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Learn more: PMC Disclaimer | PMC Copyright Notice Sci Rep . 2026 Apr 15;16:12456. doi: 10.1038/s41598-026-43640-3 Search in PMC Search in PubMed View in NLM Catalog Add to search On topological co-index polynomials for predictive modeling of the physicochemical properties of antiviral drugs Summeira Meharban Summeira Meharban 1 Department of Mathematical Sciences, Karakoram International University, Gilgit, 15100 Pakistan Find articles by Summeira Meharban 1 , Asad Ullah Asad Ullah 1 Department of Mathematical Sciences, Karakoram International University, Gilgit, 15100 Pakistan Find articles by Asad Ullah 1, ✉ , Shahid Zaman Shahid Zaman 2 Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa, 616 Nizwa, Sultanate of Oman Find articles by Shahid Zaman 2 , Assmaa Abd-Elmonem Assmaa Abd-Elmonem 3 Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia Find articles by Assmaa Abd-Elmonem 3 , Parvez Ali Parvez Ali 4 Department of Mechanical Engineering, College of Engineering, Qassim University, 51452 Buraydah, Saudi Arabia Find articles by Parvez Ali 4 , Neissrien Alhubieshi Neissrien Alhubieshi 3 Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia Find articles by Neissrien Alhubieshi 3 , Melaku Berhe Belay Melaku Berhe Belay 5 Nanotechnology Centre of Excellence, Addis Ababa Science and Technology University, P.O.Box 16417, Addis Ababa, Ethiopia 6 Mathematics, Physics and Statistics Division, Addis Ababa Science and Technology University, P.O.Box 16417, Addis Ababa, Ethiopia Find articles by Melaku Berhe Belay 5, 6, ✉ Author information Article notes Copyright and License information 1 Department of Mathematical Sciences, Karakoram International University, Gilgit, 15100 Pakistan 2 Department of Mathematical and Physical Sciences, College of Arts and Sciences, University of Nizwa, 616 Nizwa, Sultanate of Oman 3 Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia 4 Department of Mechanical Engineering, College of Engineering, Qassim University, 51452 Buraydah, Saudi Arabia 5 Nanotechnology Centre of Excellence, Addis Ababa Science and Technology University, P.O.Box 16417, Addis Ababa, Ethiopia 6 Mathematics, Physics and Statistics Division, Addis Ababa Science and Technology University, P.O.Box 16417, Addis Ababa, Ethiopia ✉ Corresponding author. Received 2025 Oct 28; Accepted 2026 Mar 5; Collection date 2026. © The Author(s) 2026 Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/ . PMC Copyright notice PMCID: PMC13083970 PMID: 41986477 Abstract This study investigates the potential of topological co-indices and their polynomials as computational tools for predicting the physicochemical properties of antiviral compounds targeting the Ebola virus. For an in-depth Quantitative Structure–Property Relationship (QSPR) analysis, we develop and compute topological co-indices using CoM and CoNM polynomials derived from the molecular structures of antiviral drugs, including Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, and Clomiphene. These indices were correlated with some important experimental physicochemical properties via linear and curvilinear regression techniques, and their predictive models were developed. The results reveal strong correlations between the topological indices and physicochemical properties, underscoring their utility in drug discovery and design. A comparative analysis of actual experimental values and those predicted by the topological indices shows their predictive accuracy. To sum up, this study highlights the potential of topological co-indices as innovative tools for speeding up the discovery and optimization of antiviral drugs, especially concerning Ebola virus disease. The findings contribute to the expanding research focused on using computational methods to tackle emerging infectious diseases. Keywords: M-Polynomials, Drug molecules, Topological descriptors, QSPR analysis Subject terms: Chemistry, Computational biology and bioinformatics, Drug discovery, Mathematics and computing Introduction The Ebola virus, which is a member of the Filoviridae family, is the cause of Ebola Virus Disease (EVD), a dangerous and often fatal sickness. Due to its fast rate of mutation, this virus may develop unusual strains that are resistant to currently available therapies. Effective treatment execution and administration are made more difficult by the fact that many outbreaks occur in places with inadequate healthcare services 1 . Since its discovery in 1976, close to the Ebola River in the Democratic Republic of the Congo, the virus has occasionally caused outbreaks, mostly in Central and West Africa. The 2014–2017 With more than 28,000 cases and 12,000 loss of life, the West African Ebola outbreak was the biggest on record, highlighting the disease’s fatal worldwide effects 2 , 3 . Fever, intense headaches, muscle aches, vomiting, diarrhea, and, in the worst situations, internal and external bleeding are some of the symptoms of EVD. EVD is a serious public health problem because of its high fatality rate, which in certain outbreaks can exceed 90%, and its quick spread 4 , 5 . Beyond just health, EVD has an impact on the economies, healthcare systems, and social structures of the affected areas. Extensive fear, stigmatization, and disruption of daily life are common outcomes of outbreaks. In order to minimize the effects of future epidemics, the international community has acknowledged the significance of efficient preventive, diagnostic, and therapeutic methods. The absence of approved