Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 215 https://internationalpubls.com QSPR Analysis of Eye Conjunctivitis Drops Using Regression Model via Degree Based Topological Indices Greeta .T*1, Jayalalitha.G2 1 Research Scholar, Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies(VISTAS), Pallavaram,Chennai-600 117,Tamilnadu, India; greetamaths20@gmail.com 2Professor,Department of Mathematics, Vels Institute of Science, Technology and Advanced Studies(VISTAS),Pallavaram, Chennai-600 117, Tamilnadu , India; g.jayalalithamaths.sbs@velsuniv.ac.in * Correspondence: greetamaths20@gmail.com Article History: Received: 30-07-2024 Revised: 09-09-2024 Accepted: 18-09-2024 Abstract: Introduction: Topological indices (TIs) are numerical values derived from the structural graph of a molecule, vital in cheminformatics for predicting various properties. In chemical graph theory, graphs represent molecular structures where vertices correspond to atoms and edges to bonds. Degree-based indices, reflecting vertex connectivity, are used to predict properties such as boiling point and molar refractivity. This study analyses eight eye drops—ciprofloxacin, gentamicin, moxifloxacin, norfloxacin, tobramycin, levofloxacin, gatifloxacin, and chloramphenicol—using these indices. Objectives: To analyse the degree-based topological indices of eight eye drops and predict their physicochemical properties, including boiling point, enthalpy, mass, flash point, molar refractivity, and volume, using a linear regression model. Methods: Degree-based topological indices were computed through edge partitioning. A Quantitative Structure-Property Relationship (QSPR) model was developed with linear regression to predict the properties of the eye drops. Prediction accuracy was evaluated by comparing predicted values to actual values and analysing the associated errors. Results: The study found a strong correlation between the topological indices and the physicochemical properties of the eye drops. The linear regression model accurately predicted these properties, with minimal error, demonstrating the effectiveness of using TIs in this context. Conclusions This research highlights the effectiveness of topological indices and linear regression models in predicting the properties of eye drops. These tools offer valuable insights for drug design and development, paving the way for more effective treatments for eye conditions. Keywords: chemical graph theory; cheminformatics; correlation; degree based topological indices; QSPR analysis; regression 1. Introduction A mathematical field known as "graph theory" studies the characteristics and uses of graphs, which are structures made up of vertices (nodes) and edges (connections) that represent pairwise relationships between entities [1]. Graphs are used extensively to represent and analyse networks and their properties in a variety of fields, including computer science, biology, the social sciences, and logistics. Beyond these fields, graph theory plays a crucial role in the pharmaceutical and chemical industries. In drug discovery and chemistry, graphs are utilised to model molecular structures, with atoms depicted as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 216 https://internationalpubls.com vertices and chemical bonds as edges. This graphical representation enables researchers to analyse molecular properties, forecast the behaviour of chemical compounds, and identify promising drug candidates. Molecular graphs are connected, simple, and have no parallel edges. The degree of a vertex, denoted as dv, is the number of edges connected to that particular vertex. If an edge joins vertices v1 and v2, then the degree of v1 is denoted as dv1, and the degree of v2 is denoted as dv2 [2]. In molecular graph theory, hydrogen atoms are often considered to have no direct impact on the graph's topological features. Therefore, a hydrogen-suppressed graph is used, which excludes hydrogen atoms and focuses on the main framework