4+312+3+112+4+313+3+113+4=498.(viii)
By using Definition7 and the above given edge partitions Em,n, we get(17)HMG1=51+32+61+42+32+32+12+42+33+32+13+42=7.72.(ix)
By using Definition8 and the above given edge partitions Em,n, we get(18)FG1=51+9+61+16+34+9+14+16+39+9+19+16=290.Theorem 2.
LetG2 be the graph of tolazamide. Various topological indices of G2 are given as follows:(i)
ABCG2=32.73.(ii)
RAG2=18.23.(iii)
SG2=18.43.(iv)
GAG2=38.61.(v)
M1G2=242.(vi)
M2G2=310.(vii)
FG2=806.(viii)
HG2=1426.(ix)
HMG2=15.98.Proof.
LetG2 be the graph of tolazamide with edge set E′E′. Let Em,n′ represent the class of edges of G2 joining vertices of degrees m and n. With E1,3′=7, E1,4′=17, E3,3′=9, E3,4′=5, and E4,4′=5, one has the following:(i)
By using Definition1 and edge partitions Em,n′, we get(19)ABCG2=71+3−21×3+171+4−21×493+3−23×3+53+4−23×4+54+4−24×4=32.73.(ii)
By using Definition2 and edge partition Em,n′, we get(20)RAG2=711×3+1711×4+913×3+513×4+514×4=18.23.(iii)
Definition3 and edge partition Em,n′ give(21)SG2=711+3+1711+4+913+3+513+4+514+4=18.43.(iv)
By using Definition4 and edge partition Em,n′, we get(22)GAG2=71×31+3+171×41+4+93×33+3+53×43+454×44+4=38.61.(v)
By using Definition5 and edge partition Em,n′, we get(23)M1G2=93+3+53+4+54+4=242.(vi)
By using Definition5 and edge partition Em,n′, we get(24)M2G2=93×3+53×4+54×4=310.(vii)
By using Definition6 and edge partition Em,n′, we get(25)HG2=1411+3+3411+4+1813+3+1013+4+1014+4=15.98.(viii)
By using Definition7 and edge partition Em,n′, we get(26)HMG=71+32+171+42+93+32+53+42+54+42=1426.(ix)
By using Definition8 and edge partition Em,n′, we get(27)FG2=71+9+171+16+99+9+59+16+516+16=806.
One can calculate topological indices of the remaining drugs by adopting a similar procedure applied in Theorem1 and Theorem 2 and using Definitions 1 to 8. Moreover, the calculated values of all drugs are listed in Table 1.
Using (1), we have calculated the following diverse linear models for all degree-based topological indices, which are given as follows:(1)
Regression models for atom bond connectivity index ABC(G):Boiling point = 103.153 + 10.44 [ABC(G)].Refractivity = 18.569 + 2.093 [ABC(G)].Flash point = 20.160 + 7.125 [ABC(G)].Polarity = 0.172 + 0.980 [ABC(G)].Molar volume = 31.785 + 6.065 [ABC(G)].Complexity = −182.330 + 19.139 [ABC(G)].(2)
Regression models for atom bond connectivity index RA(G):Boiling point = 100.512 + 18.552 [RA(G)].Refractivity = 18.020 + 3.731 [RA(G)].Flash point = 10.148 + 13.033 [RA(G)].Polarity = −0.380 + 1.759 [RA(G)].Molar volume = 29.935 + 10.820 [RA(G)].Complexity = −186.139 + 34.060 [RA(G)].(3)
Regression models for atom bond connectivity index S(G):Boiling point = 105.293 + 18.056 [S(G)].Refractive index = 18.429 + 3.638 [S(G)].Flash point = 15.429 + 12.603 [S(G)].Polarity = 0.122 + 1.710 [S(G)].Molar volume = 32.905 + 10.520 [S(G)].Complexity = −180.010 + 33.247 [S(G)].(4)
Regression models for atom bond connectivity index GA(G):Boiling point = 113.440 + 8.420 [GA(G)].Refractivity = 20.596 + 1.694 [GA(G)].Flash point = 22.547 + 5.853 [GA(G)].Polarity = 1.044 + 0.795 [GA(G)].Molar volume = 38.721 + 4.888 [GA(G)].Complexity = −465.288 + 15.519 [GA(G)].(5)
Regression models for atom bond connectivity index M1(G):Boiling point = 126.868 + 1.316 [M1(G)].Refractivity = 24.500 + 0.261 [M1(G)].Flash point = 34.273 + .908 [M1(G)].Polarity = 2.745 + 0.123 [M1(G)].Molar volume = 49.701 + 0.755 [M1(G)].Complexity = −135.869 + 2.413 [M1(G)].(6)
Regression models for atom bond connectivity index HM(G):Boiling point = 104.181 + 20.696 [HM(G)].Refractivity = 18.567 + 4.176 [HM(G)].Flash point = 10.181 + 14.677 [HM(G)].Polarity = −0.136 + 1.970 [HM(G)].Molar volume = 32.218 + 12.080 [HM(G)].Complexity = −181.037 + 38.124 [HM(G)].(7)
Regression models for atom bond connectivity index M2(G):Boiling point = 151.218 + .952 [M2(G)].Refractivity = 30.535 + 0.187 [M2(G)].Flash point = 48.647 + 0.665 [M2(G)].Polarity = 5.362 + 0.088 [M2(G)].Molar volume = 67.501 + 0.539 [M2(G)].Complexity = −86.964 + 1.741 [M2(G)].(8)
Regression models for atom bond connectivity index F(G):Boiling point = 144.280 + 0.380 [F(G)].Refractivity = 29.397 + 0.074 [F(G)].Flash point = 44.851 + 0.264 [F(G)].Polarity = 4.737 + 0.035 [F(G)].Molar volume = 63.476 + 0.215 [F(G)].Complexity = −97.844 + 0.692 [F(G)].(9)
Regression models for atom bond connectivity index H(G):Boiling point = 147.316 + 0.211 [H(G)].Refractivity = 29.878 + 0.041 [H(G)].Flash point = 46.427 + 0.147 [H(G)].Polarity = 5.004 + 0.020 [H(G)].Molar volume = 65.21.0 + 0.120 [H(G)].Complexity = −93.085 + 0.386 [H(G)].
Tables3–11 represent the statistical parameters used in the QSPR models of TIs.Table 3
Statistical parameters used in the QSPR model of ABC(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8103.15310.440.9100.82828.9470.002SignificantRefractivity1018.5692.0930.9610.92396.1860.000SignificantFlash point1020.1607.1250.9290.86350.2090.000SignificantPolarity100.1720.9800.9930.987601.8290.000SignificantMolar volume1031.7856.0650.9600.92294.6580.000SignificantComplexity10−182.33019.1390.9460.89568.0750.000SignificantTable 4
Statistical parameters used in the QSPR model of R(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8100.51218.5520.8980.80624.8690.002SignificantRefractivity1018.0203.7310.9470.89668.9520.000SignificantFlash point1010.14813.0330.9390.88259.7380.000SignificantPolarity10−0.3801.7590.9850.971269.2420.000SignificantMolar volume1029.93510.8200.9470.89769.4350.000SignificantComplexity10−186.13934.0600.9310.86651.6820.000SignificantTable 5
Statistical parameters used in the QSPR model of S(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8105.29318.0560.9040.81826.8950.002SignificantRefractivity1018.7723.6380.9530.90979.8440.000SignificantFlash point1015.42912.6030.9380.88058.4180.000SignificantPolarity100.1221.7100.9890.978361.0560.000SignificantMolar volume1032.90510.5200.9510.90475.4280.000SignificantComplexity10−180.01033.2470.9380.88058.6690.000SignificantTable 6
Statistical parameters used in the QSPR model of GA(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8113.4408.4200.9090.82728.6610.002SignificantRefractivity1020.5961.6940.9560.91585.8070.000SignificantFlash point1022.5475.8530.9380.88158.9500.000SignificantPolarity101.0440.7950.9910.981422.8130.000SignificantMolar volume1038.7214.8880.9520.90677.0890.000SignificantComplexity10−465.28815.5190.9430.89064.6440.000SignificantTable 7
Statistical parameters used in the QSPR model of M1(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8126.8681.3160.9120.83329.8510.002SignificantRefractivity1024.5000.2610.9530.90778.2610.000SignificantFlash point1034.2730.9080.9400.88460.9280.000SignificantPolarity102.7450.1230.9900.980389.9930.000SignificantMolar volume1049.7010.7550.9490.90172.5650.000SignificantComplexity10−135.8692.4130.9470.89769.8140.000SignificantTable 8
Statistical parameters used in the QSPR model of M2(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8151.2180.9520.9120.83129.5940.002SignificantRefractivity1030.5350.1870.9400.88461.7650.000SignificantFlash point1048.6470.6650.9500.90374.7260.000SignificantPolarity105.3620.0880.9830.966227.7190.000SignificantMolar volume1067.5010.5390.9350.87455.7160.000SignificantComplexity10−86.6941.7410.9430.89064.6050.000SignificantTable 9
Statistical parameters used in the QSPR model of HM(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8104.18120.6960.8960.80224.3710.003SignificantRefractivity1018.5674.1760.9440.89165.5900.000SignificantFlash point1010.18114.6770.9420.88863.2530.000SignificantPolarity10−0.1361.9700.9830.967232.0950.000SignificantMolar volume1032.21812.0800.9420.88762.9210.000SignificantComplexity10−181.03738.1240.9280.86149.6170.000SignificantTable 10
Statistical parameters used in the QSPR model of F(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8144.2800.3800.9070.82327.8600.002SignificantRefractivity1029.3970.0740.9370.87857.7420.000SignificantFlash point1044.8510.2640.9470.89669.0580.000SignificantPolarity104.7370.0350.9820.965217.8320.000SignificantMolar volume1063.4760.2150.9350.87555.8880.000SignificantComplexity10−97.8440.6920.9410.88661.9990.000SignificantTable 11
Statistical parameters used in the QSPR model of H(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8147.3160.2110.9090.82728.6270.002SignificantRefractivity1029.8780.0410.9390.88159.1680.000SignificantFlash point1046.4270.1470.9490.90071.7560.000SignificantPolarity105.0040.0200.9830.965223.4510.000SignificantMolar volume1065.2100.1200.9350.87555.8640.000SignificantComplexity10−93.0850.3860.9420.88863.2180.000Significant


