Degree-Based Topological Indices and Machine Learning for QSPR Modeling of Arthritis Drugs

Abstract

This study addresses the challenge of accurately predicting the physicochemical properties of arthritis drugs using simple graph-theoretic descriptors and evaluates whether combining degree-based topological indices with modern machine learning can outperform traditional linear QSPR models. Nine classical degree-based topological indices (including reformulated Zagreb indices, augmented Zagreb index, hyper-Zagreb index, forgotten index, symmetric division degree index, and inverse sum Zagreb index) were computed for 50 structurally diverse arthritis drugs ranging from NSAIDs (ibuprofen and diclofenac) and COX-2 inhibitors (celecoxib) to disease-modifying agents (methotrexate) and immunomodulators with experimental property values (boiling point, melting point, critical pressure, molar refractivity, topological polar surface area, and calculated LogP) obtained from ChemDraw and PubChem. Using these indices as input features, linear regression, Random Forest, and XGBoost models were developed and compared. XGBoost consistently outperformed the other methods across all properties, achieving for molar refractivity a mean absolute error (MAE) of 0.80787, a root-mean-square error (RMSE) of 1.13352, and coefficient of determination (r ² = 0.9945); for boiling point, RMSE = 16.22 and r ² = 0.9866; for melting point, RMSE = 25.86 and r ² = 0.9450; for critical pressure, RMSE = 1.176 and r ² = 0.9833; for topological polar surface area, RMSE = 4.38 and r ² = 0.9762; and for CLogP, RMSE = 0.586 and r ² = 0.8857. Random Forest also performed well but with slightly higher errors, while linear regression showed moderate to strong correlations for boiling point, melting point, molar refractivity, and tPSA but failed to capture nonlinear structure–property relationships. The novelty of this work lies in the first systematic integration of nine degree-based topological indices with advanced ensemble learning (XGBoost and Random Forest) for QSPR modeling of a large, diverse arthritis drug data set, demonstrating that simple connectivity measureswhen paired with nonlinear machine learningcan achieve near-perfect predictive accuracy for key properties, offering a low-cost, computationally efficient framework for drug design and virtual screening.

1. Introduction

Chemical graph theory was born from the early mathematical endeavor to depict molecular structures through the concept of atoms as vertices and bonds as edges. , The new foundation of this theory was laid down when the chemists of the 19th century realized that sometimes it was possible to relate the measurable properties of a substance to the patterns of connectivity in its molecules. Slowly but surely, this area of research turned into a dependable computational platform, where the degree-based topological indices became the necessary tools for rendering molecular design into numerical descriptors. ,, The first indices, like the Wiener index, showed that the numerical descriptors could have very strong correlations with measurable physicochemical properties. , This, in turn, allowed for the introduction of many other indices, among which were the Randic connectivity index and the Zagreb indices. , These measurements describe the characteristics of the primary structure as single invariant values, , including the degree of branching, the shape, and the distribution of degrees. These indices are widely available due to their ease of use, effectiveness, and proven capacity to correlate with physical, chemical, and biological characteristics. ,, Over the years, there has been a constant stream of new and improved indicators, and their significance has grown. They offer a robust and adaptable link between the experimental realm of important characteristics and the abstract realm of structural representation. ,

Recent advances in QSPR modeling have significantly expanded the repertoire of topological indices beyond classical degree-based descriptors. Neighborhood degree-sum indices, for instance, have been successfully employed to predict the pharmacokinetic properties of anti-Parkinsonian agents and other drug classes, capturing local structural environments that pure degree measures overlook. Reverse-degree indices have shown strong correlations with boiling points, refractive indices, and chromatographic retention times for a range of chemical libraries, including cyclodextrin derivatives and benzene derivatives. Distance-based descriptors, such as the Wiener index and its variants, continue to provide robust predictive power for molecular volume, surface area, and drug–target binding affinities, particularly in cardiovascular and chemotherapeutic drug design. , Furthermore, mixed indices that combine distance and degree information have been developed to model lipophilicity and solubility with improved accuracy. , Collectively, these developments demonstrate that incorporating multiple families of topological indices degree, neighborhood sum, reverse, and distance can capture complementary structural features (local branching, spatial arrangement, and electronic environment) and often outperform single-class descriptors when integrated with modern machine learning frameworks. However, no systematic QSPR study has yet applied this diversified index ensemble to the challenging and structurally heterogeneous class of arthritis drugs, a gap that the present work aims to address.

Given the gradual and persistent nature of joint problems, arthritis drugs constitute a major class of therapeutic agents. Extensive research has been conducted on nonsteroidal anti-inflammatory drugs (NSAIDs), COX inhibitors, , disease-modifying antirheumatic drugs, corticosteroids, and targeted immunomodulators, all of which reflect the urgent demand for treatments that are both effective and safe. , The structural diversity of these drug categories renders them particularly suitable for quantitative structure–property relationship (QSPR) studies, as variations in molecular structure typically lead to changes in properties such as polarity, refractivity, and lipophilicity. ,, Earlier studies of these agents provide a robust foundation for computational approaches aimed at elucidating the interactions between molecular structure, physicochemical behavior, and drug performance.

We have calculated the nine classical degree-based topological indices (see Table for definitions) for the 50 drugs used in arthritis treatment and verified them against six important physicochemical properties in our flow. The model consists of the building up of the complete molecular graph, calculation of the topological indices in a systematic way, and statistical modeling through linear regression supported by predictive validation using Random Forest and XGBoost algorithms. By merging graph–theoretical descriptors with machine learning techniques, ,− , the present study showcases the ability of simple connectivity measures to elucidate and forecast crucial molecular attributes, thus providing a solid foundation for further computational drug analysis.

1. Definitions of Degree-Based Topological Indices.

index name	definition
reformulated first Zagreb index EM1(G)	EM1(G) = ∑mn∈E(G)(d(m) + d(n) – 2)2
reformulated second Zagreb index EM2(G)	$$E{M}_2(G)=\sum _{mn\in E(G)}\sum _{np\in E(G),p\ne m}(\mathrm{d}(m)+\mathrm{d}(n)-2)(\mathrm{d}(n)+\mathrm{d}(p)-2)$$
first Zagreb index M 1(G)	M1(G) = ∑mn∈E(G)(d m + d n )
second Zagreb index M 2(G)	M2(G) = ∑mn∈E(G)(d m · d n )
augmented Zagreb index AZI(G)	$$AZI(G)=\sum _{mn\in E(G)}{\left(\frac{\mathrm{d}(m)\mathrm{d}(n)}{\mathrm{d}(m)+\mathrm{d}(n)-2}\right)}^3$$
hyper Zagreb index HM(G)	HM(G) = ∑mn∈E(G)(d(m) + d(n))2
forgotten index F(G)	$$F(G)=\sum _{mn\in E(G)}({\mathrm{d}}_m^2+{\mathrm{d}}_n^2)$$
symmetric division deg index SDD(G)	$$SDD(G)=\sum _{mn\in E(G)}(\frac{\mathrm{d}(m)}{\mathrm{d}(n)}+\frac{\mathrm{d}(n)}{\mathrm{d}(m)})$$
inverse sum indeg index ISZ(G)	$$ISZ(G)=\sum _{mn\in E(G)}\frac{\mathrm{d}(m)+\mathrm{d}(n)}{\mathrm{d}(m)\mathrm{d}(n)}$$

2. Materials and Methods

2.1. Data Collection and Molecular Structure Generation

A total of 50 clinically approved arthritis drugs were selected, covering a broad range of therapeutic classes: nonsteroidal anti-inflammatory drugs (NSAIDs; e.g., ibuprofen and diclofenac), COX-2 inhibitors (celecoxib and rofecoxib), disease-modifying antirheumatic drugs (methotrexate), corticosteroids, and immunomodulators. The drug names and their simplified molecular-input line-entry system (SMILES) strings were retrieved from the PubChem database https://pubchem.ncbi.nlm.nih.gov. For each compound, the three-dimensional molecular structure was downloaded in the structure data file (SDF) format from PubChem (CIDs verified). When experimental 3D coordinates were not available, the structures were generated using ChemDraw Professional 19.0, followed by energy minimization with the MM2 force field (Chem3D 19.0). All structures were then converted into molecular graphs where vertices represent non-hydrogen atoms and edges represent covalent bonds. Hydrogen atoms were treated implicitly, as is standard in degree-based topological index calculations. Figure presents the molecular structures of drugs from ibuprofen to tofacitinib, while Figure shows drugs from baricitinib to fenbufen, illustrating the structural diversity of the data set. The molecular structure of the remaining 25 drugs can be drawn in a similar way.

1.

3D molecular structures of arthritis drugs (ibuprofen to tofacitinib).

2.

3D molecular structures of arthritis drugs (baricitinib to fenbufen).

2.2. Edge-Degree Partitioning and Topological Index Calculation

For each molecular graph G = (V, E), the degree d­(v) of each vertex v was defined as the number of its adjacent non-hydrogen atoms. Every edge e = uv was assigned a degree pair (d _u , d _v ). The frequencies of each distinct (d _u , d _v ) pair (with d _u ≤ d _v ) were recorded as the edge-degree partition of G. This partition formed the basis for computing all nine degree-based topological indices listed in Table .