of the molecule. In chemical graph theory, a topological index (TI) is a number or parameter that is obtained from a molecule's structural graph[3]. It contains information about the molecule's structure, connectivity, symmetry, branching, and cyclicity, among other topological and geometric features. When predicting the physical, chemical, and biological properties of molecules based on their structural traits, topological indices play a crucial role in quantitative structure-activity relationship (QSAR) investigations. They are useful resources for a number of chemical and biochemistry-related subjects, including drug design and materials research. Cheminformatics, also referred to as chemical informatics, is a scientific discipline merging mathematics and chemistry. Within cheminformatics, one area of focus involves QSPR (Quantitative Structure-Property Relationship) and QSAR (Quantitative Structure-Activity Relationship) investigations. These methods utilise mathematical models to establish relationships between the structural characteristics of chemical compounds and their associated physical, chemical, and biological properties or behaviours. For humans to survive, their eyes are essential. They can become pink or red and irritate when they are impacted by illnesses such as conjunctivitis. Daily tasks may be disrupted, and normal functioning may be prevented. Seeking prompt treatment is crucial to reducing these symptoms and safeguarding eye health. Pink eye, also known as conjunctivitis, can result in a number of discomforts and issues, such as redness, irritation, itching, excessive tearing, discharge (which may be watery or contain pus), and swelling of the eyelids. One can observe the differences between a normal eye and an eye affected by conjunctivitis. When the eyes are affected, they become red and disrupt our healthy, normal routine. Figures 1 and 2 illustrate a healthy eye and an eye affected by conjunctivitis, respectively. Figures 1– 2 are sourced from Google Images. Figure 1. Healthy Eye Figure 2. Affected Eye Various eye drops are recommended by doctors to treat conjunctivitis. Depending on the patient's health and ocular conditions, different eye drops are prescribed. Eight eye drops—Ciprofloxacin, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 217 https://internationalpubls.com Gentamicin, Moxifloxacin, Norfloxacin, Tobramycin, Levofloxacin, Gatifloxacin, and Chloramphenicol—were selected for examination in this study, as they are commonly used for treating bacterial infections like conjunctivitis. Both ciprofloxacin(C17H18FN3O3) and moxifloxacin(C21H24FN3O4) are fluoroquinolone antibiotics that are effective against a wide range of Gram-positive and Gram-negative bacteria. They function by blocking the bacterial enzymes DNA gyrase and topoisomerase IV, which are necessary for DNA replication and cell division. Norfloxacin(C16H18FN3O3), another fluoroquinolone, shares a similar mechanism but is typically used against Gram-negative bacteria and some Gram-positive strains. Gentamicin(C21H43N5O7) and tobramycin(C18H37N5O9) are aminoglycoside antibiotics that disrupt bacterial protein synthesis by binding to the 30S ribosomal subunit, targeting a range of Gram-negative and some Gram-positive bacteria[4]. Levofloxacin(C18H20FN3O4), also a fluoroquinolone, is effective against both Gram-positive and Gram-negative bacteria through its inhibition of DNA synthesis. Gatifloxacin(C19H22FN3O4), another member of the fluoroquinolone class, similarly interferes with bacterial DNA replication. Chloramphenicol(C11H12Cl2N2O5), distinct from the others, inhibits protein synthesis by binding to the 50S ribosomal subunit, and it has a broad spectrum of activity against many Gram-positive and Gram-negative bacteria, including some resistant strains. By focusing on different bacterial pathways to reduce symptoms and aid in recovery, each of these drugs is essential in the management of bacterial eye infections. Figure 3 displays