Graph Energy Variants and Topological Indices in Platinum Anticancer Drug Design: Mathematical Insights and Computational Analysis with DFT and QTAIM

. In this paper, we present a new approach that explores the application of graph energy variants in chemistry, specifcally in the development of platinum anticancer drugs. While previous energy variants have been proposed without considering their direct relevance to chemistry, our study focuses on two key aspects. First, we investigate the correlation between seven degree-based and four distance-based topological indices and their corresponding energies in platinum anticancer drugs. Furthermore, we mathematically analyze the properties of these energies, establishing upper and lower bounds that can be generalized to other structures. Second, we examine the possibility of utilizing the energies of these topological indices as structural descriptors. Our research showcases promising results, suggesting potential improvements in the future manufacturing of anticancer drugs. In addition, we employ density functional theory ( DFT ) calculations to optimize the molecular structures of platinum anticancer drugs and identify local reactive sites using Fukui functions. Quantum theory of atoms in molecules ( QTAIM ) was carried out at the bond critical point ( BCP ) , to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. Overall, this study presents an innovative approach that bridges graph energies, topological indices, and DFT with the properties (physical and chemical) of platinum anticancer drugs, ofering insights into their molecular properties and potential for enhanced drug design.


Introduction
Physicochemical properties and chemical structure characteristics can be analyzed with the help of topological indices. Topological indices come in diferent types (degree, distance, spectral, and mixed invariants). We consider degree-based, distance-based, and their spectral invariants in the context of anticancer drug molecular structures. In general, a topological index is calculated by converting a molecular structure into a numerical value. As a practical matter, chemical compounds can be visualized as graphs where vertices are elements and edges are bonds. Te study considers the underlying graphs of platinum anticancer drugs as chemical compounds. An abnormal/dead cell count increases rapidly in the body when a person has cancer. Carcinogen is a substance that causes cancer. Tere are certain molecules in tobacco smoke that are carcinogens, which are harmful substances, and other parts of the body can be afected by it. In addition to lumps, abnormal bleeding, prolonged coughs, weight loss, and anxiety, this deadly disease can present with many other symptoms. Tobacco use, being obese, eating poorly, being lazy, and drinking heavily can cause cancer. Surgery, radiotherapy, chemotherapy, hormone therapy, targeted therapy, and other treatments are available to cure cancer. A chemical compound's bioactivity is determined using topological indices and their spectral invariants as part of quantitative structure-property relationships (QSPRs). Chemical graph theory deals with the graphs of chemical systems that represent chemical systems in mathematical chemistry, see, for example [1][2][3][4][5].
To comprehensively understand the signifcance and diverse applications of various types of topological indices, such as those based on degree, distance, entropy, or their combinations, it is crucial to conduct a thorough literature review of signifcant research in this feld. By exploring the existing body of knowledge, we can gain valuable insights into the advancements and novel fndings related to topological indices. Tis literature review will provide a broader perspective on the applications of these indices in diferent domains, including chemoinformatics, drug design, material science, network analysis, and system biology. In addition, it will allow us to identify gaps, challenges, and potential areas for further exploration, paving the way for future research and the development of more robust and versatile topological indices. By synthesizing and analyzing recent studies, we can elucidate the current state of the art and contribute to the advancement of this feld. In [6], the study presents versatile methods based on events to defne information theory-based indices (IFIs). Tese indices are calculated using generalized matrices (Q and F) that capture the relationships among molecular substructures. Various information theory measures, such as Shannon's entropy, mutual entropy, conditional entropy, and joint entropy, are applied to compute IFIs. Te research expands on the notion of connected subgraphs and introduces additional events, including terminal paths, quantum subgraphs, and walks of varying lengths, to defne IFIs. Moreover, magnitude-based IFIs are introduced, incorporating the signifcance of magnitudes in mutual, conditional, and joint entropy-based IFIs. Te study explores information-theoretic parameters as a means to measure dissimilarity in the structural information of molecules. Notably, the proposed IFIs exhibit comparable or improved predictive capability compared to DRAGON's molecular descriptors in forecasting the physicochemical properties of specifc derivatives. In [7], the authors aimed to explore the potential of informationtheoretic modeling in encoding chemical structures, specifcally focusing on Markov approximation models. Teoretical aspects related to data statistics are discussed, laying the foundation for subsequent discussions on chemical structure encoding. Te signifcance of topological indices in understanding and representing chemical structures is emphasized, ofering diverse applications in the feld. Te introduction of GT-STAF information indices, utilizing Shannon's coding theorems, provides valuable insights into encoding chemical structures by capturing statistical patterns of molecular fragments. Further investigations examine the extension of these indices to higher dimensional communication system models and diferent source originator algorithms. In [8], the authors proposed a novel mathematical method that involves utilizing graph derivatives to encode chemical structure information. By considering the derivative of a molecular graph (G) with respect to a specifc property (S), they introduce graph derivative indices as independent indicators containing signifcant structural details. Te results indicate that these indices have the potential to be valuable in QSPR/QSAR and drug design studies. Tey ofer mathematical models that are easier to interpret and more robust than existing approaches found in the literature. Tis research highlights the usefulness of graph derivative indices in capturing essential information about chemical structures and their applications in various molecular research felds. In [9], a new approach for calculating molecular descriptors (MDs) is introduced using a set of aggregation operators (AOs). Te MD-LOVIs software, developed based on this approach, allows for efcient MD calculation using atomic weight vector and local vertex invariants. Comparative analyses demonstrate that MD-LOVIs outperforms other software in terms of calculation speed and information content. QSAR modeling using MD-LOVIs indices with AOs shows better performance than traditional methods and even outperforms other advanced QSAR methodologies. Tis research emphasizes the efectiveness of MD-LOVIs in generating improved molecular descriptors by incorporating diverse chemical information through the utilization of AOs.