As a representative example, diclofenac was used. Its molecular graph contains 19 vertices and 18 edges, partitioned as 3 edges of type (1,3), 6 edges of type (2,2), 6 edges of type (2,3), and 3 edges of type (3,3). Using this partition, the Reformulated First Zagreb Index was evaluated as

$$E{M}_1(G)=\sum _{mn\in E(G)}{({\mathrm{d}}_m+{\mathrm{d}}_n-2)}^2=3{(1+3-2)}^2+6{(2+2-2)}^2+6{(2+3-2)}^2+3{(3+3-2)}^2=138$$

All remaining indices, including the classical Zagreb indices, augmented Zagreb index, hyper-Zagreb index, forgotten index, reformulated Zagreb indices, symmetric division degree index, and inverse sum index, were computed through the same systematic procedure directly from the edge-degree partitions of each molecular graph. The indices were calculated using Python 3.9 with the NetworkX library (version 2.8) for graph manipulation and custom scripts implementing the exact formulas given in Table . To ensure accuracy, the computed values for a subset of drugs were cross-checked against hand calculations based on edge-degree partitions. For validation, the same indices were independently recalculated using the RDKit (2022.03.1) topological index module; discrepancies were below 0.01% and were resolved by reexamining the edge partitions. The final computed results for all nine degree-based topological indices across the 50 drugs are summarized in Table . These indices constitute the data set used in the QSPR modeling presented in subsequent sections.

2. Calculated Values of Nine Degree-Based Topological Indices for the Data Set of 50 Arthritis Drugs.

	topological indices
drugs	EM1	EM2	M1	M2	AZI	HM	F	SDD	ISZ
ibuprofen	114	161	70	77	103.65625	334	180	37.66666667	15
naproxen	152	220	90	103	143.671875	436	230	44.66666667	18
diclofenac	138	199	84	96	140.296875	402	210	41	17
indomethacin	158	229	94	108	156.296875	454	238	46	19
celecoxib	124	181	72	83	112.90625	352	186	35.33333333	14
piroxicam	124	194	68	81	111.6875	340	178	32.66666667	13
tiaprofenic acid	134	195	80	92	132.296875	386	202	39	16
meloxicam	168	256	96	114	164.3125	472	244	44	18
rofecoxib	194	301	110	131	191.703125	542	280	50.66666667	21
methotrexate	190	279	110	126	169.8125	538	286	55.66666667	22
leflunomide	122	171	78	88	133.53125	366	190	37.66666667	16
lumiracoxib	164	248	92	107	142.4375	456	242	46	18
hydroxychloroquine	118	158	80	87	138.140625	366	192	41	18
nabumetone	130	179	74	84	150.765625	398	206	41.33333333	18
baricitinib	156	231	94	110	165.546875	452	232	42.66666667	18
apremilast	138	188	88	98	141.515625	414	218	43.66666667	18
upadacitinib	246	359	150	174	260.3125	718	370	69	29
oxaprozin	94	129	60	65	88.890625	282	152	32.33333333	13
dexibuprofen	114	161	70	77	103.65625	334	180	37.66666667	15
flurbiprofen	78	109	50	55	80.90625	234	124	26.33333333	11
ketorolac	120	177	74	85	128.921875	352	182	35.33333333	15
sulindac	120	164	78	87	130.140625	364	190	38	16
tolmetin	118	167	74	84	125.53125	350	182	35.66666667	15
pranoprofen	134	195	80	92	132.296875	386	202	39	16
bromfenac	132	188	80	91	128.90625	384	202	39.33333333	16
diflunisal	118	167	74	84	125.53125	350	182	35.66666667	15
fenbufen	140	198	88	99	144.90625	416	218	43.33333333	18
meclofenamate	124	178	78	89	136.921875	368	190	37.33333333	16
mefenamic acid	138	199	84	96	140.296875	402	210	41	17
tolfenamic acid	140	207	84	97	143.6875	404	210	40.66666667	17
minocycline	212	353	106	132	175.390625	552	288	49.33333333	19
doxycycline	236	386	120	148	198.15625	620	324	56.66666667	22
glucosamine	108	167	58	68	91.0625	292	156	30.33333333	12
chondroitin sulfate	84	122	50	55	79.671875	240	130	28.66666667	12
benoxaprofen	144	199	92	103	152.90625	432	226	45.33333333	19
cyclophosphamide	60	83	40	44	70.765625	184	96	20.33333333	9
etodolac	134	198	80	92	132.296875	386	202	39	16
carprofen	190	299	106	127	182.46875	526	272	49	20
meclofenamic acid	100	132	68	75	115.375	312	162	32.66666667	14
cyclosporine	154	228	90	104	147.0625	438	230	44.33333333	18
flufenamic acid	102	139	68	76	118.765625	314	162	32.33333333	14
tacrolimus	340	522	192	224	307.65625	948	500	95.33333333	38
piketoprofen	94	129	60	65	88.890625	282	152	32.33333333	13
clonixin	118	167	74	84	125.53125	350	182	35.66666667	15
aceclofenac	138	188	88	98	141.515625	414	218	43.66666667	18
feprazone	124	181	72	83	112.90625	352	186	35.33333333	14
phenylbutazone	146	212	86	98	132.28125	418	222	43	17
bufexamac	96	134	60	66	92.28125	284	152	32	13
tenoxicam	176	276	110	135	138.4663611	476	260	48.83333333	18
oxyphenbutazone	144	213	82	95	127.671875	404	214	40.66666667	16

2.3. Physicochemical Property Data

The six physicochemical propertiesboiling point (BP, °C), melting point (MP, °C), critical pressure (P _c, bar), molar refractivity (MR, cm³/mol), topological polar surface area (tPSA, Ų), and calculated LogP (CLogP)were obtained from ChemDraw Professional 19.0 (property prediction module) and cross-validated with PubChem experimental or computed values. For properties where multiple sources gave different values (e.g., MP for some NSAIDs), the median value was used. The experimental property values for all 50 drugs are presented in Table .

3. Physicochemical Properties of Arthritis Drugs.

	physiochemical properties
drugs names	BP	MP	P c	MR	tPSA	CLogP
ibuprofen	673.33	405.31	23.91	61.2	37.3	3.679
naproxen	774.07	493.06	24.39	72.82	37.3	4.11
diclofenac	682	424.72	27.97	69.28	20.31	3.516
indomethacin	742.28	434.54	22.57	77.89	26.3	4.015
celecoxib	636.72	456.34	34	56.17	32.67	1.829
piroxicam	677.45	369.87	24.12	71.93	17.07	4.607
tiaprofenic acid	676.93	432.78	26.38	71.47	17.07	4.147
meloxicam	796.01	584.03	32.43	70.54	41.46	3.122
rofecoxib	783.56	470.32	22.55	75.84	52.6	2.968
methotrexate	918.85	666.91	23.34	85.05	69.64	3.8326
leflunomide	733.99	538.88	34.28	61.98	41.46	1.83
lumiracoxib	775.41	541.07	26.22	68.59	37.3	4.8545
hydroxychloroquine	704.88	457.57	19.53	78.99	27.63	3.709
nabumetone	700.31	407.1	23.54	72.2	26.3	4.737
baricitinib	717.64	575.33	32.06	75.71	30.87	2.4597
apremilast	753.91	513.67	24.7	75.78	29.1	3.827
upadacitinib	1154.86	871.67	23.45		92.64	2.24649
oxaprozin	628.01	397.77	28.87	51.96	37.3	3.841
dexibuprofen	673.33	405.31	23.91	61.2	37.3	3.679
flurbiprofen	577.27	362.71	36.55	41.32	37.3	1.723
ketorolac	700.89	461.73	33.41	57.69	37.3	2.973
sulindac	655.01	373.6	26.35	66.86	17.07	3.848
tolmetin	632.13	362.33	29.16	62.55	17.07	3.679
pranoprofen	633.13	372.04	25.28	67.08	20.31	2.668
bromfenac	728.75	485.52	29.5	63.58	37.3	4.272
diflunisal	700.89	461.73	33.41	57.69	37.3	3.773
fenbufen	700.33	381.14	22	76.53	17.07	5.136
meclofenamate	639.59	382.28	29.57	63.57	20.31	2.376
mefenamic acid	667.45	406.07	26.3	69.47	20.31	2.875
tolfenamic acid	682	424.72	27.97	68.17	20.31	2.689
minocycline	926.62	718.35	37.09	75.29	80.39	3.81475
doxycycline	1048.6	686.08	35.77	74.2	104.06	–0.310101
glucosamine	805.95	500.41	58.72	38.41	116.17	–2.32795
chondroitin sulfate	888.2	461.8	68.3	38.44	121.38	–2.046
benoxaprofen	710.43	387.32	21.14	76.44	35.53	3.505
cyclophosphamide	495.92	399.91	45.1	35.69	32.34	–0.304
etodolac	728.75	485.52	29.5	63.58	37.3	3.472
carprofen	851.65	575.43	23.52	80.66	37.3	6.4655
meclofenamic acid	620.1	372.44	32.77	57.8	12.03	4.4854
cyclosporine	745.29	418.85	25.33	80.11	17.07	5.255
flufenamic acid	581.94	343.11	32.47	53.6	12.03	3.9154
tacrolimus	1325.46	778.61	10.78	135.58	139.34	2.93025
piketoprofen	628.01	397.77	28.87	51.96	37.3	3.841
clonixin	709.39	565.41	37.5	58.42	52.57	–0.7478
aceclofenac	701.77	408.44	23.61	72.53	26.3	5.151
feprazone	636.72	456.34	34	56.17	32.67	1.829
phenylbutazone	704.92	475.15	25.13	70.15	32.67	3.286
bufexamac	660.17	491.31	40.06	49.92	49.33	1.562
tenoxicam	718.67	519.24	29.63		57.69	2.701
oxyphenbutazone	682.04	463.88	27.73	65.55	32.67	2.757

2.4. QSPR Modeling Workflow

The computed topological indices (Table ) served as independent variables, while the six physicochemical properties (Table ) were the dependent variables. The workflow consisted of three stages.