### 3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation

#### 3.2.1. Coefficient of Physicochemical Properties

Physical properties for ten diabetes drugs are listed in Table12, and their TIs computed through the molecular structure are recorded in Table 1. The correlation coefficient between six physical properties and TIs is itemized in Table 2. The graph of TIs and physical properties is shown in Figure 2.Table 12
Physical properties of drugs.
Name of drugRefractive indexm3mol−1Flash point℃Molar volumecm3Polaritycm3Boiling point℃Complexity (bit)Acarbose139.02541.40369.8060.87675.05962Tolazamide81.34246.80237.9032.82300431Miglitol48.16284.30142.1020.81453.7179Prandin/repaglinide131.83360.80397.9051.49672.9619Metformin56.6458.10100.8013.43224132Glimepiride129.80363.20378.8053.49895Linagliptin133.43353.70338.0051.24661.2885Pioglitazone97.39301.80282.8037.91466Bromocriptine165.51492.80429.4066.44891.31230Alogliptin104.26267.80252.9035.44519.2622Figure 2
Physicochemical properties and TIs.


### 3.3. Calculation of Statistical Parameters

In this section, we find the relation between degree base TIs and physical properties of diabetes drugs such as medicines acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin, and this is done with the help of QSPR modeling, whereas TIs, N,b, and r represent regression model constant, correlation coefficient, and sample size, respectively. This will be helpful to compare and improve the model. It is noted that the value of r is greater than 0.6, and p value is less than 0.05. Hence, the calculation verifies that all properties are significant.


