Topological Descriptors and Quantum-Chemical Correlations in Skin Cancer Drug Design Using SMP -Polynomials

Topological Descriptors and Quantum-Chemical Correlations in Skin Cancer Drug Design Using SMP -Polynomials | Authorea try { document.documentElement.classList.add('js'); } catch (e) { } var _gaq = _gaq || []; _gaq.push(['_setAccount', 'G-8VDV14Y67G']); _gaq.push(['_trackPageview']); (function() { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); Skip to main content Preprints Collections Wiley Open Research IET Open Research Ecological Society of Japan All Collections About About Authorea FAQs Contact Us Quick Search anywhere Search for preprint articles, keywords, etc. Search Search ADVANCED SEARCH SCROLL This is a preprint and has not been peer reviewed. Data may be preliminary. 29 September 2025 V1 Latest version Share on Topological Descriptors and Quantum-Chemical Correlations in Skin Cancer Drug Design Using SMP -Polynomials Authors : Ali Asghar Talebi and Jaber Ramezani Tousi 0000-0001-7093-6777 [email protected] Authors Info & Affiliations https://doi.org/10.22541/au.175912045.50219731/v1 184 views 105 downloads Contents Abstract Introduction Preliminaries Main Results Application of indices for chemical compounds Conclusions Authors’ declaration References Information & Authors Metrics & Citations View Options References Figures Tables Media Share Abstract Skin cancer is one of the cancers that in addition to being dangerous, can also affect a person’s self-confidence and beauty. It is caused by the abnormal growth of skin cells. In this article, while providing a closed formula for calculating distance-based indices with the SMP -polynomial approach, the distance indices in the molecular diagram of skin cancer drugs have been calculated and then analyzed based on physical and chemical properties. Our results demonstrate strong correlations between these topological indices and critical physicochemical properties, including polarizability, molar volume, molar refractivity, boiling point, and enthalpy of vaporization. Notably, the vertex Harmonic mean Szeged index exhibited the highest correlation (r=0.981) with polarizability and molar refractivity, underscoring its predictive power. Linear regression models further validated the utility of these indices in drug property prediction, with significant statistical robustness (p<0.05). This work provides a novel mathematical framework for drug design optimization, bridging graph theory with pharmaceutical science. These findings underscore the utility of topological indices in predicting drug-like properties and offer a computationally efficient strategy for the rational design of novel anti-skin cancer agents. The methods presented here bridge graph theory with pharmaceutical science and open avenues for integrating topological descriptors with quantum-chemical calculations in future drug discovery pipelines. Ali Asghar Talebi Department of Mathematics, University of Mazandaran , Babolsar, 4741613534, Iran [email protected] Jaber Ramezani Tousi * Department of Mathematics, University of Mazandaran, Babolsar, 4741613534, Iran [email protected] (corresponding author) Abstract: Skin cancer is one of the cancers that in addition to being dangerous, can also affect a person’s self-confidence and beauty. It is caused by the abnormal growth of skin cells. In this article, while providing a closed formula for calculating distance-based indices with the SMP -polynomial approach, the distance indices in the molecular diagram of skin cancer drugs have been calculated and then analyzed based on physical and chemical properties. Our results demonstrate strong correlations between these topological indices and critical physicochemical properties, including polarizability, molar volume, molar refractivity, boiling point, and enthalpy of vaporization. Notably, the vertex Harmonic mean Szeged index exhibited the highest correlation (r=0.981) with polarizability and molar refractivity, underscoring its predictive power. Linear regression models further validated the utility of these indices in drug property prediction, with significant statistical robustness (p<0.05). This work provides a novel mathematical framework for drug design optimization, bridging graph theory with pharmaceutical science. These findings underscore the utility of topological indices in predicting drug-like properties and offer a computationally efficient strategy for the rational design of novel anti-skin cancer agents. The methods presented here bridge graph theory with pharmaceutical science and open avenues for integrating topological descriptors with quantum-chemical calculations in future drug discovery pipelines. Keywords : SMP -polynomial, Molecular graph, Topological indices, QSPR analysis. Mathematics Subject Classification 2020: 05C92. Introduction Facial skin health is very important. The skin is a very important part of the body that protects the internal organs of the body from abnormalities and damage from ex-ternal elements. Your skin is constantly exposed to various factors such as heat, humidity, pollution, dryness, etc., and how you are exposed to these factors affect your health. Unfortunately, some people take extreme precautions to care for their skin and ensure that it is soft and smooth, while they do not know the best ways to protect their skin. One of the most important ways to take care of your skin is to protect it from the sun. Prolonged exposure to the sun can lead to wrinkles, brown spots, and other skin problems. One of the worst side effects of staying in the sun is skin cancer [1, 2]. In recent times, graph theory has increasingly been applied in the medical field, with chemical graph theoreticians concentrating on calculating topological indices of drug structures [3, 4]. This approach helps them understand molecular properties and supports the process of drug development. [5-7]. Graph theory has wide applications in science, chemistry, mathematics, computer and statistics [8- 10]. Numerous investigations have focused on molecular graphs, particularly nanotubes and pharmaceutical structures, by employing topological indices [11- 16]. Topological indices are used to examine the relationships between structure and properties in chemical compounds, offering numerical parameters for Quantitative Structure-Property Relationship (QSPR) research. Numerous QSPR studies have utilized topological indices to analyze various drug structures. [17- 20, 30]. QSPR studies investigate the relationship between molecular structure and physical or chemical properties of materials. Statistical package for the social sciences ( SPSS ) software is one of the powerful tools for statistical analysis that can also be used in QSPR studies [21]. Several graph polynomials have been introduced to date, and these graph polynomials have yielded significant insights into chemical networks and their associated topological indices [22-25]. Also, the study of distance-based topological indices on molecular graphs has attracted the attention of researchers [26- 29]. In this article, various distance indices such as the Padmakar-Ivan index, the Mostar index, the sum connectivity index, the Szeged index, the harmonic mean index, and the connectivity index using SMP -polynomials were evaluated and then examined on some skin cancer drugs using linear regression so that researchers could understand their physical properties and chemical reactions. Preliminaries In the context of a molecular graph G = (V, E), the degree of a specific vertex s is indicated by d s . The relationship between two vertices z and w is represented by d s, and the typical shortest path distance between any two vertices u and v within G is expressed as d (u, v). Definition 1. In graph G , the distance from a vertex x to an edge e = uv is defined as [31]: (2.1) Moreover, if e = uv is an edge of, the following notation will be applied to certain sets of vertices and edges within : (2.2) Definition 2. In a graph G = (V, E) , the edge -SMP -polynomial and SMP -polynomial are expressed by the following relations [30]: (2.3) where Some operators, which are used further, are defined: (2.4) Definition 3. The vertex Pidmakar-Ivan index ( PI v (G) ) [15], vertex Mostar index ( Mo v (G) ) [26], vertex Szeged index ( Sz v (G) ) [27], vertex