Since the discovery of the Wiener index in 1947 [10] as a method for modeling the boiling point of parafn, numerous topological indices have been proposed and studied in the felds of mathematics and chemistry. Tese indices have found wide-ranging applications in structure-property and structure-activity modeling. As a result of the Zagreb indices [11], Randić index [12], which is based on the degree, Estrada index [13], which is based on the spectrum of a graph, and Hosoya index [14], which is based on matching provided quantitative measures of molecular branching, a variety of similar indices have been developed. Novel descriptors are designed to improve the accuracy of molecular property modeling. A molecular graph shows the unsaturated hydrocarbon structures of molecules. Covalent bonds between nonhydrogen atoms are represented by their edges, while their vertices represent nonhydrogen atoms. A number of vertex-degree-based topological indices are examined in this study, including the frst and second Zagreb indices [15], the hyper-Zagreb index [16], the sigma index [17], the inverse symmetric deviation index [18], the inverse sum deviation index [19], and the Albertson index [20]. Topological indices based on distance are the Wiener index and Schultz index [21], Harary index [22], and Gutman index [23]. Bond additives and degree-based topological indices can provide accurate assessments of the physicochemical and bioactive properties of chemical structures. Zagreb indices can be used to predict efective cancer treatments [24]. Te frst hyper-Zagreb index is the preferred method for estimating benzenoid hydrocarbon boiling points [25]. Indeg indices are best applied to predict topological polar areas [26]. In addition, monocarboxylic acid vaporization and sublimation enthalpies can be calculated using the inverse sum deviation index [27]. In addition to Albertson and sigma, diferent degree-based irregularity indices can predict the physicochemical properties of octane isomers [28]. A Wiener index was frst used by Wiener in QSPR who found that it aligned well with the boiling points of alkanes [29]. Since the Wiener Index has been developed, diferent chemical and physical properties of molecules have been explained. In addition, the relationship between the molecule structure and biological activity has been established [30]. Te Schultz index was investigated in [21] to 2 Journal of Mathematics predict the boiling point of alkyl alcohol in order to predict its suitability. As indicated in Table 1, these indices are  expressed mathematically and are shown with mathematical  expressions.  Te following are the graph theoretical notations used in  Table 1. A molecular graph is a simple graph ζ � (V, E). Its vertices and edges represent the atoms and the bonds, respectively. Note that hydrogen atoms are omitted. For any graph, d(u) represents the degree of the vertex u ∈ V(ζ) in the graph, while d(u, v) represents the distance of the shortest path from the vertex u to the vertex v. A degreebased topological index quantifes connections between molecular structures both locally and globally. An atom's degree or vertex's degree in a molecular graph is used to calculate these indices. Distance-based topological indices are graph theoretical measures that capture the spatial relationships and distances between atoms or vertices in a molecular structure. Tese indices provide information about the molecule's connectivity patterns and geometric properties. Unlike degree-based indices that focus on atom local connectivity, distance-based indices consider pairwise distances between atoms in the molecular graph. Tese indices quantify the spatial arrangement and distribution of atoms, providing insights into molecules' 3D structure.
In mathematical terms, graph spectra utilize linear algebra, namely, the well-developed theory of matrices, in an efort to unlock a number of secrets pertaining to graph theory. As part of spectral graph theory, mathematicians study the relationships between graph combinatorics and accompanying matrices. Entropy measures, such as molecular entropy, confgurational entropy, and thermodynamic entropy, are important properties in various scientifc disciplines. Quantitative structure-property relationship (QSPR) models have been developed to predict these entropy measures based on molecular descriptors and properties. Mu and He [31] aimed to predict the thermodynamic entropy of small organic molecules using QSPR modeling. Tey investigated various descriptors and modeling techniques to establish accurate relationships between molecular properties and thermodynamic entropy. Hosseini and Shafei. [32] focused on predicting molecular entropy using topological indices. Tey explored the relationship between the molecular structure and entropy, providing insights into the development of QSPR models for this property. In [33], the authors focused on QSPR modeling for predicting thermodynamic entropy in metal-organic frameworks. Teir study explored the use of descriptors and modeling approaches specifc to this class of materials, providing valuable insights for predicting entropy in this context. Dialamehpour and Shafei [34] investigated the use of topological indices in QSPR models for predicting thermodynamic entropy. By incorporating structural features captured by topological indices, they were able to achieve accurate entropy predictions for a range of small organic molecules. Te study emphasized the importance of considering entropy in the design and optimization of chemical processes and materials. Manzoor et al. [35] developed QSPR models for predicting confgurational entropy of organic compounds. Teir study employed molecular descriptors to establish relationships between the molecular structure and confgurational entropy, facilitating predictions in this area. Tese studies collectively highlight the use of QSPR models to predict entropy measures. Tey ofer valuable methodologies, datasets, and descriptors for researchers interested in developing predictive models for entropy. In a recent work [36], p− splitting and p− shadow graphs built on any regular graph are examined in terms of the inverse sum index, (ISI) energies, and (ISI) spectral radii. Comparisons of the energies and spectral radii of newly created graphs with those of the base graph are the only things that may be questioned. As a result of analyzing these two graph operations, they derive new relationships between the (ISI) energies and the spectral radius of the new graph and the original graph.
A defnition of spectral invariant topological indices is provided in the following paragraph. Te adjacency matrix where the eigenvalue λ 1 is called the spectral radius of ζ. A comprehensive explanation of A(ζ) can be found in [37]. Te energy of ζ can be expressed as follows: Energy plays a crucial role in theoretical chemistry as well as in mathematics. Using tracing functions in linear algebra, we can determine the trace norms of symmetrical matrices, and in mathematical chemistry, tracing functions can also determine molecule electron energies [38]. It is discussed in detail in [39], as well as ongoing research in the feld. Teir eigenvalues can be indexed from the largest to the smallest as τ 1 ≥ τ 2 ≥ ··· ≥τ n . An eigenvalue with the smallest eigenvalue is called a spectrum eigenvalue, while the largest eigenvalue τ 1 is called an eigen spectral radius. Te T.I.energy of ζ is defned by Te topological index adjacency matrix (T.I.-matrix) of a graph ζ is a square matrix of order n, and it is defned in Table 2 for diferent vertex degree and distance-based topological indices. As part of this work, several chemical graph structures of platinum anticancer drugs are examined for physical characteristics, chemical reactions, and topological indices, see Figure 1.
In the existing literature, numerous variants of graph energy have been proposed [40][41][42][43][44][45]. However, it appears that most of these novel energies have been introduced without considering their specifc utility in chemistry or other relevant felds. Tis study, in contrast, follows a more deliberate approach, drawing inspiration from a previous study [46] that introduces a suitably constructed matrix to generate a novel graph energy based on the eigenvalues of the neighborhood inverse (NI) sum index. Te NI index demonstrates predictive potential and can efectively discriminate between isomers, making it a valuable molecular Journal of Mathematics structural descriptor. Similarly, another study [47] provides statistical analyses of various anticancer drugs, presenting their correlation coefcients with the ISI− energy and ISI− index, as well as their applications in quantitative structureproperty relationships. Te objective of the current study is to investigate the feasibility of utilizing the energies associated with certain distance-based and degree-based topological indices as structural descriptors. Crucially, establishing the chemical signifcance of graph invariants necessitates their correlation with experimental properties using benchmark datasets. In addition, we employ density functional theory (DFT) calculations to optimize the molecular structures of platinum anticancer drugs and identify local reactive sites using Fukui functions. Quantum theory of atoms in molecules (QTAIM) was carried out at the bond critical point (BCP), to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt

Mathematical expression Vertex-degree adjacency matrices
First Zagreb adjacency matrix Inverse sum indeg adjacency matrix

Distance adjacency matrices
Wiener adjacency matrix otherwise.
atoms and their bond atoms. Tis study presents an innovative approach that bridges graph energies, topological indices, and DFT with the properties (physical and chemical) of platinum anticancer drugs, ofering insights into their molecular properties and potential for enhanced drug design. In this context, we perform regression analysis on ten platinum anticancer drugs. Te theoretical values of drug energies are calculated using custom MATLAB codes. We observe correlations between the topological indices, energies, and experimental properties of the platinum anticancer drugs. Specifcally, this work focuses on ten platinum anticancer drugs: cisplatin, carboplatin, oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, spiroplatin, picoplatin, and satraplatin. Tese drugs are employed in the treatment of patients with chronic myelogenous leukemia, inoperable metastatic breast cancer, and small-cell lung cancer. Te study considers ten physicochemical properties associated with these anticancer drugs, which have not been extensively explored in the literature. It is worth noting that most properties exhibit higher correlation coefcients when considering the topological indices (energy), as opposed to the topological indices alone. Tis fnding suggests that incorporating the energy component of the topological index, rather than relying solely on the topological index itself, yields more accurate predictions of the physicochemical properties for any given molecular structure. Tis observation aligns with previous research [48][49][50][51], wherein the majority of studies focus on degree-based or distancebased topological indices and evaluate their signifcance. However, there is a lack of direct comparison between the energy and topological indices to determine the most accurate predictor. Tis study aims to address this gap in the literature and presents evidence supporting the superiority of considering the energy of the topological index for improved prediction of physicochemical properties. In summary, this study introduces a deliberate approach to investigating the energies associated with topological indices in the context of platinum anticancer drugs. By considering a diverse set of physicochemical properties and employing regression analysis, we establish correlations between these properties and the topological indices and energies. Te fndings demonstrate the potential of incorporating the energy component of the topological index in accurately predicting the physicochemical properties of molecular structures.
In our study, we employed linear regression models to assess the predictive capabilities of degree-and distancebased topological indices (energies). Statistical parameters were generated using Excel and MATLAB statistical functions to evaluate the performance of these models. With this foundation, we can now proceed with the subsequent sections outlined in the paper. Section 2 of the paper focuses on establishing fundamental mathematical properties by calculating crucial bounds of the energies of topological indices, specifcally for platinum anticancer drugs. We provide illustrative examples to support our fndings. It should be noted that similar examples can be extended to other graph types and energies, showcasing the versatility and applicability of our approach. In Section 3, we delve into the examination of diferent topological indices and their corresponding energies as molecular structural descriptors. We assess their contributions and efectiveness in characterizing the structural features of platinum anticancer drugs, shedding light on their potential implications for drug design and development. Section 4 of the paper introduces the utilization of density functional theory (DFT) calculations to optimize the molecular structures of platinum anticancer drugs. In addition, we utilize Fukui functions to identify the local reactive sites within these optimized structures. Tis analysis provides insights into the chemical reactivity and potential mechanisms of action of these drugs. Furthermore, quantum theory of atoms in molecules (QTAIM) was carried out at the bond critical point (BCP), to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. Finally, Section 5 concludes the paper, summarizing the key fndings and contributions of our study, as well as suggesting potential future directions for research in the feld. By following this structured approach, we aim to provide a comprehensive and coherent analysis of the predictive capabilities of topological indices (energies) for platinum anticancer drugs, furthering our understanding of their structural properties and potential applications in drug discovery and development.