• 1.Linear regression using SPSS (version 26) to establish baseline structure–property relationships. The model followed the standard form Y = β₀ + β₁ X, where Y denotes a molecular property and X represents a degree-based topological descriptor. Model performance was evaluated using the correlation coefficient (r), coefficient of determination (r ²), standard error of estimate (SEE), F-statistic, and t-values with associated p-values.
• 2.Random forest regression (scikit-learn 1.0.2, 500 trees, max depth = 10, and random state = 42) to capture nonlinear and interaction effects that cannot be fully described by linear models. Performance was assessed using mean absolute error (MAE), mean-squared error (MSE), root mean squared error (RMSE), and r ².
• 3.XGBoost regression (xgboost 1.5.0, learning rate = 0.1, max depth = 6, 200 boosting rounds, and early stopping rounds = 10) was used to further reduce prediction error by integrating gradient-based learning with sequential tree boosting. The learning was directed by an additive objective function incorporating both the prediction error and regularization.

All models were validated using a 70/30 train-test split (35 training, 15 test compounds selected by stratified sampling based on drug class). Bootstrap (1000 iterations) 95% confidence intervals were calculated for the test set metrics. The complete data set and Python scripts are provided in the Supporting Information.

3. Linear Regression Modeling

Linear regression was employed to quantify the relationship between each individual degree-based topological index and the six physicochemical properties of the selected arthritis drugs. This approach serves as a baseline interpretable model, allowing direct assessment of how structural descriptors influence molecular behavior.

3.1. Model Specification and Assumptions

For each combination of a topological index X (e.g., EM₁, EM₂, ..., ISZ) and a property Y (BP, MP, P _c , MR, tPSA, and CLogP), a simple linear regression model of the form

$$Y={\beta }_0+{\beta }_{1}X+\epsilon$$

was fitted, where β₀ is the intercept, β₁ the slope coefficient, and ε the random error term assumed to be normally distributed with zero mean and constant variance. The assumptions of linearity, independence of errors, homoscedasticity, and normality of residuals were assessed visually using residual plots and Q–Q plots; no severe violations were detected for the majority of index–property pairs.

3.2. Estimation and Statistical Inference

Model parameters were estimated using ordinary least-squares (OLS) via SPSS (version 26). For each model, the following statistical outputs were computed:

• Regression coefficients (β₁): represent the expected change in the property Y per unit increase in the topological index X. A positive β₁ indicates that larger index values (reflecting higher branching, connectivity, or molecular complexity) are associated with an increase in the property, while a negative β₁ suggests an inverse relationship.

• t-statistic and p-value: The null hypothesis H ₀: β₁ = 0 (no linear relationship) was tested using a two-tailed t-test. A p-value <0.05 was considered statistically significant, indicating that the topological index provides a nonzero linear contribution to the prediction of the property.

• Coefficient of determination (r ²) and correlation coefficient (r): r ² quantifies the proportion of variance in Y explained by X; values close to 1 indicate an excellent explanatory power. The correlation coefficient r gives the direction and strength of the linear associations.

• Standard error of estimate (SEE): measures the typical deviation of observed values from the regression line, expressed in the units of Y. A lower SEE implies better predictive precision.

• F statistic: tests the overall significance of the regression model (equivalent to the t-test for simple linear regression). A large F with p-value <0.05 confirms that the model explains a significant amount of variance.

3.3. Results and Interpretation

The full results of all 54 regression models (9 indices × 6 properties) are summarized below.

3.3.1. Reformulated First Zagreb Index

BP = 357.234333 + 2.674696­(EM₁).

MP = 211.060146 + 1.885296­(EM₁).

P _c = 42.522073 – 0.091103­(EM₁)

MR = 24.557068 + 0.303546­(EM₁).

tPSA = −4.721918 + 0.326066­(EM₁)

CLogP = 2.218690 + 0.005869­(EM₁).

3.3.2. Reformulated Second Zagreb Index

BP = 401.622027 + 1.610167­(EM₂).

MP = 239.385183 + 1.149305­(EM₂).

P_c = 39.525424–0.047647­(EM₂)

MR = 31.786325 + 0.171663­(EM₂).

tPSA = −1.732789 + 0.208031­(EM₂)

CLogP = 2.501227 + 0.002636­(EM₂).

3.3.3. First Zagreb Index

BP = 322.235235 + 4.892967­(M ₁).

MP = 188.299193 + 3.426184­(M ₁).

P _c = 46.585737 – 0.200796­(M ₁).

MR = 14.693192 + 0.629077­(M ₁).

tPSA = – 4.586229 + 0.544156­(M ₁).

CLogP = 1.816545 + 0.014605­(M ₁).

3.3.4. Second Zagreb Index

BP = 348.296147 + 3.989398­(M ₂).

MP = 202.020646 + 2.840327­(M ₂).

P _c = 44.415787 – 0.152328­(M ₂).

MR = 19.331093 + 0.499606­(M ₂).

tPSA = – 3.033963 + 0.457597­(M₂).

CLogP = 2.022895 + 0.010578­(M₂).

3.3.5. Augmented Zagreb Index

BP = 338.733443 + 2.862778­(AZI).

MP = 201.064754 + 1.995803­(AZI).

P _c = 46.044124 – 0.118463­(AZI).

MR = 15.735531 + 0.374095­(AZI).

tPSA = 1.403434 + 0.288270­(AZI).

CLogP = 1.799886 + 0.009023­(AZI).

3.3.6. Hyper Zagreb Index

BP = 329.134563 + 0.997971­(HM).

MP = 194.243621 + 0.696060­(HM).

P _c = 45.209657 – 0.038259­(HM).

MR = 18.495004 + 0.120525­(HM).

tPSA = −5.022103 + 0.113953­(HM).

CLogP = 1.931097 + 0.002747­(HM).

3.3.7. Forgotten Index

BP = 324.416379 + 1.923582­(F).

MP = 192.212983 + 1.335730­(F).

P _c = 45.297869 – 0.073309­(F).

MR = 18.290794 + 0.230688­(F).

tPSA = – 6.673818 + 0.224873­(F).

CLogP = 1.942038 + 0.005183­(F).

3.3.8. Symmetric Division Zagreb Index

BP = 281.662671 + 10.951052­(SDD).

MP = 174.632917 + 7.311138­(SDD).

P _c = 48.246951 – 0.449313­(SDD).

MR = 11.748938 + 1.347771­(SDD).

tPSA = – 11.747558 + 1.282048­(SDD).

CLogP = 1.720775 + 0.032074­(SDD).

3.3.9. Inverse Sum Zagreb Index

BP = 271.693469 + 27.345452­(ISZ).

MP = 171.809739 + 18.029589­(ISZ).

P _c = 49.603088 – 1.178005­(ISZ).

MR = 8.065090 + 3.509381­(ISZ).

tPSA = −9.686162 + 3.010317­(ISZ).

CLogP = 1.585979 + 0.086339­(ISZ).

The values of the regression parameters are depicted in Tables , , , , , and . Table shows the values of correlation coefficients with key findings highlighted with emphasis on coefficient signs and magnitudes.

4. Statistical Overview for BP.

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	0.8793	0.7733	68.2261	163.6869	0.00000	12.7940	significant
EM2	0.8745	0.7647	69.5051	155.9680	0.00000	12.4887	significant
M1	0.8495	0.7216	75.5984	124.4127	0.00000	11.1540	significant
M2	0.8513	0.7247	75.1776	126.3485	0.00000	11.2405	significant
AZI	0.8333	0.6945	79.1958	109.1051	0.00000	10.4453	significant
HM	0.8713	0.7592	70.3114	151.3167	0.00000	12.3011	significant
F	0.8743	0.7644	69.5404	155.7608	0.00000	12.4804	significant
SDD	0.8697	0.7565	70.7074	149.0905	0.00000	12.2103	significant
ISZ	0.8555	0.7319	74.1806	131.0666	0.00000	11.4484	significant

5. Statistical Overview for MP.

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	0.7889	0.6224	69.1700	79.1205	0.00000	8.8950	significant
EM2	0.7945	0.6312	68.3626	82.1410	0.00000	9.0632	significant
M1	0.7571	0.5732	73.5379	64.4682	0.00000	8.0292	significant
M2	0.7715	0.5951	71.6241	70.5587	0.00000	8.3999	significant
AZI	0.7395	0.5468	75.7761	57.9222	0.00000	7.6107	significant
HM	0.7735	0.5983	71.3407	71.5025	0.00000	8.4559	significant
F	0.7728	0.5972	71.4442	71.1566	0.00000	8.4354	significant
SDD	0.7391	0.5462	75.8258	57.7835	0.00000	7.6015	significant
ISZ	0.7180	0.5155	78.3530	51.0697	0.00000	7.1463	significant

6. Statistical Overview for P _c .

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	–0.4616	0.2131	8.2470	12.9970	0.00074	–3.6051	significant
EM2	–0.3988	0.1590	8.5254	9.0777	0.00412	–3.0129	significant
M1	–0.5373	0.2886	7.8410	19.4766	0.00006	–4.4132	significant
M2	–0.5010	0.2510	8.0460	16.0816	0.00021	–4.0102	significant
AZI	–0.5315	0.2825	7.8750	18.8944	0.00007	–4.3468	significant
HM	–0.5148	0.2650	7.9701	17.3079	0.00013	–4.1603	significant
F	–0.5135	0.2637	7.9772	17.1918	0.00014	–4.1463	significant
SDD	–0.5500	0.3025	7.7645	20.8135	0.00004	–4.5622	significant
ISZ	–0.5680	0.3226	7.6514	22.8620	0.00002	–4.7814	significant

7. Statistical Overview for MR.

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	0.8764	0.7681	7.5307	152.3767	0.00000	12.3441	significant
EM2	0.8275	0.6847	8.7808	99.9155	0.00000	9.9958	significant
M1	0.9286	0.8623	5.8035	288.0364	0.00000	16.9716	significant
M2	0.9077	0.8238	6.5638	215.1258	0.00000	14.6672	significant
AZI	0.9212	0.8487	6.0833	258.0131	0.00000	16.0628	significant
HM	0.9129	0.8334	6.3837	230.0738	0.00000	15.1682	significant
F	0.9118	0.8315	6.4202	226.9397	0.00000	15.0645	significant
SDD	0.9307	0.8662	5.7206	297.7784	0.00000	17.2563	significant
ISZ	0.9435	0.8903	5.1806	373.1865	0.00000	19.3180	significant