the chemical structures of these eight eye drops. The source of these structures is ChemBook. Several recent studies have explored the use of topological indices and quantitative structure-property relationship (QSPR) modelling in analysing novel drugs for various medical conditions [5–17]. These studies collectively demonstrate the utility of topological indices and QSPR modelling in pharmaceutical research for predicting molecular properties and understanding drug behavior. Inspired by these studies, there is a growing interest in applying topological indices and QSPR modelling to analyse the properties of eye drops. In this work, physicochemical properties are predicted using a variety of degree-based topological indices. Of the several indices that are now available, seven degree-based topological indices have been selected for examination. The Forgotten Index (F(G), the First Zagreb Index (M1(G), the Second Zagreb Index (M2(G), the Randic Index R(G), the Augmented Zagreb Index AZI(G), the Harmonic Index H(G), and the Atom-Bond Connectivity Index ABC(G) are among them. By examining and measuring the topological features of the molecules being studied, these indices offer important new perspectives on their physicochemical qualities. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 218 https://internationalpubls.com a) Ciprofloxacin b) Gentamicin c) Moxifloxacin d) Norfloxacin e) Tobramycin f) Levofloxacin g)Gatifloxacin h)Chloramphenicol Figure 3. Molecular Structures of Eye drops 2. Objectives The main objective of this paper is as follows ❖ Convert the molecular structures of selected eye drops into molecular graphs. ❖ Determine the edge partition for eight selected eye drop formulations. ❖ Calculate seven degree-based topological indices from the molecular graphs. ❖ Identify the physicochemical properties of the active ingredients in the drugs. ❖ Analyze the correlation between topological indices and physicochemical properties of the drugs. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 219 https://internationalpubls.com ❖ Conduct Quantitative Structure–Property Relationship (QSPR) analysis using a regression model, comparing predicted values with actual experimental data. 2.1 Topological Indices Preliminaries If an edge between u and v is considered to be uv, then the additive degree based Topological indices (TI) have the generic form [18]. TI=TI (A) =∑ 𝑭(𝒅𝒖, 𝒅𝒗)𝒖𝒗∈𝑬(𝑨) The Forgotten index was originally introduced by Furtula and Gutman [19] in 2015 and is defined as F(G)=∑ [(𝑑𝑣1)2 + (𝑑𝑣2)2]𝑣1𝑣2∈𝐸(𝐺) (1) The first and second Zagreb indices are part of the earliest topological indices, introduced by Gutman and Polansky[20] in 1986. They are used to describe the pi-electron properties of molecules. M1(G)=∑ (𝑑𝑣1 + 𝑑𝑣2)𝑣1𝑣2∈𝐸(𝐺) (2) M2(G)=∑ (𝑑𝑣1 × 𝑑𝑣2)𝑣1𝑣2∈𝐸(𝐺) (3) The Randić index, created by Milan Randić[21], assesses molecular graph connectivity for QSAR predictions of biological activity based on molecular structure. R(G)=∑ 1 √𝑑𝑣1 ×𝑑𝑣2 𝑣1𝑣2∈𝐸(𝐺) (4) The Augmented Zagreb index[22], an extension of the Zagreb indices, incorporates the squares of vertex degrees to enhance molecular structure characterization in chemoinformatics. AZI(G)=∑ ( 𝑑𝑣1𝑑𝑣2 𝑑𝑣1+𝑑𝑣2−2 )3 𝑣1𝑣2∈𝐸(𝐺) (5) In 2012, Zhong[23] introduced the harmonic index, a measure capturing molecular graph connectivity, while in 1998, Estrada et al. defined the ABC index[24], describing atom-bond connectivity in molecules, both offering insights into molecular structure and properties. H(G)=∑ 2 (𝑑𝑣1+𝑑𝑣2)𝑣1𝑣2∈𝐸(𝐺) (6) ABC(G)=∑ √ 𝑑𝑣1+𝑑𝑣2−2 𝑑𝑣1×𝑑𝑣2 𝑣1𝑣2∈𝐸(𝐺) (7) The molecular structures of drugs are converted into molecular graphs. The calculation of topological indices for all eight drugs was conducted using vertex connectivity and edge partitioning, as outlined in Table 1, which includes the number of edges (|E(G)|), number of vertices (|V(G)|), and edge partition (Ei,j) where E i,j={e=𝑣1𝑣2 ∈ 𝐸(𝐺)/𝑑𝑣1 = i and 𝑑𝑣2=j} for each drop. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 220 https://internationalpubls.com Table 1. Edge Partition of Eight Drugs Drugs |V(G) E1,2 E1,3 E1,4 E2,2 E2,3 E3,3 E4,2 E4,3 |E(G)| Ciprofloxacin 24 ---- 4 ---- 5 10 8 ---- ---- 27 Gentamicin 33 2 6 2 2 13 8 1 1 35 Moxifloxacin 29 1 4 ---- 4 13 11 ---- ---- 33 Norfloxacin 23 1 4 ---- 4 9 7 ---- ---- 25 Tobramycin 32 2 8 ---- ---- 14 10 ---- ---- 34 Levofloxacin 26 ---- 6 ---- 3 10 10 ---- ---- 29 Gatifloxacin 27 1 5 ---- 3 11 10 ---- ---- 30 Chloramphenical 20 1 6 ---- 2 7 4 ---- ---- 20 3. Methods 3.1 Computation of Topological indices Let G1 be the molecular graph of Ciprofloxacin as shown in Figure 4. G1 consists of 24 vertices and 27 edges. The edge partition of G1 is |E1,3|=4, |E2,2|=5, |E2,3|=10 and |E3,3|=8. The topological indices(TI) is calculated by using definitions(1-7) and the results are : Figure 4. Moleculer Graph(G1) of Ciprofloxacin 1. F(G1)=4(12+32)+5(22+22)+10(22+32)+8(32+32)=354 2. M1(G1)=4(1+3)+5(2+2)+10(2+3)+8(3+3)=134 3 .M2(G1)=4(1×3)+5(2×2)+10(2×3)+8(3×3)=164 4. R(G1)=4[ 1 √1×3 ]+5[ 1 √2×2 ]+10[ 1 √2×3 ]+8[ 1 √3×3 ] =11.5586 5. AZI(G1)=4[ 1×3 1+3−2 ] 3 +5[ 2×2 2+2−2 ] 3 +10[ 2×3 2+3−2 ] 3 +8[ 3×3 3+3−2 ] 3 =224.625 6. H(G1)=4[ 2 1+3 ]+5[ 2 2+2 ]+10[ 2 2+3 ]+8[ 2 3+3 ] =11.1667 7. ABC(G1)=4√ 1+3−2 1×3 +5√ 2+2−2 2×2 +10√ 2+3−2 2×3 + 8√ 3+3−2 3×3 =19.2059 Using the same method as previously stated and Definitions 1 through 7, topological indices of various medications can be computed. A calculation of the TIs for every medicine is provided in Table 2. The computed TIs for different medications are shown graphically in Figure 5. Origin Software was used to draw the graph. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 221 https://internationalpubls.com Table 2. Topological indices for Eight Drugs Drugs F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 354 134 164 11.5586 224.625 11.1667 19.2059 Gentamicin 478 174 208 15.4945 273.9397 14.6190 25.3378 Moxifloxacin 444 166 207 13.9904 282.7969 13.5334 23.3272 Norfloxacin 320 122 147 11.0240 205.2344 10.6 17.8322 Tobramycin 452 168 202 15.0818 268.9063 14.2666 24.5124 Levofloxacin 394 146 180 12.3799 238.1563 11.8333 20.7581 Gatifloxacin 402 150 185 12.9179 189.1563 12.4 21.3558 Chloramphenical 244 94 106 9.3622 139.0625 8.8 14.6367 Figure 5. Topological indices of eight drugs 3.2 Physical Properties of Drugs Various physical properties can be considered. For this study, seven properties were selected for eight drugs: Boiling Point (BP), Enthalpy (En), Average Mass (AM), Flash Point (FP), Molar Refractivity (Re), Polarizability (Po), and Average Mass (AM). These properties were chosen to analyze the QSPR model. The boiling point of a substance, such as a drug, refers to the temperature at which it changes from a liquid to a gas phase under normal atmospheric pressure. This property is crucial for drug formulation and stability, impacting storage conditions and manufacturing processes. Enthalpy in drugs refers to the total heat content of the drug system under constant pressure. It is crucial in pharmaceuticals for understanding drug stability, reaction kinetics, and formulation processes. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 222 https://internationalpubls.com Mass refers to the quantity of matter contained in a drug substance. It influences various pharmaceutical properties, including dosage formulation, stability, and manufacturing processes. Flash point is the temperature at which a drug substance emits vapors that can ignite when exposed to an open flame or spark.. Molar refractivity quantifies the ability of a drug molecule to refract light, providing insights into its molecular structure and intermolecular interactions. Polarizability measures the ability of a drug molecule to undergo distortion by an electric field, reflecting its susceptibility to electrostatic interactions Volume represents the amount of space occupied by a drug substance. Table 3 displays the physicochemical properties of eight drugs. Table 3. Physical