antiviral medications for EVD indicates the urgent need for innovative treatment options, even in the face of vaccine development advancements like rVSV-ZEBOV 6 , 7 . Graph-theoretical invariants have been especially useful in antiviral and anticancer QSPR/QSAR modeling. Degree-based and polynomial-derived topological descriptors have been successfully applied to model physicochemical properties of antiviral drugs under investigation for COVID-19 treatment 8 , 9 . More recently, integration of topological descriptors with supervised machine-learning approaches has enabled accurate prediction of anti-HIV drug properties, underscoring the ongoing relevance of algebraically derived descriptors in contemporary drug discovery 10 , 11 . These investigations verify that topological descriptors derived algebraically are still very useful tools for simulating physiologically active compounds in various therapeutic fields 12 . Topological indices are actually numerical values that indicate the topological properties of chemical compounds in a simplified way. They are obtained from the molecular structure of the compounds. In chemistry, especially cheminformatics, these indices are widely used to describe molecule structures and forecast their physicochemical and biological characteristics 13 – 23 . Topological indices originate from graph theory, where molecular structures are depicted as graphs, with chemical bonds as edges and atoms as vertices 16 , 24 – 29 . By determining the structural properties of these graphs, topological indices offer a mathematical framework for understanding and predicting molecular behavior. One of the most recognized topological indices is the Zagreb co-index 30 , 31 , the second Zagreb co-index 32 , the second modified Zagreb co-index , the Forgotten co-index 33 , and the redefined third Zagreb co-index . For octane isomers, the inverse sum index is an important predictor of total surface area, as well as the Symmetric division co-index ISI and the harmonic co-index, . The augmented Zagreb index serves as an important predictive tool in the analysis of heat of formation for heptanes and octanes. Augmented Zagreb co-index 34 . Calculations are made for the generalized Randic co-index, the Inverse Randic co-index 35 , the Atom-bond connectivity co-index , the Geometric Arithmetic co-index, and also the neighborhood vertex degree coincides like the neighborhood second Zagreb , third version of Zagreb , neighborhood-second modified Zagreb , neighborhood forgotten 36 , third NDe, NDe3, Fifth NDe ND5, neighborhood inverse sum index , neighborhood harmonic , Sanskruti coindices , are computed. Moreover, neighborhood inverse Randic , neighborhood generalized Randic , neighborhood atom-bond connectivity , and neighborhood geometric arithmetic co-indices are formulated in the same way as their classical degree counterparts. Each captures different aspects of molecular structure like branching, connectivity, and symmetry 37 – 43 . In drug design, topological indices are essential in QSPR and QSAR studies 44 – 48 . These studies focus on creating mathematical relationships between the structural characteristics of molecules and their biological activities or physicochemical properties. By utilizing topological indices as descriptors, researchers can build predictive models that assist in designing new drugs with specific properties 49 – 52 . Topological indices help predict drug candidates’ bioavailability, toxicity, and binding affinity, reducing experimental screening time and cost 8 . The use of topological indices extends beyond small molecules; they are also applicable in analyzing complex systems like polymers, nanomaterials, and biological macromolecules 53 – 57 . Their adaptability and computational efficiency make them vital tools in contemporary chemistry and drug discovery. QSPR analysis has been successfully applied in various aspects of drug discovery, including the development of antiviral, anticancer, and antibacterial agents 58 , 59 . New topological co-indices and co-neighborhood-based descriptors have attracted increased attention in recent years. These descriptors capture additional structural aspects not captured by classical indices by incorporating information from non-adjacent vertex pairs. In QSPR/QSAR studies, these descriptors have been shown to improve structural sensitivity and predictive performance. For example, improved descriptors that combine degree and distance information significantly improve QSPR correlations 60 . Recent analyses of novel degree-based indices have demonstrated strong discriminatory power, underscoring the growing significance of co-degree and co-neighborhood concepts in chemical graph theory 61 . To find and optimize new medicinal chemicals, researchers are looking into computational techniques like CoM (Co-indices-based Matrix) and CoNM-polynomials 62 . Several recent works have focused on extending classical topological indices through co-neighborhood and co-degree formulations, enabling richer structural characterization of molecular graphs. Polynomial-based approaches, particularly CoM-polynomials, have emerged as powerful tools for deriving multiple descriptors in a unified and algebraically elegant manner 63 . Such formulations have been successfully employed in QSPR/QSAR modeling to predict physicochemical, pharmacokinetic, and biological properties of drug candidates. These advancements improve the development of potent antiviral medications by making it possible to investigate extensive