Harmonic mean Szeged index ( HMSz v (G) ), vertex Sum connectivity index ( DS v (G) ) [28] and vertex Connectivity index ( NG v (G) ) [16] are defined as follow: (2.5) Definition 4. The edge Pidmakar-Ivan index ( PI e (G) ) [15], edge Mostar index ( Mo e (G) ) [26], edge Szeged index ( Sz e (G) ) [27], edge Harmonic mean Szeged index ( HMSz e (G) ), edge Sum connectivity index ( DS e (G) ) [28] and edge Connectivity index ( NG e (G) ) [28] are defined as follow: (2.6) Proposition 1. Let G = (V, E) be a molecular graph. Then the Vertex Padmakar- Ivan, Vertex Mostar, Vertex Szeged, Vertex Harmonic mean Szeged, Vertex Sum connectivity and Vertex Connectivity indices are computed as: Proof. According to definition (2) and definition (3) we have: Similarly, we can prove the PI e (G), Mo e (G), Sz e (G), HMSz e (G), DS e (G) and NG e (G) indices for the molecular graph of G . Main Results The United States Food and Drug Administration ( FDA ) has approved several skin cancer drugs for treatment (Figure 1). In this section, after introducing the 2D -chemical structure of skin cancer drugs such as Dabrafenib, Fluorouracil, Binimetinib, Encorafenib, Dacarbazine, Trametinib, Glasdegib, Vemurafenib, Vismodegib, Imiquimod, Picato, Cobimetinib, the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are calculated using SMP -polynomial and edge SMP -polynomial. Fig. 1. 2D -chemical structure of skin cancer drugs. Theorem 1. Let B, E, Dab, F, T, Dac, G, Vem, C, I, Vis, and P be the molecular graphs of Encorafenib, Fluorouracil, Trametinib, Dacarbazine, Glasdegib, Vemurafenib, Cobimetinib, Imiquimod, Dabrafenib, Vismodegib, Picato, Binimetinib, respectively. Then the SMP -Polynomial and edge SMP -Polynomial of B, E, Dab, F, T, Dac, G, Vem, C, I, Vis, and P are calculated as follows: Proof. Using Definition (1) and according to Figure (1), for the molecular graph of Binimetinib for we have thirteen different edges that show in Table (1). Table 1. Thirteen different edges for the molecular graph of Binimetinib. Number of edges 6 1 2 3 3 2 3 (7,20) (8,19) (9,18) (11,16) (12,14) (13,13) Number of edges 1 3 1 1 2 1 Also for we have eighteen different edges that show in Table (2). Table 2. Eighteen different edges for the molecular graph of Binimetinib. Number of edges 6 1 1 2 1 1 1 2 1 (3,16) (25,2) (13,14) (20,7) (19,18) (15,11) (8,20) (3,24) (4,23) Number of edges 1 1 1 1 1 1 1 5 1 Therefore, the corresponding SMP -Polynomials are: Similarly, we can prove the SMP -Polynomial and edge SMP -Polynomial for the molecular graph of Encorafenib, Cobimetinib, Dabrafenib, Fluorouracil, Trametinib, Glasdegib, Vemurafenib, Picato, Imiquimod, Vismodegib and Dacarbazine. □ Theorem 2. Suppose SMP (B; x, y) and SMP e (B; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Binimetinib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), we have: Also by using the relations in Proposition (1) we have: Similarly, we can prove the PI e (B), Mo e (B), Sz e (B), HMSz e (B), DS e (B), and NG e (B) indices for the molecular graph of B . □ Theorem 3. Suppose SMP (E; x, y) and SMP e (E; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Encorafenib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations as in the proof of Theorem (2). □ Theorem 4. Suppose SMP (Dab; x, y) and SMP e (Dab; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Dabrafenib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 5. Suppose SMP(Dac; x, y) and SMPe(Dac; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Dacarbazine respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 6. Suppose SMP (F ; x, y) and SMP e (F ; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Fluorouracil respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 7. Suppose SMP (T ; x, y) and SMP e (T ; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Trametini respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 8. Suppose SMP (G; x, y) and SMP e (G; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Glasdegib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 9. Suppose SMP (Vem; x, y) and SMP e (Vem; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Vemurafenib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 10. Suppose SMP (I; x, y) and SMP e (I; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Imiquimed respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 11. Suppose SMP (Vis; x, y) and SMP e (Vis; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Vismodegib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations.□ Theorem 12. Suppose SMP (P ; x, y) and SMP e (P ; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Picato respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ Theorem 13. Suppose SMP (C; x, y) and SMPe(C; x, y) denotes the SMP -Polynomial and edge SMP -Polynomial of molecular graph of the Cobimetinib respectively; then the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices are computed as: Proof. According to Theorem (1), and by using the relations in Proposition (1) we can prove the above relations. □ The 3D SMP - polynomial plots plotted in Figure (2) allow us to examine the variations of this polynomial in the topological space of molecular graphs and evaluate its relationship with the structural properties of drugs. In these plots: The x and y axes represent variables related to the structure of the molecular graph, such as the number of edges, node degree, or SMP - polynomial coefficients. The z axis represents the SMP - polynomial value, which serves as a topological index for analyzing drug properties. This visualization helps to observe the variations in the SMP - polynomial among different drugs, identify similarities and differences between effective and less effective drugs, and optimize QSPR regression models for predicting the physicochemical properties of skin cancer drugs. Fig. 2. 3D SMP - polynomial of skin cancer drugs. The graphs in Figures (3, 4) provide a comparative view of topological changes in skin cancer drugs and help us better understand the relationships between the indices. Fig. 3. The relationship between topological indices Szeged and Padmakar-Ivan in the examined set of drugs. Fig. 4. A 2D graph depicting the correlation between pharmaceuticals and their topological indices. Application of indices for chemical compounds Since the correlation between any topological index with some physico-chemical characteristics of a chemical compound, should be checked in order to verify that in-dex’s efficiency (according to the regulations suggested by International academy of mathematical chemistry, hence in this article the correlation between physio-chemical features of skin cancer’s drugs and the Padmakar-Ivan, Mostar, Szeged, Harmonic mean Szeged, Sum connectivity, Connectivity indices, is investigated through QSPR calculations. The physio-chemical properties of skin cancer medications, including Enthalpy of vaporization ( EN ), Polarizability ( PO ), Molar volume ( MV ), Molar refractivity ( MR ) and Boiling point ( BP ) are listed in Table (3). Table 3. Physical properties of Skin Cancer’s drugs. Binimetinib 38.3 264.1 96.6 – – Encorafenib 53.2 371.7 134.1 – – Dabrafenib 50.5 359.9 127.4 653.7 96.3 Dacarbazine 18.3 122.6 46.2 456.3 71.6 Fluorouracil 10.2 84.6 25.9 – – Trametinib 56.1 353.1 141.5 – – Glasdegib 42.4 281 106.9 633.4 93.6 Vemurafenib 48.2 332.4 121.6 711.4 104 Imiquimod 28.2 187.7 71 456.7 71.7 Vismodegib 41.8 292.5 105.5 561.6 84.4 Picato 45.6 339.5 115.2 576.9 99.2 Cobimetinib 45.7 311.4 115.3 565.9 89.4 Table 4. the values of indices of the skin Cancer’s drugs. Binimetinib 775 447 3005 282.48 5.61 3.36 866 598 2560 209.68 5.75 1.99 Encorafenib 1188 770 5373 404.32 5.78 3.31 1301 929 4921 347.94 6.01 2.07 Dabrafenib 1120 706 4997 389.08 5.71 3.11 1317 939 4717 353.75 6.24 2.6 Dacarbazine 157 95 284 58.98 3.78 3.04 149 101 200 41.82 3.87 1.99 Fluorouracil 81 27 144 35.32 3 2.4 66 30 72 20.78 3.33 1.44 Trametinib 1331 857 6070 440.44 6.63 4.48 1423 961 6256 411.69 6.44 3.26 Glasdegib 763 401 3273 293.98 5.84 3.52 773 459 3153 274.17 5.15 1.95 Vemurafenib 1083 677 4754 383.36 6.07 3.78 1151 745 4919 363.81 5.87 2.55 Imiquimod 304 166 821 118.2 4.41 3.17 280 152 752 107.75 4.22 2.47 Vismodegib 675 377 2693 255.49 5.1 3.32 684 428 2353 221.42 5.07 2.32 Picato 912 576 3346 309.14 6.03 4.16 864 574 