### 3.4. Standard Error (SE) of Estimate and Comparison

Measure of variation for an observation calculated around the computed regression line is said to be the standard error estimate. It measures the amount of accuracy of predictions done around the computed regression line in Table13. We also compare the physicochemical properties of the experimental and theoretical calculated values of the models as presented in Tables 14–19.Table 13
Standard error of estimate.
Topological indexStd. error of the estimate for boiling pointStd. error of the estimate for refractive indexStd. error of the estimate for flash pointStd. error of the estimate for polarityStd. error of the estimate for molar volumeStd. error of the estimate for complexityABC(G)98.4304811.1746652.645052.0918832.63778121.449RA(G)104.7305113.0025148.801743.1025037.57911137.117S(G)101.4552612.1697849.284092.6890336.20431129.734GA(G)98.8358111.7766449.088032.4888435.84931124.284M1(G)97.1822013.2809248.378442.5894236.84197120.084M2(G)97.5326913.7548444.160033.3646841.42791124.317HM(G)105.5857713.2962947.582713.3338839.26701139.553F(G)99.9982314.0675045.755293.4375441.37206126.610H(G)98.8852213.9173444.974843.3955641.37987125.522Table 14
Comparison of actual and computed values for polarity from regression models.
Name of drugPolarity of drugPolarity computed from regression model for ABC(G) indexPolarity computed from regression model for R(G) indexPolarity computed from regression model for S(G) indexPolarity computed from regression model for GA(G) indexPolarity computed from regression model for M1(G) indexPolarity computed from regression model for M2(G) indexPolarity computed from regression model for F(G) indexPolarity computed from regression model for H(G) indexPolarity computed from regression model for HM(G) indexAcarbose60.87cm364.303266.778666.076765.566264.98365.37865.28766.88467.4153Tolazamide32.82cm332.247431.686531.637331.738932.51132.64232.94733.52431.3446Miglitol20.81cm322.780623.665523.292523.296024.14725.60225.59726.12423.8192Prandin51.49cm352.709852.794552.482252.114851.45350.15450.23751.36452.5221Metformin13.43cm314.62715.011214.776714.55914.55314.51414.88714.96415.0724Glimepiride53.49cm352.376651.440151.182650.946151.69950.68251.70752.40450.5915Linagliptin51.24cm349.397448.731249.387149.475448.50146.89846.59747.80449.0943Pioglitazone37.91cm334.687634.835135.211235.244934.23333.69833.01734.04435.2452Bromocriptine66.44cm365.410663.524464.366765.184666.45966.87466.40768.20463.1404Alogliptin35.44cm335.334435.468435.638735.912735.46335.63434.76735.92435.7377Table 15
Comparison of actual and computed values for molar volume from regression models.
Name of drugMolar volume of drugMolar volume from regression model for ABC(G) indexMolar volume from regression model for R(G) indexMolar volume from regression model for S(G) indexMolar volume from regression model for GA(G) indexMolar volume from regression model for M1(G) indexMolar volume from regression model for M2(G)indexMolar volume from regression model for F(G) indexMolar volume from regression model for H(G) indexMolar volume from regression model for HM(G) indexAcarbose369.8±5.0cm3428.678443.042438.661435.431431.731435.099435.426436.49446.441Tolazamide237.9±7.0cm3230.292227.183226.788227.446232.411234.591236.766236.33225.256Miglitol142.1±3.0cm3171.704177.844175.451175.536181.071191.471191.616191.93179.110Prandin397.9±3.0cm3356.929357.023355.027352.726348.681341.852342.976343.37355.116Metformin100.8±7.0cm3121.243124.61123.061121.817122.181123.557125.826124.97125.475Glimepiride378.8±5.0cm3354.867348.692347.032345.540350.191345.086352.006349.61343.278Linagliptin338±7.0cm3336.43332.029335.986336.498330.561321.909320.616322.01334.097Pioglitazone282.8±3.0cm3245.394246.551248.775249.002242.981241.059237.196239.45249.174Bromocriptine429.4±5.0cm3435.532423.025428.141433.084440.791444.262442.306444.41420.227Alogliptin252.9±5.0cm3249.397250.446251.405253.108250.531252.917247.946250.73252.194Table 16
Comparison of actual and computed values for flash point from regression models.
Name of drugFlash point of drugFlash point computed from regression model for ABC(G) indexFlash point computed from regression model for R(G) indexFlash point computed from regression model for S(G) indexFlash point computed from regression model for GA(G) indexFlash point computed from regression model for M1(G) indexFlash point computed from regression model for M2(G) indexFlash point computed from regression model for F(G) indexFlash point computed from regression model for H(G) indexFlash point computed from regression model for HM(G) indexAcarbose541.4±34.3℃486.42507.747501.526497.576493.721502.177501.571501.245513.455Tolazamide246.8±26.8℃253.361247.739247.702248.531254.009254.797257.635256.049244.719Miglitol284.3±27.4℃184.533188.309186.199186.372192.265201.597202.195201.659188.653Prandin360.8±0.0℃402.131404.135401.332398.543393.841387.132388.051387.173402.497Metformin58.1±22.6℃125.253124.186123.436122.048121.441117.807121.411119.633123.487Glimepiride363.2±34.3℃399.708394.100391.754389.939395.657391.122399.139394.817388.113Linagliptin353.7±34.3℃378.048374.029378.521379.111372.049362.527360.595361.007376.959Pioglitazone301.8±28.7℃271.102271.068274.042274.343266.721262.777258.163259.871273.779Bromocriptine492.8±34.3℃494.471483.636488.923494.767504.617513.482510.019510.947481.606Alogliptin267.8±32.9℃275.805275.760277.193279.259275.801277.407271.363273.689277.449Table 17
Comparison of actual and computed values for refractive index from regression models.
Name of drugRefractivity of drugRefractivity computed from regression model for ABC(G) indexRefractivity computed from regression model for R(G) indexRefractivity computed from regression model for S(G) indexRefractivity computed from regression model for GA(G) indexRefractivity computed from regression model for M1(G) indexRefractivity computed from regression model for M2(G) indexRefractivity computed from regression model for F(G) indexRefractivity computed from regression model for H(G) indexRefractivity computed from regression model for HM(G) indexAcarbose139.02m3⋅mol−1155.534160.469159.089158.081156.566158.069157.417156.732161.762Tolazamide81.34m3⋅mol−187.072886.036185.820386.001387.66288.50589.04188.34485.2994Miglitol48.16m3⋅mol−166.854569.022768.066968.011069.91473.54573.50173.17469.3471Prandin131.83m3⋅mol−1130.774130.808130.167129.418127.856125.718125.597124.916130.191Metformin56.64m3⋅mol−149.440750.666249.949649.39449.55649.98350.85750.29650.8057Glimepiride129.8m3⋅mol−1130.063127.935127.402126.928128.378126.84128.705127.048126.099Linagliptin133.43m3⋅mol−1123.700122.189123.582123.794121.592118.799117.901117.618122.925Pioglitazone97.39m3⋅mol−192.284492.714693.423793.471891.31690.74989.18989.4193.5679Bromocriptine165.51m3⋅mol−1157.9153.567155.451157.267159.698161.248159.785159.438152.700Alogliptin104.26m3⋅mol−193.665894.057794.333294.894893.92694.86392.88993.26494.6119Table 18
Comparison of actual and computed values for complexity from regression models.
Name of drugComplexityComplexity computed from regression model for ABC(G) indexComplexity computed from regression model for R(G) indexComplexity computed from regression model for S(G) indexComplexity computed from regression model for GA(G) indexComplexity computed from regression model for M1(G) indexComplexity computed from regression model for M2(G) indexComplexity computed from regression model for F(G) indexComplexity computed from regression model for H(G) indexComplexity computed from regression model for HM(G) indexAcarbose9621070.121114.271102.32794.2341085.101100.661099.311101.191126.23Tolazamide431444.089434.774432.732133.900448.077453.016459.908457.351428.184Miglitol179259.206279.461270.486-30.9112283.993313.736314.588314.531282.550Prandin619843.711843.494838.013531.652819.679799.475801.756801.663838.017Metformin13299.9702111.886104.916-201.46595.77994.37102.83699.143113.280Glimepiride895837.204817.268812.745508.839824.505809.921830.82821.735800.656Linagliptin885779.022764.816777.836480.129761.767735.058729.788732.955771.681Pioglitazone466491.745495.742502.218202.339481.859473.908461.292467.387503.67Bromocriptine12301091.751051.261069.08786.7841114.061130.261121.461126.671043.50Alogliptin622504.377508.003510.530215.375505.989512.21495.892503.671513.201Table 19
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(G) 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 for F(G) indexBoiling point computed from regression model for H(G) indexBoiling point computed from regression model for HM(G) indexAcarbose675.05℃at760mmHg786.34808.827801.712796.80792.764800.482801.68800.15813.8468Tolazamide300℃at760mmHg444.85438.715438.065438.53445.34446.338450.56448.20434.9031Miglitol453.7℃at760mmHg344.00354.117349.951349.11355.852370.178370.76370.13355.8444Prandin672.9℃at760mmHg662.84661.339658.167654.34648.004635.786638.28636.41657.3851Metformin224℃at760mmHg257.14262.842260.032256.58253.204250.226254.48252.39263.9541Glimepiride659.29647.053644.445641.96650.636641.498654.24647.38637.103Linagliptin661.2℃at760mmHg627.55618.483625.486626.38616.42600.562598.76598.85621.374Pioglitazone470.84471.923475.802475.66463.764457.762451.32453.68475.8812Bromocriptine891.3℃at760mmHg798.14774.506783.656792.76808.556816.666813.84814.07768.9365Alogliptin519.2℃at760mmHg477.74478.601480.316482.74476.924478.706470.32473.52481.0552