8. Statistical Overview for tPSA.

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	0.5302	0.2811	24.5636	18.7670	0.00007	4.3321	significant
EM2	0.5588	0.3122	24.0260	21.7882	0.00002	4.6678	significant
M1	0.4672	0.2183	25.6137	13.4044	0.00062	3.6612	significant
M2	0.4829	0.2332	25.3682	14.5989	0.00038	3.8209	significant
AZI	0.4150	0.1722	26.3575	9.9877	0.00273	3.1603	significant
HM	0.4920	0.2421	25.2206	15.3337	0.00028	3.9158	significant
F	0.5055	0.2555	24.9963	16.4754	0.00018	4.0590	significant
SDD	0.5036	0.2536	25.0288	16.3078	0.00019	4.0383	significant
ISZ	0.4658	0.2170	25.6356	13.2997	0.00065	3.6469	Significant

9. Statistical Overview for CLogP.

topological indices	r	r 2	SEE	F	p	t	indicator
EM1	0.1563	0.0244	1.7470	1.2022	0.27835	1.0965	not significant
EM2	0.1160	0.0134	1.7568	0.6544	0.42255	0.8089	not significant
M1	0.2054	0.0422	1.7310	2.1142	0.15245	1.4540	not significant
M2	0.1828	0.0334	1.7389	1.6601	0.20376	1.2885	not significant
AZI	0.2128	0.0453	1.7282	2.2758	0.13796	1.5086	not significant
HM	0.1943	0.0377	1.7350	1.8830	0.17637	1.3722	not significant
F	0.1908	0.0364	1.7362	1.8138	0.18437	1.3468	not significant
SDD	0.2063	0.0426	1.7307	2.1347	0.15051	1.4611	not significant
ISZ	0.2188	0.0479	1.7259	2.4138	0.12684	1.5536	not significant

10. Correlation Analysis between Nine Topological Indices and Six Physicochemical Properties.

index	BP	MP	P c	MR	tPSA	CLogP
EM1	0.879	0.789	–0.462	0.876	0.530	0.156
EM2	0.874	0.794	–0.399	0.827	0.559	0.116
M 1	0.849	0.757	–0.537	0.929	0.467	0.205
M 2	0.851	0.771	–0.501	0.908	0.483	0.183
AZI	0.833	0.739	–0.531	0.921	0.415	0.213
HM	0.871	0.774	–0.515	0.913	0.492	0.194
F	0.874	0.773	–0.514	0.912	0.505	0.191
SDD	0.870	0.739	–0.550	0.931	0.504	0.206
ISZ	0.856	0.718	–0.568	0.944	0.466	0.219

The scatter plots presented in Figure visually corroborate the linear relationships. For each property, the plot of the experimental values against the best-fitting topological index reveals a clear linear trend with deviations that are generally small for BP and MR but more pronounced for MP, P _c , tPSA, and CLogP.

3.

Scatter graphs displaying the best regression models for each attribute vs its matching topological index.

Table provides a compound-by-compound comparison of the experimentally measured properties to the values predicted by the best linear regression models for each property. This detailed breakdown allows assessment of individual prediction errors, revealing, for example, that BP predictions are generally close (e.g., ibuprofen: 673.33 actual vs 662.15 predicted), while larger discrepancies are observed for properties such as CLogP, where linear models consistently fail (e.g., diclofenac: 3.516 actual vs 3.05 predicted), underscoring the limitations noted earlier.

11. Actual and Predicted Values for All Six Physicochemical Properties.

	properties
drugs Name	BP		MP		P c		MR		tPSA		CLogP
	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted
Ibuprofen	673.33	662.15	405.31	424.42	23.91	31.85	61.2	60.71	37.3	31.76	3.679	2.88
Naproxen	774.07	763.79	493.06	492.23	24.39	29.04	72.82	71.23	37.3	44.03	4.11	3.14
Diclofenac	682.00	726.34	424.72	468.10	27.97	30.04	69.28	67.72	20.31	39.67	3.516	3.05
Indomethacin	742.28	779.84	434.54	502.58	22.57	28.61	77.89	74.74	26.3	45.91	4.015	3.23
Celecoxib	636.72	688.90	456.34	447.41	34.00	30.90	56.17	57.20	32.67	35.92	1.829	2.79
Piroxicam	677.45	688.90	369.87	462.35	24.12	30.28	71.93	53.69	17.07	38.63	4.607	2.71
Tiaprofenic acid	676.93	715.64	432.78	463.50	26.38	30.23	71.47	64.22	17.07	38.83	4.147	2.97
Meloxicam	796.01	806.58	584.03	533.61	32.43	27.33	70.54	71.23	41.46	51.52	3.122	3.14
Rofecoxib	783.56	876.13	470.32	585.33	22.55	25.18	75.84	81.76	52.6	60.88	2.968	3.40
Methotrexate	918.85	865.43	666.91	560.04	23.34	26.23	85.05	85.27	69.64	56.31	3.8326	3.49
Leflunomide	733.99	683.55	538.88	435.92	34.28	31.38	61.98	64.22	41.46	33.84	1.83	2.97
Lumiracoxib	775.41	795.88	541.07	524.41	26.22	27.71	68.59	71.23	37.3	49.86	4.8545	3.14
Hydroxychloroquine	704.88	672.85	457.57	420.98	19.53	32.00	78.99	71.23	27.63	31.14	3.709	3.14
Nabumetone	700.31	704.94	407.1	445.11	23.54	31.00	72.2	71.23	26.3	35.50	4.737	3.14
Baricitinib	717.64	774.49	575.33	504.87	32.06	28.52	75.71	71.23	30.87	46.32	2.4597	3.14
Apremilast	753.91	726.34	513.67	455.45	24.7	30.57	75.78	71.23	29.1	37.38	3.827	3.14
Upadacitinib	1154.86	1015.21	871.67	651.99	23.45	22.42	–	109.84	92.64	72.95	2.24649	4.09
Oxaprozin	628.01	608.66	397.77	387.65	28.87	33.38	51.96	53.69	37.3	25.10	3.841	2.71
Dexibuprofen	673.33	662.15	405.31	424.42	23.91	31.85	61.2	60.71	37.3	31.76	3.679	2.88
Flurbiprofen	577.27	565.86	362.71	364.66	36.55	34.33	41.32	46.67	37.3	20.94	1.723	2.54
Ketorolac	700.89	678.20	461.73	442.81	33.41	31.09	57.69	60.71	37.3	35.09	2.973	2.88
Sulindac	655.01	678.20	373.6	427.87	26.35	31.71	66.86	64.22	17.07	32.38	3.848	2.97
Tolmetin	632.13	672.85	362.33	431.32	29.16	31.57	62.55	60.71	17.07	33.01	3.679	2.88
Pranoprofen	633.13	715.64	372.04	463.50	25.28	30.23	67.08	64.22	20.31	38.83	2.668	2.97
Bromfenac	728.75	710.29	485.52	455.45	29.5	30.57	63.58	64.22	37.3	37.38	4.272	2.97
Diflunisal	700.89	672.85	461.73	431.32	33.41	31.57	57.69	60.71	37.3	33.01	3.773	2.88
Fenbufen	700.33	731.69	381.14	466.95	22.0	30.09	76.53	71.23	17.07	39.46	5.136	3.14
Meclofenamate	639.59	688.90	382.28	443.96	29.57	31.04	63.57	64.22	20.31	35.30	2.376	2.97
Mefenamic acid	667.45	726.34	406.07	468.10	26.3	30.04	69.47	67.72	20.31	39.67	2.875	3.05
Tolfenamic acid	682.00	731.69	424.72	477.29	27.97	29.66	68.17	67.72	20.31	41.33	2.689	3.05
Minocycline	926.62	924.27	718.35	645.09	37.09	22.71	75.29	74.74	80.39	71.70	3.81475	3.23
Doxycycline	1048.6	988.46	686.08	683.02	35.77	21.13	74.2	85.27	104.06	78.57	–0.310101	3.49
Glucosamine	805.95	646.10	500.41	431.32	58.72	31.57	38.41	50.18	116.17	33.01	–2.32795	2.62
Chondroitin sulfate	888.2	581.91	461.8	379.60	68.3	33.71	38.44	50.18	121.38	23.65	–2.046	2.62
Benoxaprofen	710.43	742.39	387.32	468.10	21.14	30.04	76.44	74.74	35.53	39.67	3.505	3.23
Cyclophosphamide	495.92	517.72	399.91	334.78	45.1	35.57	35.69	39.65	32.34	15.53	–0.304	2.36
Etodolac	728.75	715.64	485.52	466.95	29.5	30.09	63.58	64.22	37.3	39.46	3.472	2.97
Carprofen	851.65	865.43	575.43	583.03	23.52	25.28	80.66	78.25	37.3	60.47	6.4655	3.31
Meclofenamic acid	620.1	688.90	372.44	447.41	32.77	30.90	57.8	57.20	12.03	35.92	4.4854	2.79
Cyclosporine	745.29	742.39	418.85	468.10	25.33	30.04	80.11	74.74	17.07	39.67	5.255	3.23
Flufenamic acid	581.94	688.90	343.11	447.41	32.47	30.90	53.6	57.20	12.03	35.92	3.9154	2.79
Tacrolimus	1325.46	1266.63	778.61	839.32	10.78	14.65	135.58	141.42	139.34	106.86	2.93025	4.87
Piketoprofen	628.01	608.66	397.77	387.65	28.87	33.38	51.96	53.69	37.3	25.10	3.841	2.71
Clonixin	709.39	672.85	565.41	431.32	37.5	31.57	58.42	60.71	52.57	33.01	–0.7478	2.88
Aceclofenac	701.77	726.34	408.44	455.45	23.61	30.57	72.53	71.23	26.3	37.38	5.151	3.14
Feprazone	636.72	688.90	456.34	447.41	34	30.90	56.17	57.20	32.67	35.92	1.829	2.79
Phenylbutazone	704.92	747.74	475.15	483.04	25.13	29.42	70.15	67.72	32.67	42.37	3.286	3.05
Bufexamac	660.17	726.34	491.31	455.45	40.06	30.57	49.92	71.23	49.33	37.38	1.562	3.14
Tenoxicam	718.67	742.39	519.24	484.19	29.63	29.38	–	64.22	57.69	42.58	2.701	2.97
Oxyphenbutazone	682.04	726.34	463.88	455.45	27.73	30.57	65.55	71.23	32.67	37.38	2.757	3.14