2791 229.39 5.76 2.25 Cobimetinib 930 584 3765 327.48 5.66 3.52 1002 652 4093 303.63 5.47 2.53 4.1. The QSPR analysis of the proposed method In this section, QSPR analysis is done for the topological indices calculated for molecular graphs in Table (4), in compare with the physio-chemical properties presented in Table (3). The correlation results, obtained through SPSS software, are shown in Table (5). Correlation is a statistical tool used to assess how strongly and in what direction two variables are related. This value is usually expressed as a correlation coefficient ( r ), which ranges from -1 to +1 ( r > 0 positive correlation, r < 0 negative correlation and r = 0 no correlation). Correlation and regression are closely related. Correlation tells us whether there is a relationship between two variables, but regression allows us to model and predict this relationship. If the correlation is strong ( r close to 1), the linear regression model will make better predictions. If the correlation is weak ( r close to 0), the linear regression model will be less efficient. Linear regression is a technique used to model the connection between a dependent variable and one or more independent variables. In this approach, the association between two variables is depicted as: P=B+A(TI). (4.1.1) where P (dependent variable) is the skin cancer drug property, B is the intercept constant, the regression coefficient is represented by A , which is the slope of the regression line, while TI stands for the topological index, an independent variable. This capability is calculated using SPSS , for twelve topological indices and physio-chemical characteristics of twelve skin cancer’s drugs. Different linear models for regression of topological indices and physio-chemical properties of the drugs are obtained through equation (1) as follows: Polarizability [ PO ] Molar Volume [ MV ] Molar refract [ MR ] Boiling Point [ BP ] Enthalpy [ EN ] In QSPR analysis, a table is typically presented that includes various statistical measures to assess the model’s quality and the strength of the relationship between molecular features and drug properties. The following provides an explanation of each column in Tables (6-16) along with the appropriate value ranges: Regression Coefficients ( A and B ): These represent the slope and intercept of the linear regression equation, determining the relationship between variables. Correlation Coefficient ( R ): Ranges from -1 to +1. strong correlation (ideal for modeling). moderate correlation. weak correlation (model is less reliable). Coefficient of Determination ( R 2 ): The value ranges from 0 to 1, indicating the extent to which the model accounts for the variance in the dependent variable. strong and reliable model for prediction. acceptable but not ideal model. weak and unstable model. Standard Error of Estimate ( SE ): Reflects the model’s average error in predicting P within a linear regression frame- work. A lower SE value indicates higher model accuracy. Ideally, SE should be small relative to the data range. Fisher’s F -Test ( F ): Evaluates the statistical significance of the regression model. A higher F value indicates a more robust model. Typically, F > 10 suggests a statistically significant model. P -Value ( P ): Indicates the likelihood that the results observed are a result of random chance. If P 0.05, the model is unreliable. Significance Indicator: Determines whether the model is statistically valid. If P 0.05, the model lacks significance and cannot be trusted. Table 5. Correlation coefficients of physical properties of drugs. PI v 0.974 0.964 0.975 0.898 0.936 Mo v 0.959 0.950 0.960 0.863 0.920 Sz v 0.957 0.940 0.958 0.930 0.922 HMSz v 0.981 0.971 0.981 0.917 0.933 DS v 0.960 0.938 0.960 0.869 0.955 NG v 0.740 0.696 0.741 0.448 0.706 PI e 0.959 0.948 0.959 0.890 0.891 Mo e 0.939 0.933 0.939 0.858 0.862 Sz e 0.937 0.907 0.937 0.920 0.889 HMSz e 0.966 0.941 0.960 0.932 0.889 DS e 0.964 0.960 0.964 0.865 0.924 NG e 0.693 0.624 0.692 0.321 0.295 In Tables [6-17] bold numbers represent the highest correlation value. Table 6. The linear QSPR model for PI v (G ) is characterized by its