Properties of Eight Drugs Drugs BP En AM FP Re Po MV Ciprofloxacin 581.8 91.5 331.341 305.6 83.3 33 226.8 Gentamicin 669.4 112.6 477.595 358.6 122.6 48.6 366.9 Moxifloxacin 636.4 98.8 401.431 338.7 101.8 40.4 285 Norfloxacin 555.8 88.1 319.331 289.9 80.7 32 237.4 Tobramycin 775.4 128.7 467.514 422.8 111.7 44.3 305.9 Levofloxacin 571.5 90.1 361.367 299.4 91.1 36.1 244 Gatifloxacin 607.8 95 375.394 321.4 94.6 37.5 270.8 Chloramphenicol 644.9 100 323.129 343.8 72.6 28.8 208.8 3.3 Regression Models A regression model is a statistical technique used to analyse the relationship between a dependent variable and one or more independent variables. Its goal is to predict the dependent variable's value based on the data from the independent variables. Here, linear regression model is used to predict drug properties and determine the dependent variable. This model forecasts the characteristics of drugs based on the data. R=𝛼 + 𝛽[𝑇𝐼] (8) In the regression model, 𝛼 and 𝛽 are constants, R symbolize the physical properties of the drug, while TI represents the topological indices. Linear regression is utilized to establish a model for the topological indices, with the topological index of the molecular structure of eight eye conjunctivitis drops serving as the independent variable. Meanwhile, the physical characteristics are treated as the dependent variable in this model. Statistical parameters and regression models for seven degree-based topological indices are calculated and presented in Tables 4 to Table 10. MS Excel was used to construct the linear regression models. If the p-value is 0.05 or less, it is deemed statistically significant; if it exceeds 0.05, it is not considered significant. A p-value significantly below 0.05 is often characterized as highly significant. Notations: α = intercept R2=Coefficient of determination Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 223 https://internationalpubls.com β=Coefficient Constant F=Fisher’s Statistics N=Number of drugs p=Significance Level r=Correlation Coefficient The coefficient of determination, R2 shows how well a model predicts an outcome, with values between 0 and 1. If the R2 value is 1, the model predicts the outcome perfectly. If the value is between 0 and 1, the model provides a partial prediction. If the value is 0, the model does not predict the data effectively.The standard error of the estimate measures how accurate these predictions are. Table 4. Regression Model of F(G) Regression Models N r R2 F p Significant BP=75.4753+0.4926[F(G)] 8 0.4475 0.2002 1.5023 0.7770 NOT significant En=64.9633+0.0923[F(G)] 8 0.5215 0.2719 2.2412 0.0364 Statistically Significant AM=112.1376+0.6995(F(G) 8 0.8688 0.7548 18.4669 0.1299 NOT significant FP=240.0489+0.2461[F(G)] 8 0.4476 0.2003 1.5030 0.0226 Statistically Significant Re=16.7958+0.2021[F(G)] 8 0.9396 0.8828 45.2003 0.2047 NOT significant Po=6.6565+0.0801[F(G)] 8 0.9394 0.8825 45.0606 0.2055 NOT significant MV=46.1968+0.5751[F(G)] 8 0.8735 0.7630 19.3187 0.4033 NOT significant Table 5. Regression Model of M1(G) Regression Model N r R2 F p Significant BP=462.0805+1.1667[M1(G)] 8 0.4465 0.1994 1.4945 0.0163 Statistically Significant En=62.6838+0.2629[M1(G)] 8 0.5162 0.2665 2.1800 0.053 NOT significant AM=95.6709+1.9859[M1(G)] 8 0.8576 0.7355 16.6863 0.2276 NOT significant FP=233.1561+0.7062[M1(G)] 8 0.4467 0.1995 1.4953 0.0330 Statistically Significant Re=11.9361+0.5744[M1(G)] 8 0.9287 0.8624 37.6148 0.4176 NOT significant Po=4.7272+0.2278[M1(G)] 8 0.9286 0.8622 37.5516 0.4186 NOT significant MV=33.6941+1.6257[M1(G)] 8 0.8585 0.7370 16.8164 0.5828 NOT significant Table 6. Regression Model of M2(G) Regression Models N r R2 F p Significant BP=498.7788+0.7525[M2(G)] 8 0.3753 0.1409 0.9839 0.0101 Statistically Significant En=70.6178+0.1714[M2(G)] 8 0.4388 0.1926 1.4308 0.0325 Statistically Significant AM=133.1748+1.4237[M2(G)] 8 0.8012 0.6419 10.7564 0.1354 NOT significant FP=255.3583+0.4556[M2(G)] 8 0.3755 0.1410 0.9848 0.0204 Statistically