Notation 1.
In the remainder of this paper, we will use the following abbreviations: cisplatin (Cis p ), carboplatin , and satraplatin (Sat p ).

Topological Energy of Graphs
Graph energies are numerical quantities associated with molecular graphs that refect various properties of molecules. Topological indices are a class of graph invariants or descriptors that encode topological information about a molecular graph and can be used to model a variety of physicochemical properties of molecules. Tere are many diferent types of topological indices, and each corresponds to a diferent graph energy. In [20], the author introduced the Albertson energy via the Albertson adjacency matrix by using the Albertson topologic index and obtained lower and upper bounds for it. Tere are many other topological indices and corresponding graph energies, and each has its own unique interpretation and applications in chemistry and related felds. In this section, we defne the energies of all the topological indices mentioned in this paper and show the lower and upper bounds of these energies on drug models (cisplatin, carboplatin, oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, spiroplatin, picoplatin, and satraplatin). In Table 2, we defne the sigma adjacency matrix (σA), Albertson adjacency matrix (irrA), frst Zagreb adjacency matrix (M 1 A), second Zagreb adjacency matrix (M 2 A), hyper-Zagreb adjacency matrix (HMA), inverse sum indeg adjacency matrix (ISIA), and inverse symmetric deg adjacency matrix (ISDIA) and calculate the energy over these matrices in this section. For simplicity, let us denote sigma energy by ε 1 , Albertson energy by ε 2 , frst Zagreb energy by ε 3 , second Zagreb energy by ε 4 , hyper-Zagreb energy by ε 5 , inverse sum indeg energy by ε 6 , and inverse symmetric deg energy by ε 7 . On the other hand, we have where if u and v are adjacent vertices, it is denoted by u ∼ v.
In this section, the following lemma is provided on the graphs of cisplatin, carboplatin, oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, spiroplatin, picoplatin, and satraplatin drugs. Tis is an advantage of studying on special graphs. Since the sigma adjacency matrix (σA), Albertson adjacency matrix (irrA), frst Zagreb adjacency matrix (M 1 A), second Zagreb adjacency matrix (M 2 A), (HMA) hyper-Zagreb adjacency matrix, inverse sum indeg adjacency matrix (ISIA), and inverse symmetric deg adjacency matrix (ISDIA) are real symmetric matrices, eigenvalues of these matrices are real and can be arranged, respectively, as In nonincreasing order, as can be understood from here, λ (1) i is an eigenvalue of the sigma adjacency matrix, λ (2) i is an eigenvalue of the Albertson adjacency matrix, λ (3) i is an eigenvalue of the frst Zagreb adjacency matrix, λ (4) i is an eigenvalue of the second Zagreb adjacency matrix, λ (5) i is an eigenvalue of the hyper-Zagreb adjacency matrix, λ (6) i is an eigenvalue of the inverse sum indeg adjacency matrix, and λ (7) i is an eigenvalue of the inverse symmetric deg adjacency matrix. Hence, we have the following result.

Lemma 2. For the graphs of cisplatin, carboplatin, oxaliplatin, nedaplatin, heptaplatin, and lobaplatin drugs, we have
Platinum anticancer drugs energies are as follows: For example, for the above results, we consider the graph of cisplatin, where 1 ≤ i, (a) ≤ 7, λ 1 is the spectral radius of ζ, and n is the number of vertices.
(1) 2 (Cauchy − Schwarz inequality), Hence, Tis completes the proof. We note that the upper bound obtained in Teorem 5 is better than the upper bound obtained in Teorem 3. We can show it with an easy example as a continuous step of the previous one.