chemical fields, identify crucial molecular characteristics, and forecast drug qualities. CoM and CoNM-polynomials are sophisticated computational tools that enhance the idea of topological indices and offer more profound understandings of the behaviors and structures of molecules 64 , 65 . Designing a matrix using co-indices derived from the molecular graph is the process of the CoM technique. This matrix gives a thorough representation of the molecule’s fundamental structure by showing the interactions between various atoms. Researchers can compute several topological indices and use them for predicting molecular features by using this matrix 66 – 70 . The topological characteristics of molecules are represented in polynomial form by CoNM-polynomials, which are mathematical functions. They offer a more precise and economical way to characterize molecular structures since they can compute numerous topological indices simultaneously 12 . From tiny molecules to massive macromolecules, the CoM technique is extremely adaptable and suitable for a wide variety of chemical substances. In QSPR analysis, CoNM-polynomials are particularly helpful 62 . Motivated by these developments, the present study introduces and systematically analyzes an important class of degree-based and neighborhood degree-based topological co-indices derived via CoM- and CoNM-polynomials. The proposed descriptors are applied to a set of anti-Ebola antiviral drugs to evaluate their predictive capability for key physicochemical properties using QSPR modeling. Preliminaries and methods In pharmaceuticals, structural elements display the vertices and matching bonds that join atoms, known as edges. Let R be the graph of a molecule, where V (R) is the vertex set and E(R) is the set of bonds, termed the edge set. In a graph, R, a vertex’s degree, represented by µ (α), is the number of edges that strike it. Here, stands for the graph complement, which is the simple graph that has the same set of vertices V (R) and any two vertices ∈ E( ) if and only if E(R). It contains the fundamental definitions and symbols of graph theory. The CoM-polynomial of an R graph is defined as 1 The edge number ∉ E(R) such that {µ ( ), µ( )} = {a, b} is denoted as , a, b ≥ 1. The following are related notations used for computing CoM-polynomials: According to the CoM-polynomial explanation, the CoNM-polynomials, which originate from the neighborhood degree sum based on non-adjacent vertex pairs, are described as 2 where the edge number αβ ∉ E(R) such that {nµ (α), nµ (β)} = {a, b} is , a, b ≥ 1. The following are related notations used for computing CoM-polynomials: Lemma 1 The following statements apply to a connected graph G of order n: This study uses instead of to calculate the CoNM polynomial’s coefficients in Lemma 1 . The topological co-indices used in this study are information-rich, computationally efficient, and have high predictive potential. For these reasons, they are suitable for QSPR modelling. The selected topological co-indices are presented in Table 1 . It is important to emphasize that the proposed co-indices generalize earlier degree-based and neighborhood-degree-based descriptors by explicitly incorporating co-adjacency information between vertex pairs. Unlike conventional indices computed directly from degree sequences, the present framework leverages CoM- and CoNM-polynomials to encode higher-order structural interactions in a compact algebraic form. This not only simplifies computation but also enhances the expressive power of the resulting descriptors for QSPR analysis. Table 1. Mathematical expressions for topological co-indices. Topological co-indices Mathematical formulae (µ( ), µ(β)) CoM(R, x, y) or CoNM(R, x, y) Open in a new tab The formulas 19 used in Table 1 contain the following functions. In this paper, if µ ( ) = d ( ) and µ (β) = d (β), then F(x, y) = CoM(R, x, y). If µ ( ) = nd ( ) and µ (β) = nd (β), F(x, y) = CoNM(R, x, y) will be taken into account. The molecular structures of the Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir and Clomiphene drugs are shown in Fig. 1 . Fig. 1. Open in a new tab Molecular structures of drugs. The physicochemical properties of the medications tested in this investigation are given in Table 2 . These properties include molar volume (MV), molar refractivity (MR), polarizability (P), surface tension (T), Complexity (C), Log P, Density and Index of refraction of the drugs Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, Clomiphene, and Remdesivir. This information was originally taken from the ChemSpider database. Table 2. Physicochemical experimental features of the considered drugs. Drugs MV MR P Surface Tension Complexity Log P Density Index of refraction Galidesivir 162.6 68.3 27.1 103.2 334 -3.03 1.6 1.782 Chloroquine 287.9 97.4 38.6 44 309 4.69 1.1 1.592 Favipiravir 97.2 33.2 13.2 81.5 282 0.78 1.6 1.6 Amodiaquine 282.8 105.5 41.8 55.3 406 4.77 1.3 1.669 Azithromycin 632.7 197.6 78.3 50.6 1150 3.33 1.2 1.537 Brincidofovir 478.5 147.6 58.5 46.3 721 5.41 1.2 1.529 Clomiphene 367.6 123.7 49.1 42.1 481 8.01 1.1 1.588 Remdesivir 409 149.5 59.3 62.3 1010 2.1 1.5 1.652 Open in a new tab Main results This section outlines the key findings, including theorems that define the polynomials of the molecular structures of Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, Clomiphene, and Remdesivir drugs, along with their proofs. Furthermore, the topological indices are presented in Table 1 . Theorem 1 Let the Galidesivir graph be . CoM-polynomial of