statistical parameters ( SP ). PO 12 0.034 13.742 0.974 0.950 3.32 188.161 0.000 significant MV 12 0.224 101.054 0.964 0.929 26.544 130.508 0.000 significant MR 12 0.085 34.720 0.975 0.950 8.339 189.594 0.000 significant BP 12 0.230 406.197 0.898 0.806 42.74 24.945 0.002 significant EN 12 0.032 64.784 0.936 0.876 4.606 42.380 0.001 significant Table 7. The linear QSPR model for Mo v (G) is characterized by its SP . PO 12 0.50 16.216 0.959 0.920 4.171 115.551 0.000 significant MV 12 0.333 117.273 0.950 0.903 30.968 93.232 0.000 significant MR 12 0.126 40.954 0.960 0.921 10.486 116.221 0.000 significant BP 12 0.340 424.964 0.863 0.745 49.025 17.520 0.006 significant EN 12 0.049 66.944 0.920 0.846 5.134 32.944 0.001 significant Table 8. The linear QSPR model for Sz v (G) is characterized by its SP . PO 12 0.007 17.901 0.957 0.917 4.268 109.934 0.000 significant MV 12 0.045 129.810 0.940 0.884 33.948 75.902 0.000 significant MR 12 0.017 45.209 0.958 0.917 10.744 110.230 0.000 significant BP 12 0.49 429.294 0.930 0.865 35.661 38.452 0.001 significant EN 12 0.007 69.042 0.922 0.850 5.06 34.096 0.001 significant Table 9. The linear QSPR model for HMSz v (G ) is characterized by its SP . PO 12 0.102 11.848 0.981 0.963 2.854 258.113 0.000 significant MV 12 0.679 88.357 0.971 0.943 23.851 164.022 0.000 significant MR 12 0.257 29.948 0.981 0.963 7.17 259.980 0.000 significant BP 12 0.689 392.969 0.917 0.842 38.606 31.927 0.001 significant EN 12 0.094 63.555 0.933 0.871 4.701 40.437 0.001 significant Table 10. The linear QSPR model for DS v (G) is characterized by its SP . PO 12 12.819 -28.088 0.960 0.921 4.156 116.471 0.000 significant MV 12 84.384 -172.334 0.938 0.881 34.381 73.750 0.000 significant MR 12 32.313 -70.714 0.960 0.921 10.466 116.692 0.000 significant BP 12 93.751 77.764 0.869 0.755 48.093 18.440 0.005 significant EN 12 13.895 14.784 0.955 0.913 3.863 62.770 0.000 significant Table 11. The linear QSPR model for NG v (G) is characterized by its SP . PO 12 19.381 -26.617 0.740 0.548 9.935 12.134 0.006 significant MV 12 122.725 -146.007 0.696 0.485 71.397 9.421 0.012 significant MR 12 48.876 -67.086 0.741 0.549 25.024 12.164 0.006 significant BP 12 106.361 209.776 0.448 0.200 86.791 1.504 0.266 insignificant EN 12 22.601 10.745 0.706 0.499 9.262 5.964 0.050 significant Table 12. The linear QSPR model for PI e (G) is characterized by its SP . PO 12 0.029 15.662 0.959 0.919 4.206 113.491 0.000 significant MV 12 0.196 113.944 0.948 0.898 31.809 87.843 0.000 significant MR 12 0.074 39.566 0.959 0.919 10.588 113.794 0.000 significant BP 12 0.198 422.833 0.890 0.793 44.205 22.928 0.003 significant EN 12 0.027 67.992 0.891 0.793 5.946 23.034 0.003 significant Table 13. The linear QSPR model for Mo e (G) is characterized by its SP . PO 12 0.041 17.661 0.939 0.881 5.095 74.142 0.000 significant MV 12 0.271 126.482 0.933 0.870 35.918 66.735 0.000 significant MR 12 0.102 44.604 0.939 0.881 12.832 74.285 0.000 significant BP 12 0.271 440.019 0.858 0.736 49.895 16.707 0.006 significant EN 12 0.037 70.231 0.862 0.743 6.635 17.318 0.006 significant Table 14. The linear QSPR model for Sz e (G) is characterized by its SP . PO 12 0.007 19.684 0.937 0.878 5.165 71.895 0.000 significant MV 12 0.043 143.498 0.907 0.822 41.946 46.267 0.000 significant MR 12 0.017 49.706 0.937 0.878 13.013 71.964 0.000 significant BP 12 0.048 440.084 0.920 0.847 37.971 33.206 0.001 significant EN 12 0.006 70.963 0.889 0.790 6.001 22.507 0.003 significant Table 15. The linear QSPR model for HMSz e (G) is characterized by its SP . PO 12 0.106 14.343 0.966 0.933 3.828 139.056 0.000 significant MV 12 0.697 107.535 0.941 0.886 33.581 77.788 0.000 significant MR 12 0.268 36.247 0.966 0.933 9.652 138.988 0.000 significant BP 12 0.736 402.590 0.932 0.868 35.209 39.598 0.001 significant EN 12 0.095 66.347 0.889 0.791 5.98 22.704 0.003 significant Table 16. The linear QSPR model for DS e (G) is characterized by its SP . PO 12 13.829 -32.937 0.964 0.929 3.947 130.208 0.000 significant MV 12 92.718 -213.120 0.960 0.921 27.936 116.851 0.000 significant MR 12 34.863 -82.956 0.964 0.929 9.925 130.883 0.000 significant BP 12 95.248 81.101 0.865 0.749 48.653 17.881 0.006 