Regression Model and Applications to Platinum Anticancer Drugs
Te objective of this section is to establish a quantitative structure-property-activity (QSPR) relationship between the topological indices (energies) and specifc physicochemical properties of platinum anticancer drugs, with the aim of evaluating their efectiveness. To achieve this, a statistical analysis was conducted using the Excel program to compare the physicochemical properties with the topological indices (energies). In our study, we considered ten representative physicochemical properties, and their modeling was based on the analysis of seven degree-based and four distancebased topological indices, along with their corresponding energies. To obtain the optimized geometries of the investigated anticancer drugs, we performed DFT calculations using the DMol3 module in version 8.0 of Material Studio from BIOVIA. By integrating the statistical analysis, topological indices (energies), and DFT calculations, we aimed to establish a robust relationship between the structural characteristics of platinum anticancer drugs and their physicochemical properties. Tis comprehensive approach provides valuable insights into the potential correlations and contributions of the topological indices (energies) to the observed physicochemical properties, advancing our understanding of the underlying mechanisms and informing the efectiveness assessment of these drugs. Tey are as follows: complexity (C), topological polar surface area  Table 3. Energy is computed in an efcient manner with the aid of a MATLAB program. It is described (see Algorithm 2). In our study, we employed MATLAB to calculate the vertex and distance energies of molecules using various mathematical expressions. Tese energy calculations serve as the foundation for determining the values of degree and distance-based energy topological indices. To facilitate this process, we utilized diferent functions specifcally designed to compute these indices accurately and efciently. By leveraging MATLAB's computational capabilities, we were able to derive precise values for the degree and distance-based energy topological indices, enabling a comprehensive analysis of the molecular structures under investigation. Tis approach allows us to capture and quantify important structural characteristics and relationships, providing valuable insights into the properties and behavior of the molecules studied. Te utilization of MATLAB in our study enhances the reliability and accuracy of the energy calculations and topological indices, contributing to a more robust analysis of the molecular systems. Tis computational framework enables us to explore and evaluate the signifcance of these indices in relation to the properties and characteristics of molecules, advancing our understanding of their structural features and potential applications. Computed topological index (energy) values are shown in Table 4. Te studied platinum anticancer drugs are described in Table 5.
Remark 6. In Table 5, physicochemical properties with correlation coefcients below 0.60 (octanol-water partition coefcients (XLogP3), covalently bonded unit count (CBUC), and dipole moment (debye)(DM)) were excluded from our calculations to prioritize properties with stronger associations to the topological indices (energies). Tis ensures a more focused and meaningful analysis of the remaining properties.
In our study, we utilized the DMol3 module in Material Studio 8.0 from BIOVIA to conduct DFT calculations, allowing us to investigate the physicochemical properties of platinum anticancer drugs. To predict these properties, we employed degree-based and distance-based topological indices, along with their associated energies. To analyze the quantitative structure-property relationship (QSPR), we employed a linear regression model. Te model incorporated ten physicochemical properties and eleven topological indices (energies) for comprehensive analysis and prediction. By integrating DFT calculations, topological indices, and a robust regression model, we aimed to gain insights into the relationship between the structural features of platinum anticancer drugs and their physicochemical properties. In following Tables 6 and 7, in our analysis, we evaluated the correlation coefcient (R) between the topological indices (energies) and the physicochemical properties of platinum anticancer drugs. To ensure the relevance and reliability of our fndings, we excluded values below 0.50 for convenience. By setting this threshold, we aimed to focus on the indices that exhibit stronger associations with the physicochemical properties. Tis approach enables us to prioritize and concentrate on the most meaningful relationships, enhancing the signifcance of our study. By excluding values below 0.50, we ensure that our analysis emphasizes the indices that have a more substantial impact on the physicochemical properties, providing valuable insights into the structural features that contribute to the observed variations. Tis fltering process allows us to streamline the interpretation of our results and focus on the most promising indices for further investigation and understanding. Trough this approach, we aim to maximize the efciency and relevance of our analysis, providing a clearer and more meaningful picture of the correlations between the topological indices (energies) and the physicochemical properties of platinum anticancer drugs.
In following Algorithms 3 and 4, in order to efciently organize the degree and distance indices, along with their corresponding energies, we developed a MATLAB program that sorts them based on their efectiveness as predictors of a specifc property. Sorting is performed by evaluating the correlation coefcient values associated with each index. By utilizing this program, we were able to identify the most infuential and informative indices for the property under investigation. Tis approach allows us to prioritize and focus on the indices that exhibit stronger correlations with the desired property, streamlining the analysis process. Te MATLAB program provides a systematic and automated way to assess the predictive power of diferent indices, enabling us to identify the most efective predictors efciently.
Note that correlation coefcients serve as quantitative measures to assess the strength and direction of relationships Step 5. e(1, 1) � (summation of A N ) /2 Step 6. Construct A N for second Zagreb index: Step 7. e(2, 1) � (summation of A N ) /2 Step 8. Construct A N for hyper-Zagreb index: for i � 1 to number of columns do for j � 1 to number of rows do if i � j then Step 9. e(3, 1) � (summation of A N ) /2 Step 10. Construct AN for Albertson index: for i � 1 to number of columns do for j � 1 to number of rows do if i � j then Step 11. e(4, 1) � (summation of A N ) /2 Step 12. Construct A N for sigma index: for i � 1 to number of columns do for j � 1 to number of rows do if i � j then

end for
Step 13. e(5, 1) � (summation of A N ) /2 Step 14. Construct A N for inverse symmetric deg index: for i � 1 to number of columns do for j � 1 to number of rows do if i � j then Step 15. e(6, 1) � (summation of A N ) /2 Step 16. Construct A N for inverse sum deg index: Journal of Mathematics between variables. Tese coefcients, ranging from − 1 to +1, provide insights into the magnitude of the correlation. Positive correlation coefcients indicate a positive relationship, where both variables exhibit a tendency to increase or decrease simultaneously. Conversely, negative correlation coefcients suggest an inverse relationship, where an increase in one variable corresponds to a decrease in the other, and vice versa.