## 3.1. Regression Model

In this paper, drug computable structure analysis about ten topological indices is done for QSPR modeling tenacity. The six physical properties, flash point (FP), polarity, boiling point (BP), molar volume (MV), and refractivity (R) for ten medicine arranged in Table2 are used in diabetes treatment. We execute the regression analysis for the drugs, and the linear regression model is tested with the help of equation(9)P=A+bTI.Table 2
Correlation coefficients.
Topological indexCorrelation coefficient of boiling pointCorrelation coefficient of refractive indexCorrelation coefficient of flash pointCorrelation coefficient of polarityCorrelation coefficient of molar volumeCorrelation coefficient of complexityABC(G)0.9100.9610.9290.9930.9600.946RA(G)0.8980.9470.9390.9850.9470.931S(G)0.9040.9530.9380.9890.9510.938GA(G)0.9090.9560.9380.9910.9520.943M1(G)0.9120.9530.9400.9900.9490.947M2(G)0.9120.9400.9500.9830.9350.943HM(G)0.8960.9440.9420.9830.9420.928F(G)0.9070.9370.9470.9820.9350.941H(G)0.9090.9390.9490.9830.9350.942Here, P represents the physicochemical property of given drugs. TI is the topological index, A is constant, andb represents the regression coefficient. The IBM SPSS Statistics version-24 software is helpful to find out the results. Nine TIs of diabetes drugs and physiochemical properties are analyzed with a linear QSPR model. Equation (9) is suitable for the said calculation purpose.Theorem 1.
LetG1 be the graph metformin. Various topological indices of G1 are given as follows:(i)
ABCG1=14.75.(ii)
RAG1=8.75.(iii)
SG1=8.57.(iv)
GAG1=17.00.(v)
M1G1=96.(vi)
M2G1=104.(vii)
FG1=290.(viii)
HG1=498.(ix)
HMG1=7.72.Proof.
LetG1 be the graph of metformin with edge set E. Let Em,n represent the class of edges of G1 joining vertices of degrees m and n. With E1,3=5, E1,4=6,E2,3=3,E2,4=1,E3,3=3, and E3,4=1, one has the following: (i)
By using Definition1 and the above given edge partitions Em,n, we get(10)ABG1=51+3−21×3+61+4−21×4+32+3−22×3+12+4−22×4+33+3−23×3+13+4−23×4=14.75.(ii)
By using Definition2 and the above given edge partitions Em,n, we get(11)RAGG1=511×3+611×4+312×3+112×4+313×3+113×4=8.75.(iii)
By using Definition3 and the above given edge partitions Em,n, we get(12)SG1=511+3+611+4+312+3+112+4+313+3+113+4=8.57.(iv)
By using Definition4 and the above given edge partitions Em,n, we get(13)GAG1=51×31+3+61×41+4+32×32+3+12×42+4+33×33+3+13×43+4=17.00.(v)
By using Definition5 and the above given edge partitions Em,n, we get(14)M1G1=51+3+61+4+32+3+12+4+33+3+13+4=96.(vi)
By using Definition5 and the above given edge partitions Em,n, we get(15)M2G1=51×3+61×4+32×3+12×4+33×3+13×4=104.(vii)
By using Definition6 and the above given edge partitions Em,n, we get(16)HG1=511+3+611+4+312+3+112+4+313+3+113+4=498.(viii)
By using Definition7 and the above given edge partitions Em,n, we get(17)HMG1=51+32+61+42+32+32+12+42+33+32+13+42=7.72.(ix)
By using Definition8 and the above given edge partitions Em,n, we get(18)FG1=51+9+61+16+34+9+14+16+39+9+19+16=290.Theorem 2.
LetG2 be the graph of tolazamide. Various topological indices of G2 are given as follows:(i)
ABCG2=32.73.(ii)
RAG2=18.23.(iii)
SG2=18.43.(iv)
GAG2=38.61.(v)
M1G2=242.(vi)
M2G2=310.(vii)
FG2=806.(viii)
HG2=1426.(ix)
HMG2=15.98.Proof.
LetG2 be the graph of tolazamide with edge set E′E′. Let Em,n′ represent the class of edges of G2 joining vertices of degrees m and n. With E1,3′=7, E1,4′=17, E3,3′=9, E3,4′=5, and E4,4′=5, one has the following:(i)
By using Definition1 and edge partitions Em,n′, we get(19)ABCG2=71+3−21×3+171+4−21×493+3−23×3+53+4−23×4+54+4−24×4=32.73.(ii)
By using Definition2 and edge partition Em,n′, we get(20)RAG2=711×3+1711×4+913×3+513×4+514×4=18.23.(iii)
Definition3 and edge partition Em,n′ give(21)SG2=711+3+1711+4+913+3+513+4+514+4=18.43.(iv)
By using Definition4 and edge partition Em,n′, we get(22)GAG2=71×31+3+171×41+4+93×33+3+53×43+454×44+4=38.61.(v)
By using Definition5 and edge partition Em,n′, we get(23)M1G2=93+3+53+4+54+4=242.(vi)
By using Definition5 and edge partition Em,n′, we get(24)M2G2=93×3+53×4+54×4=310.(vii)
By using Definition6 and edge partition Em,n′, we get(25)HG2=1411+3+3411+4+1813+3+1013+4+1014+4=15.98.(viii)
By using Definition7 and edge partition Em,n′, we get(26)HMG=71+32+171+42+93+32+53+42+54+42=1426.(ix)
By using Definition8 and edge partition Em,n′, we get(27)FG2=71+9+171+16+99+9+59+16+516+16=806.
One can calculate topological indices of the remaining drugs by adopting a similar procedure applied in Theorem1 and Theorem 2 and using Definitions 1 to 8. Moreover, the calculated values of all drugs are listed in Table 1.
Using (1), we have calculated the following diverse linear models for all degree-based topological indices, which are given as follows:(1)
Regression models for atom bond connectivity index ABC(G):Boiling point = 103.153 + 10.44 [ABC(G)].Refractivity = 18.569 + 2.093 [ABC(G)].Flash point = 20.160 + 7.125 [ABC(G)].Polarity = 0.172 + 0.980 [ABC(G)].Molar volume = 31.785 + 6.065 [ABC(G)].Complexity = −182.330 + 19.139 [ABC(G)].(2)
Regression models for atom bond connectivity index RA(G):Boiling point = 100.512 + 18.552 [RA(G)].Refractivity = 18.020 + 3.731 [RA(G)].Flash point = 10.148 + 13.033 [RA(G)].Polarity = −0.380 + 1.759 [RA(G)].Molar volume = 29.935 + 10.820 [RA(G)].Complexity = −186.139 + 34.060 [RA(G)].(3)
Regression models for atom bond connectivity index S(G):Boiling point = 105.293 + 18.056 [S(G)].Refractive index = 18.429 + 3.638 [S(G)].Flash point = 15.429 + 12.603 [S(G)].Polarity = 0.122 + 1.710 [S(G)].Molar volume = 32.905 + 10.520 [S(G)].Complexity = −180.010 + 33.247 [S(G)].(4)