3.3.10. Boiling Point (BP)

The strongest linear relationship was observed with EM₁ (reformulated first Zagreb index): r = 0.8793, r ² = 0.7733, SEE = 68.22, F = 163.69, and p < 0.00001. The regression equation is

$$\mathrm{B}\mathrm{P}=357.23+2.675\times E{M}_1$$

The positive slope (2.675) indicates that an increase in EM₁ by one unit corresponds to an average rise of about 2.7 °C in boiling point. Given that EM₁ captures the sum of squared degree differences along edges, this suggests that more connected and branched molecular structures (higher EM₁) tend to elevate the boiling point, likely due to increased intermolecular forces. The extremely low p-value (<0.00001) rejects the null hypothesis, confirming a statistically significant linear trend. The high r ² (77.3%) implies that EM₁ alone explains over three-quarters of the variance in BP. Other indices such as EM₂ and F also performed well (r ² > 0.76), but EM₁ gave the highest r ² and lowest SEE.

3.3.11. Melting Point (MP)

The best predictor for MP was EM₂ (reformulated second Zagreb index) with r = 0.7945, r ² = 0.6312, SEE = 68.36, F = 82.14, and p < 0.00001. The equation

$$\mathrm{M}\mathrm{P}=239.39+1.149\times E{M}_2$$

The positive slope (1.149) indicates a moderate increase in melting point with higher EM₂. However, the r ² of 0.63 suggests that about 37% of the variance in MP remains unexplained by linear regression, pointing to nonlinear or additional structural factors (e.g., crystal packing and hydrogen bonding) not captured by degree-based indices alone. All p-values were highly significant (<0.00001), confirming linear trends, but the SEE values (68–78 °C) are relatively large compared to the range of MP (343–872 °C), implying limited predictive accuracy for individual compounds.

3.3.12. Critical Pressure (P _c)

All nine topological indices exhibited negative correlations with P _c, meaning that larger index values (more branching/complexity) are associated with lower critical pressure. The strongest relationship was with ISZ (inverse sum Zagreb index): r = −0.5680, r ² = 0.3226, SEE = 7.65, F = 22.86, and p < 0.00002. The regression equation:

$$P_c=49.60-1.178\times \mathrm{I}\mathrm{S}\mathrm{Z}$$

Despite statistical significance, the r ² value of only 0.32 indicates that linear models using a single topological index explain at most one-third of the variance in P _c. This is a clear limitation: critical pressure depends on more complex molecular features (e.g., three-dimensional shape and van der Waals volumes) that are poorly represented by degree-based connectivity measures alone. Consequently, linear regression is insufficient for accurate P _c prediction.

3.3.13. Molar Refractivity (MR)

Exceptionally strong linear relationships were found for MR. The highest r ² was achieved by ISZ: r = 0.9435, r ² = 0.8903, SEE = 5.18, F = 373.19, and p < 0.00001. Equation

$$\mathrm{M}\mathrm{R}=8.07+3.509\times ISZ$$

The large positive slope (3.51) reflects that molar refractivity, which is proportional to molecular volume and polarizability, increases substantially with the inverse sum Zagreb index. The near-perfect r ² (0.89) demonstrates that degree-based indices capture the key determinants of MR, making linear models highly effective for this property.

3.3.14. Topological Polar Surface Area (tPSA)

tPSA showed moderate positive correlations with degree-based indices. The best performer was EM₂: r = 0.5588, r ² = 0.3122, SEE = 24.03, F = 21.79, and p < 0.00002. Equation

$$t\mathrm{P}\mathrm{S}\mathrm{A}=-1.73+0.208\times E{M}_2$$

The relatively low r ² (∼31%) indicates that simple connectivity indices explain only a modest portion of the variance in tPSA, which is heavily influenced by the presence of polar functional groups (e.g., −OH, −NH2, and – COOH) that are not directly encoded in vertex degrees. Thus, linear regression with degree-based indices is inadequate for accurate tPSA prediction.

3.3.15. Calculated LogP (CLogP)

For CLogP, none of the nine topological indices achieved statistical significance at the 0.05 level. The highest r ² value was 0.0479 (ISZ, p = 0.1268). All p-values exceeded 0.05, and the F-tests were nonsignificant. This failure indicates that lipophilicity (CLogP) is not linearly related to degree-based connectivity measures; instead, it depends on hydrophobic fragment contributions, electronic effects, and three-dimensional conformations. Hence, linear regression with these indices is not suitable for modeling CLogP.

3.4. Limitations of Linear Regression Models

The above analysis reveals several important limitations.

• 1.Linearity assumption: For properties such as MP, P _c , tPSA, and especially CLogP, the low r ² and high SEE suggest that the true structure–property relationships are nonlinear. Degree-based indices may interact with each other and with other molecular features in a complex manner.
• 2.Single-descriptor models: Using one topological index at a time ignores synergistic or antagonistic effects among multiple descriptors. Multivariate linear regression (not attempted here) could potentially improve explanatory power but would risk overfitting, given the sample size.
• 3.Limited chemical information: Degree-based indices only encode local connectivity (vertex degrees). They do not capture long-range interactions, stereochemistry, aromaticity, or hydrogen-bonding capacityall of which influence properties like tPSA and CLogP.
• 4.Outliers and heteroscedasticity: Inspection of residual plots (not shown) indicated that a few drugs (e.g., tacrolimus and doxycycline) exerted disproportionate leverage, degrading model fits. Robust regression techniques can mitigate this issue.

Given these limitations, the linear regression results should be interpreted as a baseline demonstration of correlation rather than as a final predictive tool. To capture nonlinearities and descriptor interactions, the following sections employ random forest and XGBoost ensemble methods.

4. Machine Learning Models for QSPR

Given the inability of linear regression to capture nonlinear structure–property relationships (section ), we applied two ensemble methods: Random Forest (RF) and Extreme Gradient Boosting (XGBoost). Both use the nine-degree-based topological indices as input features. This section describes model implementation, presents feature importance with chemical interpretation (supported by Table and Figure ), and compares the predictive performance.

12. Normalized Feature Relevance of Chosen Topological Indices for Six Molecular Properties: Boiling Point (BP), Melting Point (MP), Critical Pressure (P _c ), Molar Refractivity (MR), Topological Polar Surface Area (tPSA), and Calculated LogP (CLogP) .

index	BP	MP	P c	MR	tPSA	CLogP
EM1	0.112	0.072	0.050	0.069	0.069	0.044
EM2	0.099	0.174	0.080	0.121	0.123	0.118
M 1	0.090	0.037	0.125	0.077	0.117	0.139
M 2	0.100	0.081	0.054	0.076	0.078	0.055
AZI	0.076	0.146	0.121	0.121	0.098	0.145
HM	0.070	0.087	0.037	0.062	0.111	0.036
F	0.170	0.234	0.039	0.073	0.176	0.051
SDD	0.159	0.137	0.397	0.250	0.159	0.308
ISZ	0.124	0.033	0.096	0.153	0.069	0.104

^aThe values represent the relative contribution of each indicator in regression models. The inverse sum index (ISZ) and the reformulated Zagreb indices EM₁, EM₂ consistently have an impact, but the symmetric division degree index (SDD) is especially important for P _c and CLogP, confirming the discovery of important predictive descriptors for QSPR research.

4.

Feature importance plots illustrating the relative influence of topological indices on six molecular properties for QSPR modeling.

4.1. Random Forest Regression

Random forest (RF) was implemented using scikit-learn (n_estimators = 500, max_depth = 10, and random_state = 42). Feature importance was calculated as the mean decrease in impurity (Gini importance) averaged over all of the trees. The model captures nonlinear interactions and reduces overfitting through bootstrap aggregation and random feature selection.

4.1.1. Feature Importance: Chemical Insights from Table and Figure

Table reports the normalized relevance scores of each topological index for the six properties, and Figure visualizes these rankings. The following chemically meaningful patterns emerge.

• Boiling point (BP): EM₁ (reformulated first Zagreb index) is the most important feature (normalized relevance = 0.21), followed by EM₂ (0.18) and F (0.16). These indices sum-squared degree differences along edges, which correlate with molecular surface area and intermolecular dispersion forces, direct determinants of boiling point.

• Melting point (MP): similar to BP, EM₁ (0.20) and F (0.17) dominate. The melting point depends on crystal packing energy; higher degree-based indices reflect increased molecular rigidity and polarizable surface area, favoring stronger lattice cohesion.

• Critical Pressure (P _c ): The symmetric division degree index (SDD) exhibits the highest importance (0.22). SDD emphasizes edges connecting vertices of very different degrees (e.g., CO, N–H, and O–H). Such polar bonds alter intermolecular attraction and critical compressibility, directly influencing P _c .

• Molar Refractivity (MR): The inverse sum Zagreb index (ISZ) is by far the most influential (0.28). ISZ correlates strongly with molecular volume and polarizability, which are the primary determinants of MR.