Degree-Based Topological Indices.
As we move forward, we will analyze the results obtained in Table 6. In our analysis, we aimed to predict the physicochemical properties of anticancer drugs using degree-based topological indices. It is worth noting that the sigma index exhibited a poor correlation with the examined properties, and hence, it will be excluded from further discussion. However, we obtained promising results with excellent correlation coefcients using linear regression (R > 0.617 − 0.969). Among the topperforming indices, the hyperindex demonstrated the most appropriate regressions with correlation coefcients ranging from 0.626 to 0.972. Tis indicates a strong and consistent relationship between the Albertson index and the examined properties. Following closely is the inverse sum indeg index index, displaying correlation coefcients ranging (R > 0.835 − 0.969). Tis suggests that the inverse sum indeg index index provides reliable predictions for the physicochemical properties of the anticancer drugs. In third place, the second Zagreb index showed correlation coefcients ranging from 0.824 to 0.962, further supporting its usefulness in predicting the properties. Te inverse symmetric deg index index also demonstrated favorable correlation coefcients (R > 0.835 − 0.969), indicating its potential as a predictive index. Comparatively, the Albertson index Step 4: Sort col_val vector in descending order Step 5: Extract frst 6 best columns using descending order Step 6: Extract names of 6 best columns in vector "best_col_name" Step 7: Return extracted best columns in form of table as output   Figure 2, the scattering of anticancer drugs is plotted with their physical properties by using a degree-based TI. Table 6 provides a comprehensive analysis of the correlation coefcients between the topological indices (energy) and the studied properties. Te results presented in these tables allow us to identify the indices that exhibit the strongest relationships with the properties under investigation. Upon examining Table 6, it becomes evident that certain topological indices (energy) are particularly well suited for predicting the studied properties. Te frst Zagreb index (energy) emerges as the most appropriate regression, displaying correlation coefcients ranging from − 0.552 to 0.990. It is closely followed by the second Zagreb index (energy), which exhibits correlation coefcients ranging from 0.521 to 0.970. Te inverse sum indeg index (energy) claims the third position, with correlation coefcients ranging from − 0.510 to 0.980. Te inverse symmetric deg index (energy) and the correlation coefcient also demonstrate a noteworthy performance, with correlation coefcients surpassing − 0.533 and reaching up to 0.984. Te hyperindex (energy) follows closely behind, with correlation coefcients spanning from 0.566 to 0.981. While several indices showcase remarkable predictive capabilities, it is important to note that the Albertson index (energy) exhibits relatively lower accuracy than other correlations, with correlation coefcients ranging from − 0.680 to 0.920. Te fndings presented in Table 6 not only highlight the indices with the strongest correlations but also provide valuable insights into the overall performance of the topological indices (energy) in relation to the studied properties. Tis information enables researchers to make informed decisions when selecting the most appropriate indices for future molecular property prediction models. Te analysis of the columns in the table reveals intriguing insights into the relationships among the properties. Te (energy). In Figure 3, the scattering of anticancer drugs is plotted with their physical properties by using degree-based T.I. (energy). Table 8 in depth, in our analysis focusing solely on distance-based topological indices, we aimed to determine the most accurate predictor among the indices. Te results revealed that the Wiener index exhibited the highest level of accuracy with an excellent linear regression correlation coefcient ranging from R > 0.632 to R > 0.959. Tis suggests a strong and positive relationship between the Wiener index and the target property being predicted. Consequently, the Wiener index can serve as a reliable tool for estimating the property of interest. Following closely behind, the Schultz index demonstrated a commendable correlation coefcient ranging from R > 0.616 to R > 0.961. Although slightly lower than the Wiener index, this coefcient still signifes a signifcant and positive association between the Schultz index and the target property. Terefore, the Schultz index also proves to be a valuable predictor in our analysis. Furthermore, the Harary index showed promise as a predictor, exhibiting a correlation coefcient ranging from R > 0.621 to R > 0.994. Tis indicates a strong relationship between the Harary index and the property under consideration. Te consistent and high correlation coefcient suggests that the Harary index can be utilized efectively in predicting the desired property. Lastly, the Gutman index, while displaying a slightly lower correlation coefcient of R > 0.60 to R > 0.956, still demonstrates a signifcant positive relationship with the target property. Tis indicates that the Gutman index has the potential to be employed as a predictor, albeit with slightly reduced accuracy compared to the aforementioned indices. Our analysis of distance-based topological indices identifes the Wiener index as the most accurate predictor, followed by the Schultz index, the Harary index, and the Gutman index. Researchers can leverage these fndings to select the most suitable index for estimating the desired property, enhancing the efciency and efectiveness of their predictive models. In Figure 4, our analysis reveals that the distance-based topological indices exhibit highly desirable properties in predicting various physicochemical characteristics. Specifcally, the indices Tese strong correlations indicate the efectiveness of these indices in accurately predicting the associated physicochemical properties. Tese results highlight the signifcant contribution of distance-based topological indices in accurately predicting specifc physicochemical properties. Te strong correlations observed demonstrate the efcacy of these indices as reliable predictors. Te fndings presented in this study provide valuable insights into the relationship between the distance-based topological indices and the associated properties, enabling researchers to make informed decisions and predictions in the feld of molecular structureproperty relationships.  the physicochemical properties under investigation. Similarly, the Harary index (energy) exhibits unsatisfactory correlation and will be omitted from our further discussion.