significant EN 12 13.705 17.425 0.924 0.854 5.004 34.998 0.001 significant Table 17. The linear QSPR model for NG e (G) is characterized by its SP . PO 12 21.517 -9.292 0.693 0.480 10.662 9.218 0.013 significant MV 12 130.555 -23.277 0.624 0.390 77.731 6.385 0.030 significant MR 12 54.216 -23.283 0.692 0.479 26,882 9.206 0.013 significant BP 12 113.967 311.159 0.321 0.103 91.94 0.687 0.439 insignificant EN 12 14.153 55.762 0.295 0.087 12.496 0.574 0.477 insignificant 4.2. Perspectives for Drug Design Using Optimized TI : The strong correlations observed between distance-based topological indices and key physicochemical properties suggest a promising avenue for rational drug design. Based on our QSPR models, the following strategies can be proposed for designing new molecular structures with optimized properties for skin cancer treatment: Index-Guided Molecular Optimization: Using the regression equations derived in Section 4, researchers can predict properties such as polarizability, molar volume, and molar refractivity for hypothetical molecular structures before synthesis. For instance, the vertex Harmonic Mean Szeged index showed the highest correlation with polarizability ( r =0.981). Designing molecules with targeted HMSz v values may allow fine-tuning of polarizability, which influences solubility and membrane permeability. Scaffold Modification via Functional Group Substitution: Molecular scaffolds of existing skin cancer drugs (e.g., Dabrafenib, Vemurafenib) can be modified by substituting functional groups to alter topological indices favorably. For example, introducing polar groups may increase the Szeged index, which correlates strongly with boiling point and enthalpy of vaporization key properties for drug stability and formulation. Virtual Screening of Chemical Libraries: The derived QSPR models can be used to screen large chemical databases (e.g., ZINC, PubChem) for compounds with desirable index ranges. Candidates showing optimal index values can be shortlisted for synthesis and biological testing, accelerating the discovery of novel anti‐skin cancer agents. Multi-Index Optimization Framework: A multi-criteria optimization approach can be employed where several indices are simultaneously optimized to balance multiple properties. For example, a compound with high HMSz v (for polarizability) and moderate PI v (for molar volume) may exhibit improved bioavailability and efficacy. Integration with Machine Learning: Topological indices can serve as features in machine learning models (e.g., random forests, neural networks) to predict not only physicochemical properties but also biological activity and toxicity profiles, enabling a more comprehensive drug design pipeline. ADMET -Oriented Design: Future work should incorporate ADMET (Absorption, Distribution, Metabolism, Excretion, and Toxicity) endpoints into QSPR models using TI . This would help in designing molecules that are not only effective but also drug-like and safe. These strategies highlight the potential of TI as low-cost, high-throughput computational tools in the early stages of drug development, providing a complementary approach to traditional molecular modeling methods. The following MATLAB code can also be used to calculate the vertex Padmakar-Ivan index ( PIv(G) ). Function [ PIv ] = calculate vertex Padmakar-Ivan Index(adjMatrix) degrees = sum(adjMatrix, 2); PIv = 0; n = size(adjMatrix, 1); for i = 1:n for j = i+1:n if adjMatrix(i,j) ~= 0 PIv = PIv + degrees(i) * degrees(j); end end end end. Conclusions In this article, a closed formula for calculating distance-based topological indices using SMP-polynomials was presented. In addition, the behavior of these polynomials in skin cancer drugs was investigated and their applications were also identified. Table (5) presents the correlated values of the physio-chemical properties of skin cancer medications with the specified distance-based topological indices. A positive relationship has been identified between the physical and chemical characteristics of skin cancer medications and topological indices based on distance. Distance-based topological indices, especially the harmonic mean Szeged