Regression models for atom bond connectivity index GA(G):Boiling point = 113.440 + 8.420 [GA(G)].Refractivity = 20.596 + 1.694 [GA(G)].Flash point = 22.547 + 5.853 [GA(G)].Polarity = 1.044 + 0.795 [GA(G)].Molar volume = 38.721 + 4.888 [GA(G)].Complexity = −465.288 + 15.519 [GA(G)].(5)
Regression models for atom bond connectivity index M1(G):Boiling point = 126.868 + 1.316 [M1(G)].Refractivity = 24.500 + 0.261 [M1(G)].Flash point = 34.273 + .908 [M1(G)].Polarity = 2.745 + 0.123 [M1(G)].Molar volume = 49.701 + 0.755 [M1(G)].Complexity = −135.869 + 2.413 [M1(G)].(6)
Regression models for atom bond connectivity index HM(G):Boiling point = 104.181 + 20.696 [HM(G)].Refractivity = 18.567 + 4.176 [HM(G)].Flash point = 10.181 + 14.677 [HM(G)].Polarity = −0.136 + 1.970 [HM(G)].Molar volume = 32.218 + 12.080 [HM(G)].Complexity = −181.037 + 38.124 [HM(G)].(7)
Regression models for atom bond connectivity index M2(G):Boiling point = 151.218 + .952 [M2(G)].Refractivity = 30.535 + 0.187 [M2(G)].Flash point = 48.647 + 0.665 [M2(G)].Polarity = 5.362 + 0.088 [M2(G)].Molar volume = 67.501 + 0.539 [M2(G)].Complexity = −86.964 + 1.741 [M2(G)].(8)
Regression models for atom bond connectivity index F(G):Boiling point = 144.280 + 0.380 [F(G)].Refractivity = 29.397 + 0.074 [F(G)].Flash point = 44.851 + 0.264 [F(G)].Polarity = 4.737 + 0.035 [F(G)].Molar volume = 63.476 + 0.215 [F(G)].Complexity = −97.844 + 0.692 [F(G)].(9)
Regression models for atom bond connectivity index H(G):Boiling point = 147.316 + 0.211 [H(G)].Refractivity = 29.878 + 0.041 [H(G)].Flash point = 46.427 + 0.147 [H(G)].Polarity = 5.004 + 0.020 [H(G)].Molar volume = 65.21.0 + 0.120 [H(G)].Complexity = −93.085 + 0.386 [H(G)].
Tables3–11 represent the statistical parameters used in the QSPR models of TIs.Table 3
Statistical parameters used in the QSPR model of ABC(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8103.15310.440.9100.82828.9470.002SignificantRefractivity1018.5692.0930.9610.92396.1860.000SignificantFlash point1020.1607.1250.9290.86350.2090.000SignificantPolarity100.1720.9800.9930.987601.8290.000SignificantMolar volume1031.7856.0650.9600.92294.6580.000SignificantComplexity10−182.33019.1390.9460.89568.0750.000SignificantTable 4
Statistical parameters used in the QSPR model of R(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8100.51218.5520.8980.80624.8690.002SignificantRefractivity1018.0203.7310.9470.89668.9520.000SignificantFlash point1010.14813.0330.9390.88259.7380.000SignificantPolarity10−0.3801.7590.9850.971269.2420.000SignificantMolar volume1029.93510.8200.9470.89769.4350.000SignificantComplexity10−186.13934.0600.9310.86651.6820.000SignificantTable 5
Statistical parameters used in the QSPR model of S(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8105.29318.0560.9040.81826.8950.002SignificantRefractivity1018.7723.6380.9530.90979.8440.000SignificantFlash point1015.42912.6030.9380.88058.4180.000SignificantPolarity100.1221.7100.9890.978361.0560.000SignificantMolar volume1032.90510.5200.9510.90475.4280.000SignificantComplexity10−180.01033.2470.9380.88058.6690.000SignificantTable 6
Statistical parameters used in the QSPR model of GA(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8113.4408.4200.9090.82728.6610.002SignificantRefractivity1020.5961.6940.9560.91585.8070.000SignificantFlash point1022.5475.8530.9380.88158.9500.000SignificantPolarity101.0440.7950.9910.981422.8130.000SignificantMolar volume1038.7214.8880.9520.90677.0890.000SignificantComplexity10−465.28815.5190.9430.89064.6440.000SignificantTable 7
Statistical parameters used in the QSPR model of M1(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8126.8681.3160.9120.83329.8510.002SignificantRefractivity1024.5000.2610.9530.90778.2610.000SignificantFlash point1034.2730.9080.9400.88460.9280.000SignificantPolarity102.7450.1230.9900.980389.9930.000SignificantMolar volume1049.7010.7550.9490.90172.5650.000SignificantComplexity10−135.8692.4130.9470.89769.8140.000SignificantTable 8
Statistical parameters used in the QSPR model of M2(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8151.2180.9520.9120.83129.5940.002SignificantRefractivity1030.5350.1870.9400.88461.7650.000SignificantFlash point1048.6470.6650.9500.90374.7260.000SignificantPolarity105.3620.0880.9830.966227.7190.000SignificantMolar volume1067.5010.5390.9350.87455.7160.000SignificantComplexity10−86.6941.7410.9430.89064.6050.000SignificantTable 9
Statistical parameters used in the QSPR model of HM(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8104.18120.6960.8960.80224.3710.003SignificantRefractivity1018.5674.1760.9440.89165.5900.000SignificantFlash point1010.18114.6770.9420.88863.2530.000SignificantPolarity10−0.1361.9700.9830.967232.0950.000SignificantMolar volume1032.21812.0800.9420.88762.9210.000SignificantComplexity10−181.03738.1240.9280.86149.6170.000SignificantTable 10
Statistical parameters used in the QSPR model of F(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8144.2800.3800.9070.82327.8600.002SignificantRefractivity1029.3970.0740.9370.87857.7420.000SignificantFlash point1044.8510.2640.9470.89669.0580.000SignificantPolarity104.7370.0350.9820.965217.8320.000SignificantMolar volume1063.4760.2150.9350.87555.8880.000SignificantComplexity10−97.8440.6920.9410.88661.9990.000SignificantTable 11
Statistical parameters used in the QSPR model of H(G).
Physiochemical propertyNAbrr2FpIndicatorBoiling point8147.3160.2110.9090.82728.6270.002SignificantRefractivity1029.8780.0410.9390.88159.1680.000SignificantFlash point1046.4270.1470.9490.90071.7560.000SignificantPolarity105.0040.0200.9830.965223.4510.000SignificantMolar volume1065.2100.1200.9350.87555.8640.000SignificantComplexity10−93.0850.3860.9420.88863.2180.000Significant