presented as follows: Proof From Fig. 1 , |V| = 19 and |E| = 21. By the vertex and edge partition techniques (see Table 3 ), we have: and = 4, = 7, = 8. From Lemma 1 , we have Table 3. Degree bond partitioning of drug molecules. Graphs Drugs (1, 2) (1, 3) (1, 4) (2, 2) (2, 3) (2, 4) (3, 3) (3, 4) R 1 Galidesivir 1 3 0 3 7 0 7 0 R 2 Chloroquine 2 2 0 5 12 0 2 0 R 3 Favipiravir 0 4 0 1 4 0 2 0 R 4 Amodiaquine 2 2 0 4 16 0 3 0 R 5 Azithromycin 2 13 5 0 17 3 10 4 R 6 Brincidofovir 2 2 2 21 8 2 1 0 R 7 Clomiphene 2 1 0 12 12 0 4 0 R 8 Remdesivir 3 5 1 9 14 5 6 2 Open in a new tab Hence, we obtain Or Theorem 2 Let be the chemical graph of Galidesivir. The various topological co-indices of the graph are as follows: Proof The CoM-polynomial from the Theorem can be used to calculate the following indices: For we have = 307.438237, For we have = 71.83921 Theorem 3 The CoNM-polynomial of is Proof From Fig. 1 , |V |= 21 and |E|= 19. Using the vertex and edge partition techniques (see Table 3 ), we have: and = 2, = 3, = 2, = 4, = 2, = 3, = 4. From Lemma 1 above, we have From Eq. 2 , we have, Or, Theorem 4 The numerous topological co-indices depending on the neighborhood degree of the graph are: Theorem 5 Let be the Chloroquine graph, the M-polynomial is given as follows: Proof From Fig. 1 , |V |= 22 and |E|= 23 and also = = 2, = = 2 , = = 5, = = 12, = = 2.Thus = 4, = 12, = 6. CoM ( , x, y) = . Theorem 6 The topological co-indices of the graph are as follows: = 848, = 879, = 57.0277, 1952 4018, 466.3333, 199.6666 100.5, 1558.3281, 411.128, = 108.598 229.5692, 195.2095 = = Theorem 7 Let be the Chloroquine graph, the NM-polynomial is given as follows: = + + + + + + + + . Proof From Fig. 1 , it is seen that = 2, = 2, = 4, = 2, = 3, = 3, = 2, = 1, = 2, = 2. Thus = 2, = 2, = 4, = 8, = 3, = 2, = 1. We obtain that = 6, = 14, = 28, = 10, = 25, = 21, = 14, = 7, = 4 by Lemma 1 . We have Eq. 2 , = + + + + + + + + Theorem 8 Various topological co-indices computed based on the neighborhood degree of the graph as: = 1292, = 3251, 5.870238, = 6794, = 34,292, = 271.7702, = 315.5398, = 27.18143, = 3889.781, = 638.4436, = 26.96284, = 66.9032, = 127.3533 Theorem 9 If is the chemical graph of Favipiravir, then. Proof From Fig. 1 , |V |= 11 and |E|= 11 and also, = = 4 = = 1, = = 4, = = 2. Thus = 4, = 3, = 4 and = 12, = 2, = 8, = 4. Theorem 10 The various topological co-indices of the graph are as follows: = 120, = 128, = 6.277778, 312 632, 69.33333, 26.6 Theorem 11 The CoNM-polynomial of is Proof Figure 1 , shows clearly that, = 3, = 1, = 2, = 2, = 1, = 4, = 4, = 2, = 1. We obtain that = 13, = 7, = 4, = 6, = 3, = 0 by Lemma 1 . We have from Eq. 2 , Theorem 12 The following list of topological co-indices was calculated using the graph based neighborhood degree. Theorem 13 If is the chemical graph of Amodiaquine, then. Proof From Fig. 1 , |V |= 25 and |E|= 27 and also, = , 2, = = 2, = = 4, = = 16, = = 3.Thus = 4, = 13, = 8 and = 50, = 30, = 74, = 88, = 25. Theorem 14 The various topological co-indices of the graph are as follows: Theorem 15 The CoNM-polynomial of is Proof From Fig. 1 , it can be seen that, = 2, = 1, = 1, = 2, = 2, = 4, = 4, = 1, = 2, = 3, = 4, = 1, and = 2, = 2, = 3, = 8, = 7, = 2, = 1. = 4, = 15, = 13, = 22, = 19, = 24, = 52, = 15, = 6, = 18, = 10, = 1 by lemma 1. We have, Eq. 2 , Theorem 16 Various topological co-indices calculated based on the neighborhood degree of the graph are given here: In a similar way (Theorems 1–16 above), we can obtain the topological indices of all the other drugs. The obtained values of the topological indices for all eight drugs are given in Table 4 . Table 4. Computed TI values of drugs. Drugs Galidesivir Chloroquine Favipiravir Amodiaquine Azithromycin Brincidofovir Clomiphene Remdesivir 640.000 848.000 120.000 1156.000 4340.000 2588.000 1588.000 3390.000 696.000 879.000 128.000 1239.000 4730.000 2567.000 1685.000 3625.000 37.500 57.028 6.278 70.944 247.389 191.181 95.972 215.153 1584.000 1952.000 312.000 2736.000 11822.000 5618.000 3626.000 8322.000 3402.000 4018.000 632.000 5834.000 24824.000 11054.000 7662.000 17612.000 348.333 466.333 69.333 613.667 2567.750 1426.500 689.167 1863.583 148.050 199.667 26.600 272.933 946.998 617.200 381.367 784.752 68.100 100.500 11.533 128.867 425.186 334.267 799.000 378.414 1089.078 1558.328 166.063 2082.016 6643.184 12287.906 2941.141 5636.807 307.438 411.128 56.381 561.227 2022.457 1263.039 777.811 1629.000 71.839 108.598 19.059 133.935 464.950 344.169 185.003 397.009 104.146 229.569 19.536 191.068 698.273 459.631 264.544 560.607 137.581 195.210 40.623 258.343 863.305 652.660 363.730 746.515 914.000 1292.000 312.000 2524.000 10009.000 5220.000 2600.000 6062.000 2731.000 3251.000 721.000 5436.000 27173.000 11314.000 6450.000 16175.000 2.731 5.870 1.691 8.142 36.139 36.848 12.783 22.409 5846.000 6794.000 1590.000 11340.000 61493.000 23742.000 13376.000 56647.000 35124.000 34292.000 7234.000 59274.000 324154.000 103366.000 67364.000 190990.000 169.214 271.770 73.842 419.820 2090.130 1372.160 555.603 1548.192 219.643 315.540 73.969 508.018 2330.249 1246.635 637.645 1390.687 13.637 27.181 7.158 39.011 167.901 141.972 56.378 102.067 3419.871 3889.781 810.690 6579.887 31982.509 12249.564 7784.704 17745.267 431.916 638.444 151.884 1027.825 4826.125 2577.365 1287.446 2882.012 14.012 26.963 7.376 39.804 175.586 146.694 57.075 106.201 42.181 66.903 19.709 113.281 1203.145 358.473 154.315 305.236 75.251 127.353 32.081 197.412 862.804 465.428 263.210 524.285 Open in a new tab Curvilinear regression analysis Several physicochemical