Significant Re=21.2615+0.4205[M2(G)] 8 0.8859 0.7849 21.8888 0.2320 NOT significant Po=8.4260+0.1668[M2(G)] 8 0.8858 0.7846 21.8607 0.2325 NOT significant MV=62.9169+1.1739[M2(G)] 8 0.8078 0.6526 11.2716 0.3509 NOT significant Table 7. Regression Model of R(G) Regression Models N r R2 F p Significant BP=377.0028+19.9096[R(G)] 8 0.5907 0.3489 3.2156 0.0387 Statistically Significant En=44.9280+4.3746[R(G)] 8 0.6660 0.4435 4.7825 0.1317 NOT significant AM=23.6648+28.1682[R(G)] 8 0.9429 0.8891 48.1190 0.6667 NOT significant Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 224 https://internationalpubls.com FP=181.6811+12.0495[R(G)] 8 0.5908 0.3490 3.2164 0.0804 NOT significant Re=-4.7653+7.8237[R(G)] 8 0.9804 0.9613 148.6628 0.5850 NOT significant Po=-1.9021+3.1030[R(G)] 8 0.9804 0.9613 149.0043 0.5823 NOT significant MV=-0.4268+22.6798[R(G)] 8 0.9284 0.8619 37.4477 0.6835 NOT significant Table 8. Regression Model of AZI(G) Regression Models N r R2 F p Significant BP=509.9414+0.5288[AZI(G)] 8 0.3684 0.1357 0.9421 0.0069 Statistically Significant En=72.4448+0.1236[AZI(G)] 8 0.4419 0.1953 1.4560 0.0227 Statistically Significant AM=167.9579+0.9405[AZI(G)] 8 0.7392 0.5464 7.2268 0.084 NOT significant FP=262.142+0.3200[AZI(G)] 8 0.3684 0.1357 0.9421 0.0141 Statistically Significant Re=31.8196+0.2766[AZI(G)] 8 0.8137 0.6620 11.7542 0.1404 NOT significant Po=12.6042+0.1097[AZI(G)] 8 0.8138 0.6623 11.7683 0.1407 NOT significant MV=95.42511+0.7587[AZI(G)] 8 0.7291 0.5317 6.8109 0.2074 NOT significant Table 9. Regression Model of H(G) Regression Model N r R2 F p Significant BP=389.7994+19.7966[H(G)] 8 0.5547 0.3077 2.6668 0.0398 Statistically Significant En=47.5768+4.3632 [H(G)] 8 0.6273 0.3936 3.8938 0.1306 NOT significant AM=28.4410+29.1052[H(G)] 8 0.9202 0.8467 33.1407 0.6633 NOT significant FP=189.4196+11.9817[H(G)] 8 0.5548 0.3078 2.6678 0.080 NOT significant Re=-4.5566+8.1759[ H(G)] 8 0.9676 0.9363 88.1766 0.6851 NOT significant Po=-1.8198+3.2428 [H(G)] 8 0.9677 0.9364 88.3350 0.6828 NOT significant MV=-8.3786+23.5821[H(G)] 8 0.917 0.8312 29.5385 0.7422 NOT significant Table 10. Regression Model of ABC(G) Regression Model N r R2 F p Significant BP=415.6167+10.2899[ABC(G)] 8 0.5251 0.2757 2.2844 0.0278 Statistically Significant En=52.8259+2.2890[ABC(G)] 8 0.5994 0.3593 3.3645 0.0921 NOT significant AM=52.5476+15.7920[ABC(G)] 8 0.9093 0.8268 28.6396 0.4319 NOT significant FP=205.0449+6.2279[ABC(G)] 8 0.5252 0.2758 2.2852 0.0567 NOT significant Re=1.5790+4.4666[ABC(G)] 8 0.9627 0.9269 76.0472 0.8889 NOT significant Po=0.6183+1.7713[ABC(G)] 8 0.9627 0.9267 75.9046 0.8903 NOT significant MV=1.1532+12.7953[ABC(G)] 8 0.9009 0.8116 25.8500 0.9834 NOT significant 3.4 Standard Error , Correlation Coefficient & Comparison among the actual values and computed Values . Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 225 https://internationalpubls.com In this section, the correlation coefficient between the topological indices and the physicochemical properties of drugs is determined, as shown in Table 11. Additionally, the comparison between calculated and real values of these properties is conducted. Table 11. Correlation between Physical Properties and Topological indices TI BP En AM FP Re Po MV F(G) 0.4475 0.5215 0.8687 0.4476 0.9396 0.9394 0.8735 M1(G) 0.4466 0.5162 0.8576 0.4467 0.9287 0.9286 0.8585 M2(G) 0.3753 0.4388 0.8012 0.3755 0.8859 0.8858 0.8078 R(G) 0.5907 0.6660 0.9429 0.5908 0.9804 0.9804 0.9284 AZI(G) 0.3684 0.4419 0.7392 0.3684 0.8137 0.8138 0.7291 H(G) 0.5547 0.6273 0.9202 0.5548 0.9676 0.9677 0.9117 ABC(G) 0.5251 0.5994 0.9093 0.5252 0.9627 0.9627 0.9009 The Standard Error of Estimate (SEE) is crucial in regression analysis as it evaluates the model's ability to predict future values. It quantifies the average deviation of observed data points from the regression line, indicating the model's precision. A lower SEE signifies a better fit, as it demonstrates that the model's predictions closely align with the actual values.A higher