index ( HMSz ), showed very strong correlations with properties such as polarizability ( PO ), molar volume ( MV ), and molar refraction ( MR ) (correlation value r > 0.97). The linear regression models presented for the different indices were statistically significant (p < 0.05). The regression model for the PI v index with the polarizability feature had a coefficient of determination ( R² = 0.950) and a low standard error ( SE = 3.32), indicating the high accuracy of the model in predicting this feature. This highlights the potential significance of topological indices in the QSPR analysis of skin cancer medications. The findings of this study hold promise for more effectively producing, developing, and improving skin cancer drugs. While TI provide a powerful graph-theoretical framework for predicting physicochemical properties, their significance can be further illuminated through a quantum-chemical lens. The molecular structures studied in this work such as Dabrafenib, Vemurafenib, and Trametinib contain conjugated π -systems, heteroatoms, and functional groups that directly influence electronic properties like charge distribution, frontier molecular orbital energies, and reactivity. These electronic features are inherently quantum-mechanical in nature and can be computationally evaluated using methods such as density functional theory ( DFT ) . The strong correlations observed between TI (e.g., HMSz v ) and properties like polarizability ( PO ) suggest that these indices may serve as low-cost proxies for quantum-mechanical descriptors . For instance: The harmonic mean Szeged index ( HMSz v ), which captures global distance-based symmetry and branching, may indirectly reflect the electron density distribution and molecular softness key concepts in DFT -based reactivity theory. The Szeged index ( Sz v ), related to molecular volume and shape, may correlate with quantum-chemical volume descriptors such as the molecular electrostatic potential ( MEP ) surface area. We propose that future studies integrate topological indices with quantum-chemical descriptors such as HOMO-LUMO gap, dipole moment, Fukui functions, or molecular electrostatic potential ( MEP ) maps to build multi-scale QSPR models. Such hybrid models could enhance predictive accuracy while providing deeper insight into the electronic underpinnings of drug-receptor interactions. Furthermore, the SMP -polynomial framework introduced here can be extended to encode quantum-topological features by incorporating weights derived from atomic partial charges or bond orders computed at the DFT level. This would bridge discrete graph theory with continuous electronic structure theory, offering a more comprehensive tool for in silicone drug design. Authors’ declaration manuscript are ours. • No human studies are present in the manuscript. Data Availability Statement Data is contained within the article. Acknowledgments The authors are very grateful to the editor and the anonymous referee(s) for their comments and suggestions which led to the present improved version of the manuscript. Conflicts of Interest The authors declare no conflict of interest. References 1. [1] Simoes, M.C.F., Simoes, j.j.S., Pais, A.A.C.C. 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Keywords qspr analysis smp-polynomial graph theory molecular graph topological indices Authors Affiliations Ali Asghar Talebi University of Mazandaran Faculty of Basic Science View all articles by this author Jaber Ramezani Tousi 0000-0001-7093-6777 [email protected] University of Mazandaran View all articles by this author Metrics & Citations Metrics Article Usage 184 views 105 downloads .FvxKWukQNSOunydq8rnd { width: 100px; } Citations Download citation Ali Asghar Talebi, Jaber Ramezani Tousi. Topological Descriptors and Quantum-Chemical Correlations in Skin Cancer Drug Design Using SMP -Polynomials. Authorea . 29 September 2025. DOI: https://doi.org/10.22541/au.175912045.50219731/v1 If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download. For more information or tips please see 'Downloading to a citation manager' in the Help menu . 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