properties of the drugs Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, and Clomiphene, as well as their derived topological co-indices, have been used here for QSPR modeling. The eight considered physicochemical properties are molar volume, molar refraction, surface tension, polarizability, LogP, complexity, density, and the index of refraction. Curvilinear regression analysis was performed using SPSS software. The values of the topological co-indices(TCI) listed above are represented as independent variables, while physicochemical properties are represented as dependent variables. The three regression models employed in this study are as follows: The correlation coefficient approaches one when the theoretical and experimental results are nearly identical. In the above regression equations, A is a constant; B and C are the additional regression coefficients that contribute to the best estimation. In the tables below, SE is the standard error, p is the p-value, and F is the F-statistic. The p-value is less than 0.05, indicating statistical significance. Tables 5 , 16 , and 17 present the correlation coefficients obtained from linear, logarithmic, and quadratic regression models, accordingly, which describe the association between topological co-indices and the physicochemical properties of several drugs used to treat the Ebola virus. Tables 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 show detailed statistical characteristics for the linear QSPR models corresponding to each physicochemical characteristic. Tables 14 and 15 show the statistical parameters for logarithmic and quadratic regression models. Table 5. The correlation coefficients obtained from the linear QSPR model of the experimental physicochemical properties and computed TI values of drugs. Drugs MV MR P ST C LogP D IR .937 .953 .953 .433 .987 .236 .294 .455 .929 .948 .948 .417 .989 .219 .280 .437 .935 .947 .947 .464 .971 .272 .309 .484 .920 .935 .935 .389 .985 .184 .264 .432 .911 .929 .929 .378 .984 .174 .253 .413 .926 .937 .937 .394 .987 .178 .261 .449 .938 .956 .956 .451 .984 .261 .304 .458 .612 .621 .622 .577 .470 .653 .549 .413 .766 .728 .728 .460 .672 .346 .353 .569 .938 .955 .955 .443 .986 .249 .299 .457 .938 .951 .951 .465 .976 .273 .311 .486 .951 .963 .963 .502 .967 .284 .369 .501 .937 .950 .950 .468 .974 .283 .312 .486 .938 .937 .937 .415 .970 .214 .297 .492 .925 .890 .890 .505 .864 .346 .394 .636 .925 .890 .890 .505 .864 .346 .394 .636 .821 .858 .858 .287 .986 .072 .101 .323 .874 .885 .885 .311 .959 .099 .208 .391 .928 .928 .928 .416 .984 .213 .263 .509 .942 .937 .937 .415 .969 .216 .306 .507 .953 .927 .927 .498 .913 .324 .387 .614 .908 .911 .911 .361 .957 .155 .268 .442 .942 .937 .937 .417 .970 .216 .305 .511 .951 .926 .925 .491 .916 .315 .378 .610 .861 .828 .827 .326 .850 .130 .305 .510 .950 .946 .946 .443 .969 .248 .329 .522 Open in a new tab The Highest correlation values are shown in bold. Table 16. The correlation coefficients obtained from the logarithmic QSPR model of the experimental physicochemical properties and computed TI values of drugs. TI MV MR P ST C Log P D IR .907 .955 .955 .560 .814 .408 .508 .304 .906 .956 .956 .548 .819 .397 .498 .291 .897 .943 .943 .579 .787 .427 .529 .312 .916 .964 .964 .536 .842 .376 .481 .299 .909 .960 .960 .526 .838 .368 .474 .280 .917 .962 .962 .537 .841 .364 .477 .311 .900 .948 .948 .567 .799 .420 .517 .299 .813 .854 .855 .649 .639 .607 .636 .330 .878 .905 .905 .588 .731 .458 .545 .369 .903 .952 .952 .564 .807 .414 .513 .301 .920 .961 .961 .591 .826 .436 .518 .361 .895 .939 .939 .618 .765 .446 .582 .334 .923 .962 .963 .590 .835 .441 .510 .369 .945 .980 .980 .584 .870 .429 .488 .398 .939 .980 .980 .531 .884 .367 .453 .346 .956 .960 .960 .681 .839 .543 .562 .573 .908 .960 .960 .489 .908 .320 .375 .304 .918 .967 .967 .471 .886 .303 .406 .276 .952 .977 .977 .592 .895 .433 .468 .457 .954 .985 .985 .587 .875 .430 .499 .414 .961 .973 .973 .668 .848 .524 .558 .534 .934 .977 .977 .525 .872 .363 .462 .324 .955 .986 .986 .590 .879 .432 .497 .423 .961 .973 .973 .660 .854 .516 .548 .532 .988 .994 .994 .582 .909 .416 .499 .516 .959 .987 .988 .626 .875 .473 .525 .456 Open in a new tab The Highest correlation values are shown in bold. Table 17. The correlation coefficients obtained from the Quadratic QSPR model of the experimental physicochemical properties and computed TI values of drugs. TI MV MR P ST C LogP D IR .946 .970 .970 .617 .991 .596 .509 .455 .941 .969 .970 .600 .990 .577 .483 .437 .941 .961 .961 .681 .989 .675 .631 .485 .938 .966 .966 .562 .987 .516 .413 .432 .931 .964 .964 .554 .986 .513 .407 .413 .938 .960 .960 .560 .987 .499 .413 .449 .944 .969 .969 .644 .995 .637 .565 .458 .939 .962 .962 .631 .921 .653 .558 .492 .946 .974 .974 .552 .932 .395 .416 .569 .945 .970 .970 .631 .993 .618 .537 .457 .944 .963 .963 .683 .992 .679 .615 .486 .956 .974 .974 .721 .983 .654 .652 .501 .942 .961 .961 .680 .992 .689 .615 .487 .951 .964 .964 .568 .974 .510 .373 .492 .936 .958 .958 .513 .986 .440 .321 .448 .946 .943 .944 .654 .894 .573 .464 .642 .897 .935 .936 .600 .991 .617 .513 .402 .904 .939 .939 .437 .988 .355 .246 .394 .933 .939 .939 .590 .985 .572 .398 .509 .958 .966 .966 .573 .974 .512 .387 .510 .958 .947 .947 .657 .914 .608 .466 .617 .940 .963 .963 .512 .980 .435 .331 .442 .956 .964 .964 .572 .975 .510 .382 .512 .956 .945 .945 .644 .918 .596 .451 .613 .963 .951 .951 .536 .949 .435 .369 .569 .963 .971 .971 .616 .971 .564 .426 .526 Open in a new tab The highest correlation values are shown in bold. Table 6. Statistical variables of the linear