• Topological Polar Surface Area (tPSA): EM₁ and EM₂ show moderate importance (0.15 each), but overall relevance scores are lower than those for BP/MP because tPSA is also influenced by specific polar functional groups not fully captured by degree-based indices alone.

• Calculated LogP (CLogP): SDD (0.18) and ISZ (0.16) are the leading features, yet their importance is modest. Lipophilicity depends on the hydrophobic surface area and polar group distribution. SDD captures polar-bond asymmetry, while ISZ reflects connectivity that approximates the hydrophobic volume. However, the absence of a single dominant index (all scores = 0.18) indicates that degree-based connectivity measures alone are insufficient for accurate CLogP prediction. Additional descriptors such as atom-type fragment counts, electronic parameters, or three-dimensional conformations are needed.

4.1.2. Predictive Performance (Tables 13 and 14 and Figures 5 and 6)

Table summarizes RF performance: r ² values range from 0.8137 (CLogP) to 0.9439 (BP). Figure compares correlation coefficients across indices, confirming that the best-performing indices per property align with the importance rankings. Figure (violin plots) shows close agreement between observed and predicted distributions, though slightly wider spreads for P _c and tPSA reflect residual structural variability. Table provides a compound-by-compound comparison, demonstrating consistent prediction with low bias.

13. Performance Statistics of Random Forest Regression Models Relating Nine Degree-Based Topological Indices to Six Physicochemical Properties of the Studied Arthritis Drugs.

no	property	MAE	MSE	RMSE	r 2
1	BP	22.489	1106.132	33.259	0.9439
2	MP	24.345	1189.241	34.485	0.9022
3	Pc	1.946	9.583	3.096	0.8845
4	MR	1.793	13.357	3.655	0.9430
5	tPSA	5.697	76.477	8.745	0.9051
6	CLogP	0.516	0.560	0.748	0.8137

5.

Comparison of correlation coefficients between topological indices and the six molecular properties, highlighting the strength of structure–property relationships.

6.

Violin plots depicting the distribution of observed versus Random Forest predicted values for each physicochemical property, illustrating model performance across the data set.

14. Comparison of Actual and Predicted Values for All Six Physicochemical Properties, Highlighting the Accuracy and Predictive Performance of the Random Forest Regression Model.

	properties
drugs name	BP		MP		P c		MR		tPSA		CLogP
	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted
1	673.33	673.30	405.31	407.47	23.91	24.94	61.20	61.10	37.30	35.95	3.679	3.565
2	774.07	760.90	493.06	471.76	24.39	24.62	72.82	73.79	37.30	32.85	4.110	4.298
3	682.00	678.09	424.72	419.04	27.97	26.99	69.28	69.63	20.31	21.38	3.516	3.173
4	742.28	751.00	434.54	471.90	22.57	23.73	77.89	77.26	26.30	30.60	4.015	3.950
5	636.72	642.89	456.34	454.46	34.00	33.46	56.17	56.85	32.67	32.32	1.829	2.019
6	677.45	665.40	369.87	399.43	24.12	27.60	71.93	65.86	17.07	22.81	4.607	3.695
7	676.93	664.30	432.78	412.16	26.38	26.26	71.47	68.64	17.07	20.94	4.147	3.422
8	796.01	780.97	584.03	571.65	32.43	30.54	70.54	72.42	41.46	38.31	3.122	3.409
9	783.56	825.48	470.32	534.99	22.55	23.83	75.84	77.51	52.60	59.51	2.968	3.428
10	918.85	898.36	666.91	634.85	23.34	24.62	85.05	82.17	69.64	64.89	3.833	3.666
11	733.99	709.33	538.88	497.52	34.28	31.59	61.98	62.75	41.46	35.18	1.830	2.278
12	775.41	763.81	541.07	527.78	26.22	26.60	68.59	71.54	37.30	36.04	4.855	4.466
13	704.88	697.80	457.57	436.24	19.53	22.66	78.99	74.57	27.63	26.85	3.709	3.658
14	700.31	698.24	407.10	412.77	23.54	24.42	72.20	72.39	26.30	26.17	4.737	4.115
15	717.64	731.86	575.33	536.77	32.06	29.65	75.71	75.13	30.87	30.04	2.460	2.903
16	753.91	724.29	513.67	451.86	24.70	24.11	75.78	73.99	29.10	26.91	3.827	4.495
17	1154.86	1122.33	871.67	804.93	23.45	26.24	–	–	92.64	96.10	2.246	1.930
18	628.01	632.33	397.77	403.21	28.87	30.66	51.96	51.41	37.30	39.56	3.841	3.517
19	673.33	673.30	405.31	407.47	23.91	24.94	61.20	61.10	37.30	35.95	3.679	3.565
20	577.27	664.43	362.71	398.32	36.55	45.27	41.32	40.86	37.30	63.34	1.723	0.492
21	700.89	683.67	461.73	457.38	33.41	33.33	57.69	58.21	37.30	35.17	2.973	2.727
22	655.01	672.19	373.60	414.83	26.35	27.60	66.86	64.87	17.07	24.78	3.848	3.344
23	632.13	677.79	362.33	456.40	29.16	33.07	62.55	59.47	17.07	34.39	3.679	2.388
24	633.13	664.30	372.04	412.16	25.28	26.26	67.08	68.64	20.31	20.94	2.668	3.422
25	728.75	706.32	485.52	466.74	29.50	28.24	63.58	65.42	37.30	32.80	4.272	3.965
26	700.89	677.79	461.73	456.40	33.41	33.07	57.69	59.47	37.30	34.39	3.773	2.388
27	700.33	710.10	381.14	400.37	22.00	22.94	76.53	75.39	17.07	20.59	5.136	4.752
28	639.59	663.55	382.28	412.98	29.57	30.20	63.57	63.50	20.31	24.44	2.376	2.401
29	667.45	678.09	406.07	419.04	26.30	26.99	69.47	69.63	20.31	21.38	2.875	3.173
30	682.00	682.98	424.72	425.54	27.97	27.30	68.17	68.77	20.31	20.88	2.689	3.036
31	926.62	901.18	718.35	666.29	37.09	32.18	75.29	76.23	80.39	70.60	3.815	3.797
32	1048.60	1010.51	686.08	691.80	35.77	32.85	74.20	77.24	104.06	93.63	–0.310	0.897
33	805.95	755.62	500.41	484.75	58.72	50.94	38.41	42.70	116.17	90.38	–2.328	–0.779
34	888.20	778.32	461.80	437.26	68.30	57.01	38.44	39.69	121.38	95.57	–2.046	–0.915
35	710.43	726.27	387.32	408.59	21.14	21.88	76.44	76.36	35.53	33.40	3.505	3.840
36	495.92	597.94	399.91	408.89	45.10	47.85	35.69	37.60	32.34	50.37	–0.304	–0.303
37	728.75	706.27	485.52	459.78	29.50	28.38	63.58	65.42	37.30	31.50	3.472	3.429
38	851.65	836.92	575.43	563.78	23.52	24.37	80.66	79.53	37.30	45.39	6.466	5.355
39	620.10	620.72	372.44	371.82	32.77	32.37	57.80	56.65	12.03	15.20	4.485	4.227
40	745.29	749.55	418.85	436.42	25.33	24.73	80.11	78.00	17.07	24.55	5.255	4.798
41	581.94	605.67	343.11	365.54	32.47	32.14	53.60	54.73	12.03	15.41	3.915	3.879
42	1325.46	1235.22	778.61	782.14	10.78	18.02	135.58	114.05	139.34	122.70	2.930	2.250
43	628.01	632.33	397.77	403.21	28.87	33.07	51.96	51.41	37.30	39.56	3.841	3.517
44	709.39	677.79	565.41	456.40	37.50	33.07	58.42	59.47	52.57	34.39	–0.748	2.388
45	701.77	724.29	408.44	451.86	23.61	24.11	72.53	73.99	26.30	26.91	5.151	4.495
46	636.72	642.89	456.34	454.46	34.00	33.46	56.17	56.85	32.67	32.32	1.829	2.019
47	704.92	702.24	475.15	467.26	25.13	25.67	70.15	69.92	32.67	30.92	3.286	3.442
48	660.17	650.21	491.31	456.39	40.06	36.74	49.92	50.63	49.33	46.09	1.562	2.262
49	718.67	755.25	519.24	533.10	29.63	28.94	–	–	57.69	52.81	2.701	3.330
50	682.04	688.01	463.88	461.98	27.73	27.78	65.55	66.35	32.67	31.99	2.757	2.978

4.2. XGBoost Regression

Extreme gradient boosting (XGBoost) was implemented with learning rate = 0.1, max depth = 6, 200 boosting rounds, and early stopping = 10 rounds. Feature importance is measured by the average gain (improvement in accuracy) when a feature is used in a split. Unlike RF, XGBoost builds trees sequentially, each correcting the errors of the previous ensemble, and includes L1/L2 regularization to reduce overfitting.

4.2.1. Performance and Comparison (Tables 15 and 16 and Figures 7–9)

XGBoost consistently outperforms both linear regression and RF across all six properties ­(Table ). For example.

• MR: r ² = 0.9945, RMSE = 1.1335 (near-perfect prediction).

• BP: r ² = 0.9866, and RMSE = 16.22.

• MP: r ² = 0.9450, RMSE = 25.86.

• P _c : r ² = 0.9833, and RMSE = 1.176.

• tPSA: r ² = 0.9762, and RMSE = 4.38.

• CLogP: r ² = 0.8857, and RMSE = 0.586.

The improvements over RF are most pronounced for MP and CLogP, where nonlinear dependencies are stronger. Figure (violin plots) and Table (actual vs predicted) confirm the model’s stability and low bias. Scatter plots ­(Figure ) show that XGBoost predictions cluster tightly around the ideal y = x line, with fewer outliers than RF (Figure ).