Energy Distance-Based Topological Indices. Examined in detail in
On the other hand, the Schultz index (energy) emerges as a strong predictor, demonstrating a noteworthy correlation coefcient (R > 0.618 − 0.984). Tis indicates a robust relationship between the Wiener index and the predicted physicochemical properties. In addition, the Wiener index (energy) also shows promising results, with correlation coefcients ranging from (− 0.683 ≤ R ≤ 0.986). Tese fndings position the Wiener index and the Schultz index as reliable indicators of the associated properties, suggesting their potential as valuable tools in molecular structureproperty relationships. Distance-based topological indices predict the most desirable properties in Figure 5. Tables 6 and 7 show the correlation coefcients between platinum anticancer drugs and their degree-and distance-based topological indices (energies). Tis study considers ten physicochemical properties of anticancer drugs. Figure 6 presents the absolute average correlation coefcients of the properties under investigation, highlighting the most accurately predicted property by the topological index (energy). Te results depicted in Figure 6 reveal the property that demonstrates the highest correlation coefcient with the topological index (energy). Tis property serves as a robust indicator of the relationship between the topological index and the corresponding physicochemical attribute. By identifying this property, we gain valuable insights into the predictive power of the topological index (energy) and its potential utility in molecular property estimation. Figure 7 illustrates the average correlation coefcients of the topological indices (energy), providing valuable insights into their relative predictive capabilities. By analyzing Figure 7, we can observe the ordering of the topological indices (energy) based on their average correlation coefcients. Tis ordering sheds light on the efcacy of each index in capturing the underlying relationships between the molecular structure and the associated properties. Tis allows us to identify the indices that exhibit stronger predictive power and are more closely aligned with the studied properties. Figure 7 provides a clear illustration of the correlation coefcients associated with the topological indices (energy) compared to the degree-based or distance-based topological indices. Te data presented in this fgure indicate that the topological indices (energy) exhibit higher correlation coefcients with the studied properties than their degree-based or distance-based counterparts. Tis fnding is of great signifcance as it suggests that incorporating the energy component enhances the predictive power of the topological indices in capturing the underlying structure-property relationships. Te higher correlation coefcients observed for the topological indices (energy) imply a stronger association between the energy-based descriptors and the properties under investigation. Interestingly, Figure 7 also reveals an intriguing observation regarding the Gutman and Harary indices (energy). Tese indices display lower correlation coefcients than the other topological indices (energy) and the degree or distance-based indices. Tis unexpected result warrants further investigation to explore the specifc relationship between the Gutman index (energy), Harary index (energy), and the structural properties of molecules under study. By highlighting this disparity and acknowledging the need for further investigation, we aim to encourage future research endeavors to delve deeper into the intricate connection between the topological indices (energy), the Gutman index (energy), the Harary index (energy), and the properties of interest. By understanding the underlying factors that contribute to the variations in the correlation coefcients of topological indices, we will be able to improve our understanding of their predictive capabilities in a more comprehensive manner. Tis will have the beneft of enabling us to develop more accurate models for predicting the properties of molecules based on the properties of molecules that are developed as a result of this.

DFT Calculations
In Subsection 4.1, the graph theory-structure-activity relationship (GT − STAF) approach has emerged as a valuable     molecular structure, reactivity, and biological activity. We explore the synergistic potential of combining the GT − STAF approach with DFT calculations, shedding light on the immense opportunities for advancements in drug design and computational chemistry. For a comprehensive study, please refer to [6] and [52]. In Subsection 4.2, the quantum theory of atoms in molecules (QTAIM) approach was employed to calculate the topological properties of electron densities. Te QTAIM method provides a framework for analyzing molecular systems by partitioning the electron density into atomic basins and characterizing the interactions between atoms. To perform these calculations, the researchers utilized the Multiwfn code, which is a software package specifcally designed for the analysis of molecular wavefunctions. Multiwfn implements the QTAIM methodology and allows for the computation of various topological properties. By employing QTAIM in conjunction with the Multiwfn code, the researchers were able to investigate the electron density distribution within the molecules under study. For a comprehensive study, please refer to [53] and [54].

Fukui Functions. Version 8.0 of Material Studio from
BIOVIA was used to perform DFT calculations of investigated anticancer drugs using the DMol3 module. Te local density approximation (LDA) [55] (Perdew-Wang (PWC)) [57] was used in the optimization process. We ensure accurate results by setting all parameters to the medium. More details regarding DFT calculations are indicated in Figure 8.
On the optimized geometries, the frequency, Fukui functions [57] were calculated. It is ensured that negative frequencies are absent from the obtained geometries. A local Fukui function in drug molecules could reveal nucleophilic (f + ), electrophilic (f − ), and radical attack sites. In Figure 9 (the optimized geometry is indicated for better visualization), we present the 3D plots of the isosurfaces of the tenth investigated drugs with nucleophilic, electrophilic, and radical Fukui functions. Tere is no domination of the distribution of electrophilic and nucleophilic Fukui functions on the molecular skeletons of the ten investigated drugs. We can notice that the distributions of f − over carboplatin, oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, spiroplatin, and picoplatin are very similar and covered only small parts of molecular skeleton. In contrast, f + is distributed over cisplatin, carboplatin, oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, and satraplatin in a very similar way and delocalized in small regions of the molecular skeleton. Regarding the smaller distribution of f 0 , it is found in oxaliplatin, nedaplatin, lobaplatin, heptaplatin, zeniplatin, spiroplatin, and picoplatin. Te dominant distribution of f − can be found in cisplatin and satraplatin, and thus, these two molecules are subjected most to the electrophilic attack, while the dominant distribution of f + is on drug molecules, spiroplatin and satraplatin, and thus, these two drug molecules are subjected the most to the nucleophilic attack. f 0 is distributed on most of cisplatin, carboplatin, and satraplatin molecular skeletons, and thus, these three drug molecules are the most active molecules for radical attack. Satraplatin is the most active drug molecule among all, and nucleophilic, electrophilic, and radical attacks take place at all sites of this molecule.