## 3.2. Quantitative Structure Analysis and Comparison between Topological Indices and Correlation

### 3.2.1. Coefficient of Physicochemical Properties

Physical properties for ten diabetes drugs are listed in Table12, and their TIs computed through the molecular structure are recorded in Table 1. The correlation coefficient between six physical properties and TIs is itemized in Table 2. The graph of TIs and physical properties is shown in Figure 2.Table 12
Physical properties of drugs.
Name of drugRefractive indexm3mol−1Flash point℃Molar volumecm3Polaritycm3Boiling point℃Complexity (bit)Acarbose139.02541.40369.8060.87675.05962Tolazamide81.34246.80237.9032.82300431Miglitol48.16284.30142.1020.81453.7179Prandin/repaglinide131.83360.80397.9051.49672.9619Metformin56.6458.10100.8013.43224132Glimepiride129.80363.20378.8053.49895Linagliptin133.43353.70338.0051.24661.2885Pioglitazone97.39301.80282.8037.91466Bromocriptine165.51492.80429.4066.44891.31230Alogliptin104.26267.80252.9035.44519.2622Figure 2
Physicochemical properties and TIs.


## 3.2.1. Coefficient of Physicochemical Properties

Physical properties for ten diabetes drugs are listed in Table12, and their TIs computed through the molecular structure are recorded in Table 1. The correlation coefficient between six physical properties and TIs is itemized in Table 2. The graph of TIs and physical properties is shown in Figure 2.Table 12
Physical properties of drugs.
Name of drugRefractive indexm3mol−1Flash point℃Molar volumecm3Polaritycm3Boiling point℃Complexity (bit)Acarbose139.02541.40369.8060.87675.05962Tolazamide81.34246.80237.9032.82300431Miglitol48.16284.30142.1020.81453.7179Prandin/repaglinide131.83360.80397.9051.49672.9619Metformin56.6458.10100.8013.43224132Glimepiride129.80363.20378.8053.49895Linagliptin133.43353.70338.0051.24661.2885Pioglitazone97.39301.80282.8037.91466Bromocriptine165.51492.80429.4066.44891.31230Alogliptin104.26267.80252.9035.44519.2622Figure 2
Physicochemical properties and TIs.


## 3.3. Calculation of Statistical Parameters

In this section, we find the relation between degree base TIs and physical properties of diabetes drugs such as medicines acarbose, tolazamide, miglitol, prandin, metformin, glimepiride, linagliptin, pioglitazone, bromocriptine, and alogliptin, and this is done with the help of QSPR modeling, whereas TIs, N,b, and r represent regression model constant, correlation coefficient, and sample size, respectively. This will be helpful to compare and improve the model. It is noted that the value of r is greater than 0.6, and p value is less than 0.05. Hence, the calculation verifies that all properties are significant.