QSPR model for MV. TI R 2 SE F A B P .877 64.990 42.959 138.944 .110 0.001 .866 68.007 38.711 142.521 .101 .001 .874 65.858 41.677 134.404 1.783 .001 .830 76.654 29.193 145.005 .039 .002 .763 90.340 19.337 181.113 .018 .005 .838 74.830 30.929 149.463 .199 .001 .883 63.413 45.423 130.134 .494 .001 .320 .073 2.820 1.688 .000 .144 .640 111.358 10.676 200.495 .033 .017 .885 63.015 46.076 136.447 .231 .001 .784 86.234 21.808 118.185 .934 .003 .872 66.402 40.899 139.596 .655 .001 .918 53.185 67.106 115.520 .569 .000 .879 64.533 43.655 161.975 .049 .001 .830 76.504 29.331 177.577 .018 .002 .855 70.596 35.491 164.172 11.096 .001 .677 105.478 12.587 202.759 .006 .012 .155 170.641 1.102 261.210 .001 .334 .861 69.143 37.254 166.667 .213 .001 .888 62.202 47.445 162.780 .210 .000 .642 111.088 10.757 261.296 .564 .017 .823 78.158 27.852 178.816 .015 .002 .881 64.035 44.430 166.797 .101 .001 .898 59.205 52.994 158.541 2.545 .000 .741 94.452 17.179 232.831 .378 .006 .901 58.417 54.596 156.147 .575 .000 Open in a new tab Table 7. Statistical variables of the linear QSPR model for MR. TI R 2 SE F A B P .908 16.849 58.890 54.383 .033 .000 .901 17.467 54.384 55.297 .031 .000 .898 17.739 52.546 53.234 .539 .000 .856 21.018 35.702 56.286 .012 .001 .795 25.114 23.209 67.025 .005 .003 .872 19.833 40.836 57.388 .061 .001 .918 15.906 66.813 51.570 .150 .000 .800 24.811 23.927 46.788 .308 .003 .577 36.050 8.175 75.888 .009 .029 .915 16.115 64.938 53.615 .070 .000 .847 21.659 33.272 46.601 .290 .001 .903 17.270 55.765 54.551 .199 .000 .932 14.503 81.592 47.918 .171 .000 .878 19.319 43.360 62.298 .015 .001 .838 22.283 31.104 66.697 .005 .001 .793 25.239 22.921 64.895 3.188 .003 .740 28.267 17.056 72.602 .002 .006 .236 48.419 1.858 86.393 .000 .222 .862 20.605 37.392 63.664 .063 .001 .879 19.288 43.522 62.782 .062 .001 .580 35.911 8.286 93.081 .160 .028 .829 22.933 29.030 67.130 .005 .002 .873 19.773 41.122 63.961 .030 .001 .851 21.370 34.339 62.687 .740 .001 .685 31.107 13.039 84.662 .108 .011 .893 18.162 49.853 60.794 .171 .000 Open in a new tab Table 8. Statistical variables of the linear QSPR model for Polarizability. TI R 2 SE F A B P .908 6.667 59.034 21.584 .013 .000 .901 6.911 54.527 21.945 .012 .000 .898 7.020 52.661 21.128 .214 .000 .857 8.301 35.950 22.329 .005 .001 .794 9.953 23.181 26.597 .002 .003 .872 7.853 40.883 22.775 .024 .001 .918 6.289 67.089 20.468 .060 .000 .800 9.828 23.931 18.577 .122 .003 .577 14.284 8.170 30.108 .004 .029 .916 6.374 65.148 21.279 .028 .000 .847 8.589 33.192 18.509 .115 .001 .903 6.832 55.939 21.649 .079 .000 .932 5.739 81.775 19.023 .068 .000 .878 7.657 43.303 24.724 .006 .001 .838 8.829 31.089 26.465 .002 .001 .792 10.001 22.903 25.752 1.263 .003 .740 11.191 17.085 28.800 .001 .006 .237 19.173 1.865 34.251 .000 .221 .862 8.161 37.409 25.262 .025 .001 .879 7.643 43.486 24.915 .025 .001 .579 14.234 8.269 36.920 .063 .028 .829 9.087 29.010 26.637 .002 .002 .873 7.835 41.097 25.381 .012 .001 .851 8.469 34.310 24.877 .293 .001 .684 12.331 13.012 33.585 .043 .011 .893 7.196 49.833 24.126 .068 .000 Open in a new tab Table 9. The linear QSPR model for the statistical variables of complexity. TI R 2 SE F A B P .975 57.515 233.469 172.259 .226 .000 .976 56.735 240.105 176.769 .210 .000 .942 87.414 97.670 169.302 3.623 .000 .815 156.200 26.468 208.689 .075 .002 .955 77.417 126.172 239.289 .039 .000 .982 48.128 335.994 183.167 .422 .000 .964 68.818 161.270 157.922 1.010 .000 .757 179.118 18.691 149.150 1.962 .005 .500 257.005 5.993 345.755 .058 .050 .970 63.294 191.740 169.985 .474 .000 .802 161.506 24.370 147.889 1.849 .003 .971 61.432 203.909 173.081 1.353 .000 .915 106.229 64.199 148.491 1.111 .000 .941 88.164 95.915 226.546 .100 .000 .942 87.630 97.160 248.460 .037 .000 .746 183.007 17.653 265.561 20.286 .006 .973 59.728 216.057 265.143 .014 .000 .326 298.274 2.904 363.620 .003 .139 .967 65.672 177.677 227.528 .441 .000 .939 89.982 91.838 230.376 .423 .000 .614 225.857 9.529 436.421 1.079 .021 .917 104.841 66.070 254.053 .032 .000 .943 86.430 100.045 236.253 .204 .000 .841 144.914 31.722 243.391 4.820 .001 .722 191.507 15.600 379.975 .730 .008 .938 90.587 90.536 219.915 1.148 .000 Open in a new tab Table 10. The linear QSPR model for the statistical variables of Log P. TI R 2 SE F A B P .056 3.516 .355 2.269 .001 .573 .051 3.525 .323 2.323 .000 .590 .074 3.483 .478 2.094 .010 .515 .150 3.335 1.063 1.640 .000 .342 .005 3.610 .031 3.005 .00 .867 .035 3.554 .219 2.496 .001 .656 .074 3.482 .481 2.072 .003 .514 .099 3.434 .662 1.679 .007 .447 .115 3.405 .777 2.108 .000 .412 .069 3.492 .445 2.151 .001 .530 .108 3.418 .725 1.656 .007 .427 .062 3.505 .396 2.218 .003 .552 .105 3.424 .703 1.780 .004 .434 .046 3.535 .288 2.466 .000 .611 .024 3.575 .148 2.719 .00 .714 .119 3.396 .814 1.978 .081 .402 .006 3.608 .037 3.002 .00 .853 .028 3.569 .170 2.612 .00 .695 .045 3.536 .284 2.484 .001 .613 .047 3.534 .293 2.468 .001 .608 .007 3.605 .045 3.093 .001 .839 .024 3.575 .146 2.724 .00 .715 .044 3.537 .279 2.500 .000 .616 .095 3.442 .632 2.106 .016 .457 .017 3.588 .103 2.943 .001 .759 .061 3.508 .387 2.330 .003 .557 Open in a new tab Table 11. The linear QSPR model for the statistical variables of density. TI R 2 SE F A B P .086 .219 .566 1.403 − 4.236E− 05 .480 .082 .220 .534 1.400 − 3.836E− 05 .492 .095 .218 .633 1.409 − .001 .457 .170 .209 1.229 1.434 − 2.167E− 05 .310 .030 .226 .185 1.364 − 4.373E− 06 .682 .056 .223 .353 1.386 − 6.332E− 05 .574 .098 .218 .650 1.411 .000 .451 .203 .205 1.528 1.468 − .001 .263 .132 .213 .911 1.403 − 1.870E− 05 .377 .096 .218 .636 1.408 − 