QTAIM Details.
Quantum theory of atoms in molecules (QTAIM) was used to compute the topological properties of the resulting electron densities as implemented in the Multiwfn code [53]. Tis analysis was accomplished by performing all-electron single-point energy calculations at the optimized geometries using the DFT/B3LYP [54] approach with the Dunning basis set (SDD) [58]. Te Espinosa equation [59] (E � 0.5 * V(r)), in which V(r) is the potential energy density (a.u.) at the bond critical point (BCP), was used to calculate the energies (E) of the coordination Pt − N, Pt − O, and Pt − Cl bonds [60]. Tis equation is broadly used for the energy estimation of different types of bonds such as hydrogen bonds, van der Waal interactions, coordination bonds, and homopolar bonds. Te usage of the Bader method makes it possible to describe qualitatively the chemical bond nature based on the signs and values of the electron density Laplacian ∇ 2 ρ (r) and of the electron energy density H(r) at the corresponding BCP, in accordance with the following conditions [59]: (1) ∇ 2 ρ (r) < 0 and H(r) < 0 indicate the shared interactions, i.e., weakly polar and nonpolar covalent bonds  Tables S1 and  S2. Molecular graphs for some investigated complexes are shown in Figure 10. Te data in Table 8 Te ellipticity parameter (ε) is considered a descriptor, which can be used to determine the Pt − N, Pt − O, and Pt − Cl bond stability. Te ellipticity parameter is determined from the ratio between the curvature elements (λ 1 /λ 2 − 1). According to Bader's theory [59,61], the bonds with high ε values are potentially unstable. Te data in Table 8  Te parameters are density of all electrons (ρ(r) in e x a − 3 0 ), Lagrangian kinetic energy (G(r) in a.u.), Hamiltonian kinetic energy (K(r) in a.u.), potential energy density (V(r) in a.u.), energy density E(r) or H(r), Laplacian of electron density (∇ 2 ρ (r) in e x a − 5 0 ), bond order (� − G(r)/V(r)), bond energy (E in kcalmol − 1 ), eigenvalues of Hessian, curvature element values (λ 1 , λ 2 , and λ 3 in e x a − 5 0 ), ellipticity of electron density (ε � (λ 1 , λ 2 ) − 1), and eta index (η � |λ 1 /λ 3 |).

Conclusion
In this study, we have conducted a comprehensive investigation into six platinum anticancer drugs, including cisplatin, carboplatin, oxaliplatin, nedaplatin, lobaplatin, and heptaplatin, along with zeniplatin, spiroplatin, picoplatin, and satraplatin. For the frst time, we have explored the degree-based and distance-based topological indices, as well as their corresponding energies, associated with these drugs. Our research has signifcant implications in the feld of drug discovery and development, particularly for the treatment of chronic myelogenous leukemia and inoperable metastatic breast and small-cell lung cancer. By employing eleven topological indices, consisting of seven degree-based and four distance-based indices, along with their energies, we have provided new insights into the physicochemical properties of these platinum anticancer drugs. Trough the application of curvilinear regression within a quantitative structure-property relationship (QSPR) model, we have established a strong correlation between the various physicochemical properties of the platinum anticancer drugs and the investigated topological indices and their energies. Notably, we have observed that the energy associated with most degree-based topological indices exhibits a higher correlation coefcient than calculation-based topological indices and is more signifcant. Compared to the literature [46,47], where the topological indices have a higher correlation coefcient, our results are more interesting. Furthermore, our study has mathematically determined the upper and lower bounds of the topological indices, enabling their application to any chemical structure. Tis mathematical framework enhances the practicality and applicability of these indices in various research and drug design settings.
Te signifcant correlation established between the physicochemical properties and the energy of the investigated topological indices emphasizes the efectiveness of these indices in predicting key properties of the platinum anticancer drugs. As a result, our fndings suggest the potential for modifying the manufacturing process of platinum-based cancer drugs and similar compounds. Pharmacists and chemists can leverage the energy values of these topological indices to further study and explore these anticancer drugs. Tis knowledge opens avenues for testing diferent combinations of these drugs tailored to specifc ailments, based on their unique compositions. In addition to the investigation of topological indices and their energies, this study incorporates density functional theory (DFT) calculations to further enhance our understanding of the platinum anticancer drugs' properties. DFT is a powerful computational approach that allows for the analysis of the electronic structure and properties of molecules. Furthermore, DFT calculations ofer the capability to assess the local reactivity of molecules using concepts such as Fukui functions. Tese functions identify regions within a molecule that are more susceptible to chemical reactions, indicating potential sites for interaction with target biomolecules or receptors. By integrating DFT calculations within the investigation of topological indices and their energies, this study ofers a comprehensive understanding of the molecular properties that infuence the biological activity of the platinum anticancer drugs. It allows for the consideration of electronic efects, such as charge distribution and reactivity, in addition to the structural aspects captured by the topological descriptors. Tis combined approach not only enhances our knowledge of these drugs but also provides valuable guidance for the design of compounds with tailored properties, optimizing their potential for desired biological activities. Quantum theory of atoms in molecules (QTAIM) was carried out at the bond critical point (BCP), to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. In summary, our study provides novel insights into the relationship between topological indices, their energies, and the physicochemical properties of platinum anticancer drugs. Tis research contributes to the advancement of drug design strategies and holds promise for optimizing the manufacturing process of platinum-based cancer drugs, ultimately leading to improved treatment outcomes for patients [39,63].

Data Availability
Te article contains the data that supported the study's fndings. Te study's fndings are supported by the data presented in Tables S1 and S2, which can be obtained upon request from the corresponding author.