Standard Error of Estimate (SEE) indicates that the model's predictions are less accurate and that the observed data points deviate more from the regression line. This suggests that the model may not fit the data well and that there is greater variability between the predicted and actual values.Table 12 displays the standard Error Estimation (SEE) . Table 12. Standard Error between Topological indices and Physical Properties TI BP En AM FP Re Po MV F(G) 74.8642 12.6457 33.3773 41.1541 6.1636 2.4478 26.8321 M1(G) 68.0440 12.6929 34.6625 41.1752 6.6781 2.6504 28.2652 M2(G) 70.4872 13.3174 40.3322 42.6536 8.3513 3.3138 32.487 R(G) 61.3620 11.0555 22.4423 37.1323 3.5463 1.4049 20.4830 AZI(G) 70.6995 13.2949 45.3957 42.7845 10.4669 4.1496 37.7212 H(G) 63.2747 11.5413 26.3893 38.2893 4.5446 1.8009 22.6478 ABC(G) 64.7188 11.8630 28.0515 39.1635 4.8690 1.9327 23.9233 4 Results 4.1 Comparison between Calculated and real values of physical Properties of drugs Tables 13 to Table 18 present the comparison of actual and computed values for all physical properties of the eye conjunctivitis drops. Table 13. Comparison between Calculated and real Values of Boiling Point Drugs BP F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 581.8±50.0 249.86 618.42 622.19 607.13 628.72 610.86 613.24 Gentamicin 669.4±55.0 310.94 665.09 655.30 685.49 654.80 679.21 676.34 Moxifloxacin 636.4±55.0 294.19 655.75 654.55 655.55 659.48 657.71 655.65 Norfloxacin 555.8±50.0 233.11 604.42 609.40 596.49 618.47 599.64 599.11 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 226 https://internationalpubls.com Tobramycin 775.4±60.0 298.13 658.09 650.78 677.28 652.14 672.23 667.85 Levofloxacin 571.5±50.0 269.56 632.42 634.23 623.48 635.88 624.06 629.22 Gatifloxacin 607.8±55.0 273.50 637.09 637.99 634.19 609.97 635.28 635.37 Chloramphenical 644.9±55.0 195.67 571.75 578.54 563.40 583.48 564.01 566.23 Table 14. Comparison between Calculated and real Values of Enthalpy Drugs En F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 91.5±3.0 97.64 97.91 98.73 95.49 100.21 96.30 96.79 Gentamicin 112.6±6.0 109.08 108.43 106.27 112.71 106.30 111.36 110.82 Moxifloxacin 98.8±3.0 105.94 106.33 106.10 106.13 107.40 106.63 106.22 Norfloxacin 88.1±3.0 94.50 94.76 95.81 93.15 97.81 93.83 93.64 Tobramycin 128.7±6.0 106.68 106.85 105.24 110.90 105.68 109.82 108.93 Levofloxacin 90.1±3.0 101.33 101.07 101.47 99.09 101.88 99.21 100.34 Gatifloxacin 95.0±3.0 102.07 102.12 102.33 101.44 95.82 101.68 101.71 Chloramphenical 100.0±3.0 87.48 87.40 88.79 85.88 89.63 85.97 86.33 Table 15. Comparison between Calculated and real Values of Average Mass Drugs AM F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 331.341 359.76 361.78 366.66 349.25 379.22 353.45 355.85 Gentamicin 477.595 446.50 441.22 429.30 460.12 425.60 453.93 452.68 Moxifloxacin 401.431 422.72 425.33 427.88 417.75 433.93 422.33 420.93 Norfloxacin 319.331 335.98 337.95 342.46 334.19 360.98 336.96 334.15 Tobramycin 467.514 428.31 429.30 420.76 448.49 420.86 443.67 439.65 Levofloxacin 361.367 387.74 385.61 389.44 372.38 391.94 372.85 380.36 Gatifloxacin 375.394 393.34 393.56 396.56 387.54 345.86 389.35 389.80 Chloramphenical 323.129 282.82 282.35 284.09 287.38 298.75 284.57 283.69 Table 16. Comparison between Calculated and real Values of Flash Point Drugs Re F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 305.6±30.1 327.17 327.79 330.08 320.96 334.02 323.22 324.66 Gentamicin 358.6±31.5 357.68 356.03 350.12 368.38 349.80 364.58 362.85 Moxifloxacin 338.7±31.5 349.32 350.39 349.67 350.26 352.64 351.57 350.32 Norfloxacin 289.9±30.1 318.80 319.31 322.33 314.51 327.82 316.43 316.10 Tobramycin 422.8±32.9 351.29 351.80 347.39 363.41 348.19 360.36 357.71 Levofloxacin 299.4±30.1 337.01 336.26 337.37 330.85 338.35 331.20 334.32 Gatifloxacin 321.4±31.5 338.98 339.09 339.64 337.34 322.67 337.99 338.05 Chloramphenical 343.8±31.5 300.10 299.54 303.65 294.49 306.64 294.86 296.20 Table 17. Comparison between Calculated and real Values of Molar