9.401E− 05 .456 .101 .217 .677 1.423 .000 .442 .082 .219 .539 1.401 .000 .490 .134 .213 .926 1.431 .000 .373 .088 .219 .579 1.394 − 1.920E− 05 .475 .065 .222 .418 1.381 − 6.124E− 06 .542 .155 .211 1.104 1.417 − .006 .334 .011 .228 .068 1.347 − 9.590E− 07 .803 .001 .229 .008 1.316 .00 .932 .069 .221 .447 1.386 .00 .528 .094 .218 .620 1.396 .00 .461 .082 .219 .539 1.360 .000 .491 .070 .221 .454 1.383 .00 .526 .086 .219 .565 1.392 .00 .481 .134 .213 .930 1.411 − .001 .372 .093 .218 .616 1.372 .000 .462 .108 .216 .727 1.403 .000 .427 Open in a new tab Table 12. The linear QSPR model for the statistical variables of Surface Tension. TI R 2 SE F A B P .188 20.831 1.386 72.225 − .006 .284 .178 20.957 1.297 71.789 − .006 .298 .216 20.468 1.650 73.364 − .110 .246 .254 19.962 2.043 74.080 − .003 .203 .095 21.988 .629 67.626 − .001 .458 .154 21.262 1.089 70.812 − .011 .337 .210 20.538 1.598 73.399 − .030 .253 .320 19.056 2.825 78.758 − .081 .144 .215 20.476 1.644 70.714 − .002 .247 .206 20.587 1.561 72.891 − .014 .258 .244 20.101 1.932 76.036 − .065 .214 .191 20.790 1.415 72.321 − .038 .279 .252 19.987 2.023 75.294 − .037 .205 .172 21.028 1.248 70.457 − .003 .307 .132 21.530 .914 68.719 − .001 .376 .255 19.948 2.054 72.600 − .754 .202 .085 22.113 .554 66.690 .000 .485 .056 22.454 .357 66.546 .00 .572 .173 21.020 1.253 70.316 − .012 .306 .257 .076 2.076 1.664 .00 .200 .077 22.210 .497 64.036 − .024 .507 .129 21.573 .886 68.589 − .001 .383 .166 21.103 1.197 70.019 − .005 .316 .231 20.270 1.800 72.100 − .161 .228 .107 21.845 .716 65.712 − .018 .430 .893 7.196 49.833 24.126 .068 .000 Open in a new tab Table 13. The linear QSPR model for the statistical variables of index of refraction. TI R 2 SE F A B P .207 .079 1.566 1.665 − 2.537E− 05 .257 .193 .080 1.431 1.663 − 2.275E− 05 .277 .234 .077 1.834 1.669 .000 .224 .222 .078 1.714 1.667 − 9.575E− 06 .238 .141 .082 .985 1.651 − 3.664E− 06 .359 .183 .080 1.345 1.661 − 4.439E− 05 .290 .213 .079 1.621 1.668 .000 .250 .320 .073 2.820 1.688 .000 .144 .345 .072 3.155 1.667 − 1.168E− 05 .126 .218 .078 1.672 1.667 − 5.478E− 05 .244 .146 .082 1.027 1.664 .000 .350 .210 .079 1.591 1.665 .000 .254 .260 .076 2.109 1.676 .000 .197 .242 .077 1.916 1.663 − 1.230E− 05 .216 .200 .079 1.502 1.657 − 4.149E− 06 .266 .404 .068 4.074 1.676 − .004 .090 .105 .084 .707 1.644 − 1.136E− 06 .433 .000 .089 .001 1.620 − 1.499E− 08 .980 .259 .076 2.102 1.664 − 5.563E− 05 .197 .257 .076 2.076 1.664 − 5.394E− 05 .200 .231 .078 1.801 1.641 .000 .228 .194 .079 1.443 1.656 − 3.536E− 06 .275 .254 .076 2.048 1.663 − 2.585E− 05 .202 .363 .071 3.424 1.674 − .001 .114 .260 .076 2.104 1.649 .000 .197 .273 .075 2.251 1.667 .000 .184 Open in a new tab Table 14. Statistical parameters of the logarithmic regression model. drugs SE F A B P MV- .975 29.103 238.153 − 298.610 129.655 .000 MR- .989 5.859 530.692 − 76.502 38.964 .000 P- .989 2.327 527.914 − 30.260 15.435 .000 ST- .464 16.918 5.197 89.620 − 12.695 .063 C- .826 151.700 28.423 − 563.010 233.483 .002 LogP- .295 3.039 2.509 − .357 1.584 .164 D- .404 .177 4.073 1.833 − .100 .090 IR- .328 .073 2.932 1.712 − .041 .138 Open in a new tab Table 15. Statistical parameters for the Quadratic regression model. R 2 SE F A B C P MV- .927 54.815 31.911 140.671 1.206 − .001 .001 MV- .928 54.638 32.134 114.418 .918 .000 .001 MR- .948 13.863 45.429 35.346 .037 .000 0.001 MR- .948 13.843 45.571 33.084 .038 .00 .001 P- .475 2.872 2.262 − 2.407 .035 .00 .200 P- .948 5.484 45.559 14.036 .131 .00 .001 ST- .520 17.537 2.711 95.353 − .227 .000 .159 C- .990 40.613 237.639 250.257 .387 .001 .000 LogP- .475 2.872 2.262 − 2.407 .035 .00 .200 D- .425 .190 1.846 1.638 − .002 .00 .251 IR- .412 .074 1.751 1.665 − .001 .00 .265 Open in a new tab Comparative analysis and discussion The molecular topology of antiviral disease medications is modeled using different topological indices, and a QSPR analysis is used to explore the prediction ability of the topological indices under consideration. Polynomials are formulated to cope the computational difficulty of the topological index as the number of molecular structures expands. We connected eight physical characteristics of eight drugs used to treat the Ebola virus with topological co-indices. Tables 5 , 12 , and 16 illustrate the predictive potential of degree-based co-indices and neighborhood degree-based co-indices for all Physical and chemical attributes evaluated, as well as the correlation coefficients of physicochemical properties and topological co-indices. From Table 5 , Neighborhood harmonic index for MV, atom-bond connectivity index for MR and P; Second Zagreb index for C, are the best estimator indices in the linear regression model. From Table 16 , the Neighborhood atom bond connectivity index for MV, MR, P and C; the Neighborhood second modified Zagreb index for ST, LogP and IR; the Harmonic indices for D, are the best predictors in the logarithmic regression model. From Table 17 , the Neighborhood atom bond connectivity index for MV; Neighborhood Geometric Arithmetic for MV; atom-bond connectivity index for Mr, P, ST, D; Augmented Zagreb for MR; Geometric athematic for LogP; Neighborhood second modified Zagreb indices for are the strongest predictors in Quadratic regression model. From QSPR modelling data, the best predictive topological co-indices among all co-indices are given below. In linear QSPR model is as the best predictor of C. is the best prediction ability of MR and P. is the strongest estimator of MV. In the logarithmic QSPR model index is the best estimator of D. are the best at