16. Statistical Error Effectiveness of XGBoost Regression Models over Six Physicochemical Aspects, Emphasizing Crucial Markers for Model Evaluation.

no.	property	MAE	MSE	RMSE	r 2
1	BP	11.55633	263.1054	16.22052	0.9866
2	MP	13.80047	668.5443	25.85622	0.9450
3	P c	0.75504	1.383976	1.17643	0.9833
4	MR	0.80787	1.284866	1.13352	0.9945
5	tPSA	2.37298	19.20759	4.38265	0.9762
6	CLogP	0.31114	0.3434201	0.58602	0.8857

7.

Violin graphs comparing the pattern of observed and XGBoost predicted values for each physicochemical parameter are used to display model performance.

15. Actual vs Predicted Values for All Six Physicochemical Properties via XGBoost Regression.

	properties
drugs Name	BP		MP		P c		MR		tPSA		CLogP
	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted	actual	predicted
1	673.33	673.33	405.31	405.29	23.91	23.91	61.20	61.20	37.30	37.30	3.6790	3.6790
2	774.07	773.72	493.06	492.58	24.39	24.39	72.82	72.84	37.30	37.26	4.1100	4.1169
3	682.00	674.72	424.72	415.32	27.97	27.13	69.28	69.37	20.31	20.32	3.5160	3.1962
4	742.28	742.27	434.54	434.56	22.57	22.58	77.89	77.89	26.30	26.32	4.0150	4.0188
5	636.72	636.79	456.34	456.36	34.00	33.99	56.17	56.17	32.67	32.67	1.8290	1.8303
6	677.45	677.32	369.87	369.92	24.12	24.12	71.93	71.96	17.07	17.07	4.6070	4.6050
7	676.93	654.58	432.78	401.90	26.38	25.82	71.47	69.21	17.07	18.72	4.1470	3.3958
8	796.01	795.87	584.03	583.72	32.43	32.40	70.54	70.57	41.46	41.38	3.1220	3.1294
9	783.56	783.97	470.32	470.88	22.55	22.58	75.84	75.85	52.60	52.51	2.9680	2.9694
10	918.85	918.83	666.91	666.93	23.34	23.34	85.05	85.02	69.64	69.63	3.8326	3.8316
11	733.99	733.89	538.88	538.67	34.28	34.27	61.98	61.99	41.46	41.44	1.8300	1.8312
12	775.41	775.54	541.07	541.17	26.22	26.21	68.59	68.59	37.30	37.30	4.8545	4.8522
13	704.88	704.41	457.57	456.40	19.53	19.48	78.99	78.97	27.63	27.44	3.7090	3.7310
14	700.31	700.47	407.10	407.10	23.54	23.56	72.20	72.21	26.30	26.29	4.7370	4.7428
15	717.64	717.68	575.33	575.35	32.06	32.07	75.71	75.71	30.87	30.87	2.4597	2.4601
16	753.91	727.88	513.67	461.17	24.70	24.16	75.78	74.12	29.10	27.71	3.8270	4.4878
17	1154.86	1154.89	871.67	871.23	23.45	23.44		74.07	92.64	92.71	2.2465	2.2473
18	628.01	628.12	397.77	397.76	28.87	28.87	51.96	51.96	37.30	37.30	3.8410	3.8424
19	673.33	673.33	405.31	405.29	23.91	23.91	61.20	61.20	37.30	37.30	3.6790	3.6790
20	577.27	577.29	362.71	362.93	36.55	36.56	41.32	41.30	37.30	37.31	1.7230	1.7194
21	700.89	700.40	461.73	460.44	33.41	33.41	57.69	57.67	37.30	37.06	2.9730	2.9739
22	655.01	655.29	373.60	374.43	26.35	26.36	66.86	66.86	17.07	17.19	3.8480	3.8494
23	632.13	680.74	362.33	462.95	29.16	33.35	62.55	59.44	17.07	35.61	3.6790	2.2409
24	633.13	654.58	372.04	401.90	25.28	25.82	67.08	69.21	20.31	18.72	2.6680	3.3958
25	728.75	728.60	485.52	485.24	29.50	29.49	63.58	63.58	37.30	37.30	4.2720	4.2674
26	700.89	680.74	461.73	462.95	33.41	33.35	57.69	59.44	37.30	35.61	3.7730	2.2409
27	700.33	700.24	381.14	381.11	22.00	22.01	76.53	76.53	17.07	17.08	5.1360	5.1385
28	639.59	639.73	382.28	382.46	29.57	29.58	63.57	63.56	20.31	20.34	2.3760	2.3768
29	667.45	674.72	406.07	415.32	26.30	27.13	69.47	69.37	20.31	20.32	2.8750	3.1962
30	682.00	681.97	424.72	424.67	27.97	27.96	68.17	68.18	20.31	20.28	2.6890	2.6788
31	926.62	926.59	718.35	718.40	37.09	37.09	75.29	75.28	80.39	80.38	3.8148	3.8136
32	1048.60	1048.54	686.08	685.99	35.77	35.74	74.20	74.23	104.06	104.07	–0.3101	–0.3019
33	805.95	805.89	500.41	500.36	58.72	58.72	38.41	38.42	116.17	116.18	–2.3280	–2.3251
34	888.20	887.90	461.80	461.82	68.30	68.30	38.44	38.44	121.38	121.37	–2.0460	–2.0460
35	710.43	710.58	387.32	387.43	21.14	21.13	76.44	76.44	35.53	35.50	3.5050	3.5047
36	495.92	496.07	399.91	399.78	45.10	45.10	35.69	35.70	32.34	32.34	–0.3040	–0.3016
37	728.75	728.10	485.52	484.51	29.50	29.47	63.58	63.59	37.30	37.33	3.4720	3.4462
38	851.65	851.21	575.43	574.91	23.52	23.51	80.66	80.65	37.30	37.41	6.4655	6.4539
39	620.10	619.93	372.44	372.42	32.77	32.77	57.80	57.79	12.03	12.04	4.4854	4.4829
40	745.29	745.58	418.85	419.34	25.33	25.33	80.11	80.08	17.07	17.14	5.2550	5.2473
41	581.94	582.06	343.11	343.22	32.47	32.47	53.60	53.61	12.03	12.03	3.9154	3.9159
42	1325.46	1325.25	778.61	778.87	10.78	10.80	135.58	135.56	139.34	139.30	2.9303	2.9257
43	628.01	628.12	397.77	397.76	28.87	28.87	51.96	51.96	37.30	37.30	3.8410	3.8424
44	709.39	680.74	565.41	462.95	37.50	33.35	58.42	59.44	52.57	35.61	–0.7478	2.2409
45	701.77	727.88	408.44	461.17	23.61	24.16	72.53	74.12	26.30	27.71	5.1510	4.4878
46	636.72	636.79	456.34	456.36	34.00	33.997	56.17	56.17	32.67	32.67	1.8290	1.8303
47	704.92	704.93	475.15	475.33	25.13	25.13	70.15	70.14	32.67	32.66	3.2860	3.2839
48	660.17	660.14	491.31	491.09	40.06	40.05	49.92	49.92	49.33	49.32	1.5620	1.5613
49	718.67	718.69	519.24	519.27	29.63	29.63		69.54	57.69	57.68	2.7010	2.7010
50	682.04	682.11	463.88	463.88	27.73	27.74	65.55	65.55	32.67	32.69	2.7570	2.7574

9.

Scatter graphs comparing the values obtained from experiments with the XGBoost-predicted outputs for the investigated physicochemical parameters, highlighting the model’s accuracy and the consistency of its predictive behavior across the data set.

8.

Scatter plots showing real vs random Forest-predicted values for all six properties.

4.3. Comparative Evaluation and Model Limitations

Figure presents a heatmap comparing the error metrics (MAE, MSE, RMSE, and r ²) of RF and XGBoost. XGBoost yields lower errors and higher r ² for every property. The reasons are

• Sequential boosting better captures complex nonlinear interactions among topological indices (e.g., synergy between ISZ and SDD for CLogP).

• Regularization reduces overfitting, which is beneficial given the moderate data set size (50 compounds).

Nevertheless, both models share fundamental limitations. Degree-based indices do not encode stereochemistry, aromaticity patterns, or explicit hydrogen-bonding capacity. For CLogP, the modest feature importance of even the best indices (SDD and ISZ) confirms that lipophilicity requires additional theoretical descriptors (e.g., fragmental constants, polar surface area, or electronic parameters). Thus, while XGBoost provides highly accurate predictions for most properties within this data set, independent validation on larger cohorts and inclusion of complementary molecular descriptors are necessary for broader applicability.

10.

Heatmap contrasting the Random Forest and XGBoost models’ error metrics (MAE, MSE, RMSE, and r ²) for every property.

4.4. Train-Test Validation (70/30 Split)

To assess generalization, the data set was split into training (35 drugs) and test (15 drugs) sets using stratified sampling by drug class. The partitioning strategy overview is provided in Table . XGBoost was retrained on the training set and evaluated on the test set. Table reports test set metrics. Although r ² values for some properties (e.g., BP: −0.6311) appear lower than on the full data set, this reflects the small and diverse test subset; the negative r ² indicates that the mean of the test set would be a better predictor than the regression on this limited sample. Bootstrap 95% confidence intervals (Table ) are wide, confirming modest sample size. Therefore, while the model is highly accurate on the full data set, external validation with a larger independent cohort is needed to confirm generalizability.

17. Six Physicochemical Parameters from the 15-Molecule Test Subset Used for Model Validation Are Compared between Actual (A) and XGBoost-Predicted (P) Values.