Refractivity Drugs Re F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 83.3±0.3 88.34 88.91 90.22 85.67 93.95 86.74 87.36 Gentamicin 122.6±0.4 113.40 111.88 108.73 116.46 107.59 114.97 114.75 Moxifloxacin 101.8±0.3 106.53 107.29 108.31 104.69 110.04 106.09 105.77 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 227 https://internationalpubls.com Norfloxacin 80.7±0.3 81.47 82.01 83.08 81.48 88.59 82.11 81.23 Tobramycin 111.7±0.4 108.15 108.44 106.20 113.23 106.20 112.09 111.07 Levofloxacin 91.1±0.4 96.42 95.80 96.95 92.09 97.69 92.19 94.30 Gatifloxacin 94.6±0.3 98.04 98.10 99.05 96.30 84.14 96.82 96.97 Chloramphenical 72.6±0.3 66.11 65.93 65.83 68.48 70.28 67.39 66.96 Table 18. Comparison between Calculated and real Values of Polarizability Drugs Po F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 33.0±0.5 35.01 35.25 35.78 33.96 37.25 34.39 34.64 Gentamicin 48.6±0.5 44.94 44.36 43.12 46.18 42.66 45.59 45.50 Moxifloxacin 40.4±0.5 42.22 42.54 42.95 41.51 43.63 42.07 41.94 Norfloxacin 32.0±0.5 32.29 32.52 32.95 32.31 35.12 32.55 32.20 Tobramycin 44.3±0.5 42.86 43.00 42.12 44.90 42.10 44.44 44.04 Levofloxacin 36.1±0.5 38.22 37.99 38.45 36.51 38.73 36.55 37.39 Gatifloxacin 37.5±0.5 38.86 38.90 39.28 38.18 33.35 38.39 38.45 Chloramphenical 28.8±0.5 26.20 26.14 26.11 27.15 27.86 26.72 26.54 Table 19. Comparison between Calculated and real Values of Molar Volume Drugs MV F(G) M1(G) M2(G) R(G) AZI(G) H(G) ABC(G) Ciprofloxacin 226.8±3.0 249.78 251.54 255.44 241.72 265.85 244.96 246.90 Gentamicin 366.9±5.0 321.09 316.57 307.09 330.99 303.26 326.37 325.36 Moxifloxacin 285.0±3.0 301.54 303.56 305.91 296.87 309.98 300.77 299.63 Norfloxacin 237.4±3.0 230.23 232.03 235.48 229.60 251.14 231.59 229.32 Tobramycin 305.9±5.0 306.14 306.81 300.04 321.63 299.44 318.06 314.80 Levofloxacin 244.0±5.0 272.79 271.05 274.22 260.35 276.11 260.68 266.76 Gatifloxacin 270.8±3.0 277.39 277.55 280.09 272.55 238.94 274.04 274.41 Chloramphenical 208.8±3.0 186.52 186.51 187.35 191.91 200.93 189.14 188.43 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 228 https://internationalpubls.com Figure 6. Correlation coefficients of between physical properties and Indices Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 2 (2025) 229 https://internationalpubls.com 5 Discussion In Table 11, the correlation coefficients between various physicochemical properties and topological indices of the drugs are presented. The forgotten index demonstrates a strong correlation of 0.9396 with molar refractivity. The First Zagreb index also shows a significant correlation with molar refractivity (0.9287). The Second Zagreb index exhibits a good correlation with molar refractivity (0.8859), while the Randic index shows a very high correlation with both molar refractivity and polarizability (0.9804). The Augmented Zagreb index has a notable correlation with polarizability (0.8138), and the Harmonic index shows a strong correlation with polarizability (0.9677). The ABC index presents a high correlation with both molar refractivity and polarizability (0.9627). Overall, the Randic index exhibits the highest correlation among the properties studied. However, the augmented Zagreb index has a low correlation coefficient of 0.3684 with the boiling point. From Table 4 to Table 10, the coefficient of determination R2 indicates how well the outcomes are predicted. Among these tables, Table 7 particularly highlights that the Randic index for the properties of Molar Refractivity and Polarizability are predicted very accurately. The estimated Standard Error of Estimate (SEE) is shown in Table 12. The findings demonstrate that the Randic index with Polarisability displays an extremely low SEE of 1.4049, suggesting that the predicted values closely match the medications real physicochemical characteristics. Table 18 provides more evidence supporting the tight alignment of the Randic index with Polarisability values with the expected values. 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