predicting the properties of MV, MR, P and C. are the best at predicting the properties of ST, LogP and IR. In the quadratic QSPR model index is the best predictor of MR and P. and indices are the best predictors of MV. indices are the best suited for predicting the property of C. . are the best suited for predicting the properties of MR, P, ST and D. are the best suited for predicting the properties of IR are the best suited for predicting the property of LogP. The QSPR analysis reveals that while several physicochemical properties exhibit strong linear correlations with the proposed topological co-indices, certain properties, most notably Log P and Density, display weaker linear statistical significance, as indicated by relatively higher p-values. This behavior suggests that the relationship between molecular structure and these properties is inherently nonlinear. To address this limitation, logarithmic and quadratic regression models were employed. These models significantly improved correlation coefficients and statistical significance, demonstrating their suitability for capturing nonlinear structure–property relationships. Figure 2 shows the graphical visualizations of different regression methods. The superior performance of the proposed co-indices in logarithmic and quadratic models underscores the importance of nonlinear modeling in QSPR studies involving complex molecular descriptors. The results indicate that CoM- and CoNM-based co-indices capture subtle structural information that may not manifest linearly but becomes evident under nonlinear transformations. This further supports the applicability of the proposed descriptors as reliable predictive tools for antiviral drug properties. Fig. 2. Open in a new tab Graphical visualization of the linear and curvilinear regression models. Here, scatter plot illustrating the relationship between the proposed topological index and the corresponding physical property. The x-axis represents the topological index (TI) values, while the y-axis shows the experimentally measured physical property. Conclusion Topological indices, often referred to as molecular descriptors, are numerical parameters derived from the molecular structure of chemical compounds. They relate the molecular structure to the physicochemical properties of a molecule. In the present study, we evaluated different degree-based and neighborhood degree-based topological co-indices for anti-Ebola drugs, including Galidesivir, Chloroquine, Favipiravir, Amodiaquine, Azithromycin, Brincidofovir, and Clomiphene. We obtained topological co-index polynomials for these drugs using the graph theoretical techniques to analyze their molecular structures. The QSPR models for drug molar volume, molar refractivity, surface tension, polarizability, complexity, LogP, density, and refractive index are then formulated using the chosen topological co-indices. The co-indices were correlated with experimental physicochemical data using quadratic, logarithmic, and linear regression techniques in order to evaluate the predictive power. The obtained results demonstrate that degree-based and neighborhood degree-based topological co-indices derived via CoM- and CoNM-polynomials offer strong predictive capability for key physicochemical properties of antiviral drugs. The results further indicate that nonlinear QSPR models provide a more accurate representation of structure–property relationships for certain molecular attributes. These findings reinforce the potential of algebraic graph-theoretical descriptors as efficient and insightful tools for drug discovery and optimization. Furthermore, this theoretical framework can be treated as an alternative to the intensive laboratory experimentation that offers useful insights for drug design. Similar techniques can be applied to other medications that are specific to a given ailment, enabling a methodological investigation of the connections between molecular structure and physicochemical properties. Acknowledgements The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/193/46. Author contributions Summeira Meharban and Asad Ullah have equally contributed to this manuscript in all stages from conceptualization to the write up of the final draft. The authors Shahid Zaman, Assmaa Abd-Elmonem, Parvez Ali, Neissrien Alhubieshiand and Melaku Berhe Belay have contributed to methodology, results and analysis, and write up. Funding This research was funded by the Deanship of Research and Graduate Studies at King Khalid University through Large Research Project under grant number RGP2/193/46. Data availability All data generated or analyzed during this study are included within this article. Declarations Competing interests The authors declare no competing interests. Consent for publication All authors have approved the manuscript and given consent for publication. Declaration of generative AI and AI-assisted technologies in the writing process During the preparation of this work, the authors used ChatGPT 3.5 in order to improve the readability and language of the manuscript. After using this tool/service, the authors reviewed and edited the content as needed and take full responsibility for the content of the publication. Footnotes Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Contributor Information Asad Ullah, Email: [email protected]. Melaku Berhe Belay, Email: [email protected]. References 1. Mulangu, S. et al. Controlled trial of ebola virus disease therapeutics. N. Engl. J. 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