## 3.4. Standard Error (SE) of Estimate and Comparison

Measure of variation for an observation calculated around the computed regression line is said to be the standard error estimate. It measures the amount of accuracy of predictions done around the computed regression line in Table13. We also compare the physicochemical properties of the experimental and theoretical calculated values of the models as presented in Tables 14–19.Table 13
Standard error of estimate.
Topological indexStd. error of the estimate for boiling pointStd. error of the estimate for refractive indexStd. error of the estimate for flash pointStd. error of the estimate for polarityStd. error of the estimate for molar volumeStd. error of the estimate for complexityABC(G)98.4304811.1746652.645052.0918832.63778121.449RA(G)104.7305113.0025148.801743.1025037.57911137.117S(G)101.4552612.1697849.284092.6890336.20431129.734GA(G)98.8358111.7766449.088032.4888435.84931124.284M1(G)97.1822013.2809248.378442.5894236.84197120.084M2(G)97.5326913.7548444.160033.3646841.42791124.317HM(G)105.5857713.2962947.582713.3338839.26701139.553F(G)99.9982314.0675045.755293.4375441.37206126.610H(G)98.8852213.9173444.974843.3955641.37987125.522Table 14
Comparison of actual and computed values for polarity from regression models.
Name of drugPolarity of drugPolarity computed from regression model for ABC(G) indexPolarity computed from regression model for R(G) indexPolarity computed from regression model for S(G) indexPolarity computed from regression model for GA(G) indexPolarity computed from regression model for M1(G) indexPolarity computed from regression model for M2(G) indexPolarity computed from regression model for F(G) indexPolarity computed from regression model for H(G) indexPolarity computed from regression model for HM(G) indexAcarbose60.87cm364.303266.778666.076765.566264.98365.37865.28766.88467.4153Tolazamide32.82cm332.247431.686531.637331.738932.51132.64232.94733.52431.3446Miglitol20.81cm322.780623.665523.292523.296024.14725.60225.59726.12423.8192Prandin51.49cm352.709852.794552.482252.114851.45350.15450.23751.36452.5221Metformin13.43cm314.62715.011214.776714.55914.55314.51414.88714.96415.0724Glimepiride53.49cm352.376651.440151.182650.946151.69950.68251.70752.40450.5915Linagliptin51.24cm349.397448.731249.387149.475448.50146.89846.59747.80449.0943Pioglitazone37.91cm334.687634.835135.211235.244934.23333.69833.01734.04435.2452Bromocriptine66.44cm365.410663.524464.366765.184666.45966.87466.40768.20463.1404Alogliptin35.44cm335.334435.468435.638735.912735.46335.63434.76735.92435.7377Table 15
Comparison of actual and computed values for molar volume from regression models.
Name of drugMolar volume of drugMolar volume from regression model for ABC(G) indexMolar volume from regression model for R(G) indexMolar volume from regression model for S(G) indexMolar volume from regression model for GA(G) indexMolar volume from regression model for M1(G) indexMolar volume from regression model for M2(G)indexMolar volume from regression model for F(G) indexMolar volume from regression model for H(G) indexMolar volume from regression model for HM(G) indexAcarbose369.8±5.0cm3428.678443.042438.661435.431431.731435.099435.426436.49446.441Tolazamide237.9±7.0cm3230.292227.183226.788227.446232.411234.591236.766236.33225.256Miglitol142.1±3.0cm3171.704177.844175.451175.536181.071191.471191.616191.93179.110Prandin397.9±3.0cm3356.929357.023355.027352.726348.681341.852342.976343.37355.116Metformin100.8±7.0cm3121.243124.61123.061121.817122.181123.557125.826124.97125.475Glimepiride378.8±5.0cm3354.867348.692347.032345.540350.191345.086352.006349.61343.278Linagliptin338±7.0cm3336.43332.029335.986336.498330.561321.909320.616322.01334.097Pioglitazone282.8±3.0cm3245.394246.551248.775249.002242.981241.059237.196239.45249.174Bromocriptine429.4±5.0cm3435.532423.025428.141433.084440.791444.262442.306444.41420.227Alogliptin252.9±5.0cm3249.397250.446251.405253.108250.531252.917247.946250.73252.194Table 16
Comparison of actual and computed values for flash point from regression models.
Name of drugFlash point of drugFlash point computed from regression model for ABC(G) indexFlash point computed from regression model for R(G) indexFlash point computed from regression model for S(G) indexFlash point computed from regression model for GA(G) indexFlash point computed from regression model for M1(G) indexFlash point computed from regression model for M2(G) indexFlash point computed from regression model for F(G) indexFlash point computed from regression model for H(G) indexFlash point computed from regression model for HM(G) indexAcarbose541.4±34.3℃486.42507.747501.526497.576493.721502.177501.571501.245513.455Tolazamide246.8±26.8℃253.361247.739247.702248.531254.009254.797257.635256.049244.719Miglitol284.3±27.4℃184.533188.309186.199186.372192.265201.597202.195201.659188.653Prandin360.8±0.0℃402.131404.135401.332398.543393.841387.132388.051387.173402.497Metformin58.1±22.6℃125.253124.186123.436122.048121.441117.807121.411119.633123.487Glimepiride363.2±34.3℃399.708394.100391.754389.939395.657391.122399.139394.817388.113Linagliptin353.7±34.3℃378.048374.029378.521379.111372.049362.527360.595361.007376.959Pioglitazone301.8±28.7℃271.102271.068274.042274.343266.721262.777258.163259.871273.779Bromocriptine492.8±34.3℃494.471483.636488.923494.767504.617513.482510.019510.947481.606Alogliptin267.8±32.9℃275.805275.760277.193279.259275.801277.407271.363273.689277.449Table 17