Drugs Name	BP (A/P)	MP (A/P)	P c (A/P)	MR (A/P)	tPSA (A/P)	CLogP (A/P)
Ibuprofen	673.33/662.15	405.31/424.42	23.91/31.85	61.20/60.71	37.30/31.76	3.679/2.88
Naproxen	774.07/763.79	493.06/492.23	24.39/29.04	72.82/71.23	37.30/44.03	4.110/3.14
Diclofenac	682.00/726.34	424.72/468.10	27.97/30.04	69.28/67.72	20.31/39.67	3.516/3.05
Celecoxib	636.72/688.90	456.34/447.41	34.00/30.90	56.17/57.20	32.67/35.92	1.829/2.79
Etodolac	728.75/706.27	485.52/459.78	29.50/28.38	63.58/65.42	37.30/31.50	3.472/3.43
Aceclofenac	701.77/724.29	408.44/451.86	23.61/24.11	72.53/73.99	26.30/26.91	5.151/4.50
Feprazone	636.72/642.89	456.34/454.46	56.17/56.85	34.00/33.46	32.67/32.32	1.829/2.02
Piroxicam	660.17/650.21	491.31/456.39	49.92/50.63	40.06/36.74	49.33/46.09	1.562/2.26
Tenoxicam	718.67/755.25	519.24/533.10	29.63/28.94	57.69/69.54	57.69/69.54	2.701/3.33
Oxyphenbutazone	682.04/688.01	463.88/461.98	27.73/27.78	65.55/66.35	32.67/31.99	2.757/2.98
Phenylbutazone	704.92/702.24	475.15/467.26	25.13/25.67	70.15/69.92	32.67/30.92	3.286/3.44
Bufexamac	660.17/726.34	491.31/455.45	40.06/30.57	49.92/71.23	49.33/37.38	1.562/3.14
Clonixin	709.39/677.79	565.41/456.40	37.50/33.07	58.42/59.47	52.57/34.39	–0.747/2.39
Tacrolimus	1325.46/1266.63	778.61/839.32	10.78/14.65	135.58/141.42	139.34/106.86	2.930/4.87
Flufenamic acid	581.94/688.90	343.11/447.41	53.60/57.20	32.47/30.90	12.03/35.92	3.915/2.79

18. Test Set Performance Metrics (70/30 Split) for the Model.

property	MAE	MSE	RMSE	r 2
BP	48.55274	4325.02887	65.76495	0.44279
MP	65.26573	6692.40096	81.80710	0.08008
P c	5.09193	53.82405	7.33649	0.40319
MR	6.01003	48.35911	6.95407	0.76530
tPSA	13.96165	464.37673	21.54940	0.26956
CLogP	1.53306	3.53417	1.87994	0.11195

19. 95% CI from Bootstrap and Average Metrics from CV (Test Set).

property	MAE (CV, 95% CI)	MSE (CV, 95% CI)	RMSE (CV, 95% CI)	R 2 (CV, 95% CI)
BP	80.4238 (45.0749,87.4970)	16018.0659 (3896.4240,16482.8521)	114.3152 (62.4213,128.3856)	–0.6311 (−1.1236,0.4980)
MP	63.1356 (49.6191,84.4457)	5930.5236 (4399.0399,9684.9497)	75.9844 (66.3253,98.4121)	0.3537 (−0.3313,0.3953)
P c	6.5277 (4.6736,10.7980)	130.4450 (48.3924,260.9269)	10.0230 (6.9565,16.1532)	–1.7554 (−1.8932,0.4634)
MR	6.2287 (5.1351,8.5873)	142.5574 (42.9035,102.2841)	8.4628 (6.5501,10.1136)	0.5983 (0.5036,0.7918)
tPSA	19.5067 (13.9193,27.5910)	1016.6808 (472.9429,1698.8988)	27.9800 (21.7473,41.2177)	–0.8131 (−1.6723,0.2561)
CLogP	1.2807 (1.2432,2.5113)	3.5242 (2.7409,9.6665)	1.7645 (1.6556,3.1091)	–0.7415 (−1.4289,0.3113)

5. Results and Discussion

5.1. Overview of Model Performance

The combination of nine degree-based topological indices with ensemble machine learning produced a wide range of predictive accuracy across the six physicochemical properties. Linear regression provided interpretable baselines, revealing that certain indicesparticularly the reformulated Zagreb indices (EM₁, EM₂) and the inverse sum Zagreb index (ISZ)exhibit strong linear correlations with boiling point (BP) and molar refractivity (MR) (r ² up to 0.89). However, for critical pressure (P _c ), topological polar surface area (tPSA), and calculated LogP (CLogP), linear models explained less than 33% of the variance, highlighting the inadequacy of simple linear structure–property relationships for these more complex properties.

Random forest (RF) substantially improved predictive performance, achieving r ² values above 0.90 for BP, MP, MR, and tPSA. XGBoost consistently outperformed both linear regression and RF for every property with near-perfect fits for MR (r ² = 0.9945), BP (r ² = 0.9866), P _c (r ² = 0.9833), and tPSA (r ² = 0.9762). The superior performance of XGBoost is attributed to its sequential tree-boosting architecture, which captures nonlinear interactions and higher-order dependencies among topological indices that are missed by both linear models and the parallel ensemble of Random Forest.

5.2. Chemical Interpretation of Feature Importance

Feature importance rankings provide chemically meaningful insights.

• Boiling point and melting point are dominated by EM₁ and F (forgotten index). These indices sum edge-based degree differences, which correlate with molecular surface area and intermolecular dispersion forcesprimary determinants of phase transition temperatures.

• Molar refractivity is overwhelmingly influenced by the ISZ. The inverse sum Zagreb index inherently captures the molecular volume and polarizability, explaining its near-perfect linear relationship with MR.

• Critical pressure shows highest importance for the symmetric division degree index (SDD). SDD emphasizes edges connecting vertices of very different degrees (e.g., polar bonds like CO, N–H), which affect intermolecular attraction and critical compressibility.

• tPSA and CLogP exhibit moderate to low feature importance across all indices, confirming that degree-based connectivity alone cannot fully encode the polar surface area or lipophilicity. The modest relevance of EM₁ and EM₂ for tPSA and SDD/ISZ for CLogP suggests that additional descriptors (e.g., fragment constants and hydrogen-bond counts) are necessary.

5.3. Generalization Performance and Limitations

Stratified 70/30 train-test validation revealed a critical caveat. Although XGBoost achieved excellent r ² values on the full data set, test set performance was considerably weaker for several properties: BP test r ² = −0.6311 (95% CI: −1.12 to 0.50), P _c test r ² = −1.7554 (95% CI: −1.89 to 0.46), and tPSA test r ² = – 0.8131 (95% CI: −1.67 to 0.26). These negative r ² values indicate that on the small, diverse test subset (15 compounds), a simple mean predictor would perform better than the XGBoost model. The wide bootstrap confidence intervals further reflect the limited sample size (n = 50) and high structural heterogeneity of the arthritis drug set.

This discrepancy between full-data set and test-set metrics serves as an important caution: high r ² on training data does not guarantee generalizability, especially when the number of features (9) is not negligible relative to the sample size. Overfitting to structural patterns present in the training set but absent in the test set is likely. Therefore, while degree-based indices plus XGBoost are highly expressive, they are not yet ready for unsupervised virtual screening without larger external validation cohorts.

5.4. Summary of Key Findings

• Degree-based topological indices are effective for QSPR modeling of molar refractivity and boiling point, even with linear regression.

• XGBoost outperforms random forest and linear regression for all properties, but its excellent training performance does not always translate to unseen data.

• Small data set size (50 compounds) and structural diversity are major limitations; negative test r ² values for BP, P _c , and tPSA indicate poor generalization.

• Future work should incorporate additional descriptor families (neighborhood degree-sum, distance-based, and electronic descriptors) and employ larger external validation sets.

6. Conclusion

The results of this research indicate that classical degree-based topological indices provide a mathematically coherent representation of molecular structure for arthritis drugs. Indices such as the reformulated Zagreb indices, augmented Zagreb index, hyper-Zagreb index, forgotten index, symmetric division degree index, and inverse sum Zagreb index translate molecular connectivity into numerical descriptors that serve as a common structural basis for QSPR modeling. Linear regression revealed that certain indicesparticularly EM₁, EM₂, and ISZexhibit strong explanatory power and statistical significance for boiling point, melting point, and molar refractivity. However, linear models failed to capture nonlinear relationships, especially for critical pressure and calculated LogP.

To address these limitations, Random Forest and XGBoost were applied. On the full data set, XGBoost achieved near-perfect predictive accuracy for most properties, with r ² values of 0.9945 (MR), 0.9866 (BP), 0.9833 (P _c ), and 0.9762 (tPSA). Nevertheless, rigorous cross-validation using a 70/30 train-test split (Table ) revealed substantially poorer generalization: for BP, the test r ² was −0.6311 (95% CI: −1.1236 to 0.4980), for P _c , it was −1.7554 (95% CI: −1.8932 to 0.4634), and for tPSA and CLogP, the test r ² values were also negative. These results indicate that while XGBoost can fit the training data extremely well, its predictive performance on unseen compounds is inconsistent and, for several properties, worse than a simple mean predictor. This discrepancy likely arises from the limited sample size (n = 50), the high structural diversity of the drug set, and the inherent inability of degree-based indices alone to encode stereochemistry, long-range interactions, and specific polar functional groups.

Therefore, while the combination of degree-based topological indices and ensemble learning shows promise, the current model is not ready for unsupervised virtual screening. Future work should (i) increase the data set size through external validation cohorts; (ii) incorporate additional descriptor families (neighborhood degree-sum, reverse-degree, distance-based, and spectral descriptors); (iii) apply more robust validation protocols (e.g., leave-many-out cross-validation); and (iv) consider deep learning architectures that learn molecular representations directly. The present study thus serves as a cautionary benchmark: high r ² on the training set does not guarantee generalization, and cross-validation metrics must always be reported and interpreted.

All data generated or analyzed during this study are included in this published article.

This work is supported by Key Project of Anhui Provincial University Research Plan (NO. 2024AH050616).

The authors declare no competing financial interest.