Comparison of actual and computed values for refractive index from regression models.
Name of drugRefractivity of drugRefractivity computed from regression model for ABC(G) indexRefractivity computed from regression model for R(G) indexRefractivity computed from regression model for S(G) indexRefractivity computed from regression model for GA(G) indexRefractivity computed from regression model for M1(G) indexRefractivity computed from regression model for M2(G) indexRefractivity computed from regression model for F(G) indexRefractivity computed from regression model for H(G) indexRefractivity computed from regression model for HM(G) indexAcarbose139.02m3⋅mol−1155.534160.469159.089158.081156.566158.069157.417156.732161.762Tolazamide81.34m3⋅mol−187.072886.036185.820386.001387.66288.50589.04188.34485.2994Miglitol48.16m3⋅mol−166.854569.022768.066968.011069.91473.54573.50173.17469.3471Prandin131.83m3⋅mol−1130.774130.808130.167129.418127.856125.718125.597124.916130.191Metformin56.64m3⋅mol−149.440750.666249.949649.39449.55649.98350.85750.29650.8057Glimepiride129.8m3⋅mol−1130.063127.935127.402126.928128.378126.84128.705127.048126.099Linagliptin133.43m3⋅mol−1123.700122.189123.582123.794121.592118.799117.901117.618122.925Pioglitazone97.39m3⋅mol−192.284492.714693.423793.471891.31690.74989.18989.4193.5679Bromocriptine165.51m3⋅mol−1157.9153.567155.451157.267159.698161.248159.785159.438152.700Alogliptin104.26m3⋅mol−193.665894.057794.333294.894893.92694.86392.88993.26494.6119Table 18
Comparison of actual and computed values for complexity from regression models.
Name of drugComplexityComplexity computed from regression model for ABC(G) indexComplexity computed from regression model for R(G) indexComplexity computed from regression model for S(G) indexComplexity computed from regression model for GA(G) indexComplexity computed from regression model for M1(G) indexComplexity computed from regression model for M2(G) indexComplexity computed from regression model for F(G) indexComplexity computed from regression model for H(G) indexComplexity computed from regression model for HM(G) indexAcarbose9621070.121114.271102.32794.2341085.101100.661099.311101.191126.23Tolazamide431444.089434.774432.732133.900448.077453.016459.908457.351428.184Miglitol179259.206279.461270.486-30.9112283.993313.736314.588314.531282.550Prandin619843.711843.494838.013531.652819.679799.475801.756801.663838.017Metformin13299.9702111.886104.916-201.46595.77994.37102.83699.143113.280Glimepiride895837.204817.268812.745508.839824.505809.921830.82821.735800.656Linagliptin885779.022764.816777.836480.129761.767735.058729.788732.955771.681Pioglitazone466491.745495.742502.218202.339481.859473.908461.292467.387503.67Bromocriptine12301091.751051.261069.08786.7841114.061130.261121.461126.671043.50Alogliptin622504.377508.003510.530215.375505.989512.21495.892503.671513.201Table 19
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(G) 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 for F(G) indexBoiling point computed from regression model for H(G) indexBoiling point computed from regression model for HM(G) indexAcarbose675.05℃at760mmHg786.34808.827801.712796.80792.764800.482801.68800.15813.8468Tolazamide300℃at760mmHg444.85438.715438.065438.53445.34446.338450.56448.20434.9031Miglitol453.7℃at760mmHg344.00354.117349.951349.11355.852370.178370.76370.13355.8444Prandin672.9℃at760mmHg662.84661.339658.167654.34648.004635.786638.28636.41657.3851Metformin224℃at760mmHg257.14262.842260.032256.58253.204250.226254.48252.39263.9541Glimepiride659.29647.053644.445641.96650.636641.498654.24647.38637.103Linagliptin661.2℃at760mmHg627.55618.483625.486626.38616.42600.562598.76598.85621.374Pioglitazone470.84471.923475.802475.66463.764457.762451.32453.68475.8812Bromocriptine891.3℃at760mmHg798.14774.506783.656792.76808.556816.666813.84814.07768.9365Alogliptin519.2℃at760mmHg477.74478.601480.316482.74476.924478.706470.32473.52481.0552


## 4. Conclusions

It is obvious from statistical parameters used in linear QSPR models and topological indices that ABC(G) index provides a maximum high correlated value for polarityr = 0.993, refractive index r = 0.961, and molar volume r = 0.960. M1 index provides a high correlated value of boiling point; that is, r = 0.912 and complexity r = 0.947. M2 index depicts the utmost correlation coefficient of flash point r = 0.950 and boiling point r = 0.912.In this paper, we have computed topological indices and related them to the linear QSPR model for the drugs used to cure diabetes. The results gained in the following means will be supportive for designing various new drugs to attain defensive measures for the said disease in the pharmaceutical industry. The correlation coefficient plays a substantial impact on the range of topological indices of the drugs and may be considered for the combination to designing novel drugs. The outcomes are helpful to research workers on drugs science in the pharmaceutical industry and offer a pathway to approximate physical properties for novice discoveries of diabetes medicines to cure other specific diseases [25].



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