Face reverse degree topological analysis of TP-COFs, existence of isentropic COFs and spectral characteristics

Abstract

The recent synthesis of triple-pore covalent organic frameworks (TP-COFs), specifically TP-COF-DAB and TP-COF-BZ, represents a significant milestone in framework chemistry, as replicating such multi-pore architectures remains highly challenging. Despite their structural novelty, the mathematical perspective of their topology and associated properties is still limited. In this study, closed-form expressions are derived for reverse degree topological indices of these TP-COFs based on a bitrapezium topological arrangements. Scaled face reverse degree indices are employed to explicitly characterize pore geometry in triple-pore architectures. Furthermore, reverse degree based entropy measures are computed to quantify structural orderness and to explore isentropic configurations within the TP-COF family. Using these structural parameters, quantitative structure-property models are constructed for spectral energy, demonstrating strong predictive capability. In addition, the spectral diameter and HOMO–LUMO gap are analyzed using graph spectral techniques, providing insights into the spectral characteristics of TP-COFs.

1 Introduction

Covalent organic frameworks have emerged as highly promising materials over the past decades due to their remarkable crystallinity, extensive surface areas, and wide-ranging applications, including catalysis, sensing, energy storage, drug delivery, optoelectronic devices, separation processes, and gas adsorption and storage (Bai et al., 2016; Ding et al., 2016; Kandambeth et al., 2017; Liao et al., 2016; Liu et al., 2026). As an emerging class of crystalline porous organic materials, the properties and performance of COFs strongly depend on the characteristics of their pores, which are determined by the topological structures of the networks and their sizes (Carrington et al., 2022; Liang et al., 2020; Yusran et al., 2024). In particular, appropriate pore architectures and functionalization enable COFs to act as efficient adsorbents for the capture of hazardous metal ions, organic and biological pollutants, as well as greenhouse gases (Ghazi et al., 2018; Li et al., 2021; Xiao et al., 2021; Xin et al., 2021). Traditionally, COFs have been constructed through the principles of reticular chemistry, which enable the precise prediction of their structures based on the symmetry and geometry of the building blocks used for condensation reactions (Abuzeid et al., 2021; Yaghi, 2016; Yu et al., 2022). The design strategies for COFs have therefore focused mainly on combining building units with compatible symmetries. Using this approach, various COFs with tetragonal, hexagonal, or triangular pores have been designed and synthesized since the first two COFs were reported in 2005 (Chen et al., 2025; Côté et al., 2005; Gao et al., 2018; Geng et al., 2020; Jing et al., 2023). In these traditional COFs, there is usually only one kind of pore in a given framework, which limits their structural complexity and functional diversity.

Recent advances have introduced a paradigm shift with the development of heteropore COFs that incorporate two or three distinct types of pores within a single continuous network (Dalapati et al., 2016; Du et al., 2016; Han et al., 2020). These structures enrich the structural diversity of the COF family and open new avenues for multifunctionality. A pioneering design strategy combining vertex truncation and multiple linking site approaches has enabled the fabrication of triple-pore COFs (Qian et al., 2017). In this design, the building block (TPTCA) is shown in Figure 1a, while the linear linkers 1,4-diaminobenzene (DAB) and benzidine (BZ) are shown in Figures 1b,c. Condensation of TPTCA with these linkers yields the structural units for TP-COF-DAB and TP-COF-BZ, presented in Figure 2. These COFs exhibit an fxt topology, where the V-shaped geometry of the TPTCA unit is crucial for generating the connectivity that gives rise to three distinct pore types (Bhambri et al., 2022). This advancement reflects a broader challenge in polymer and materials science, moving beyond primary connectivity toward precisely controlled higher-order structures. While biological polymers achieve remarkable hierarchical organization, replicating such structural precision in synthetic systems remains a significant challenge (Kricheldorf, 2006). The design of multi-pore COFs such as TP-COF-DAB and TP-COF-BZ represents a meaningful step toward this goal, embodying the idea of tailor-made frameworks in materials design. Analyzing their intricate architectures is therefore not only of theoretical interest but also essential for understanding and exploiting their full functional potential.

FIGURE 1

FIGURE 2

Since the pore architecture is the primary factor governing the physical and chemical properties of COFs, a topological analysis that quantitatively characterizes the pore environment is essential. Topological indices, which are numerical descriptors derived from graph theory, have been widely employed to characterize the structure of molecular and extended frameworks (Arockiaraj et al., 2025a; Junias and Clement, 2024; Kalaam et al., 2024; Mondal et al., 2026; Tang et al., 2025). In the context of COFs, several studies have applied degree-based and related topological indices to correlate structural features with properties (Arockiaraj et al., 2025b; Augustine and Roy, 2022; Kurian et al., 2025; Tu et al., 2025; Zhang X. et al., 2025). These indices provide a powerful way to understand how the structure of the framework influences its physicochemical properties (Arockiaraj et al., 2025c; Manuel and Angamuthu, 2025; Paul et al., 2025; Raza et al., 2024). However, conventional degree-based topological indices mainly describe atom connectivity, which may not fully capture the complexity of the pores themselves. To address this limitation, a face-degree-based topological framework is employed, in which indices are defined in terms of face degrees, thereby providing a more direct and accurate representation of the pore structure. Notably, scaled face reverse-degree based indices have been shown to be effective for benzenoid hydrocarbons compared with conventional degree-based indices (Arockiaraj et al., 2026a). In parallel, reverse-degree topological indices offer an alternative perspective for capturing structural information by emphasizing complementary connectivity patterns within the network (Ahmad et al., 2023; Kalaam and Greeni, 2024; Kalaam and Greeni, 2025; Rao et al., 2024; Youssef et al., 2024). Consequently, the reverse-degree framework provides additional structural information that can be used for subsequent QSPR modeling (Aq et al., 2026; Ravi, 2024). Thus, in this study, we analyze TP-COFs with reverse-degree indices, and scaled face reverse-degree indices. We further investigate the existence of isentropic structures by means of reverse-degree based graph entropies, which have attracted considerable attention in the literature as measures of structural complexity and information content (Kavitha et al., 2021; Naeem et al., 2024; Zhang G. et al., 2025). In addition, entropy measures are relevant to both thermodynamics and information theory, which provide valuable insights into the behaviour of complex systems across scientific and engineering domains (Abraham et al., 2022; Hussain et al., 2025; Jawahar and Clement, 2026; Peter et al., 2025; Yu et al., 2024). Building upon these analyses, we further investigate spectral descriptors and develop predictive models for spectral graph energy using the computed topological indices.

2 Computational techniques

The triple-pore covalent organic frameworks investigated in this study are modeled as hydrogen-suppressed graph structures , where represents the set of vertices corresponding to atoms, and represents the set of edges corresponding to bonds. In this work, we aim to explore the degree parameter of each vertex , defined as the number of edges incident on the vertex, to its full potential. Accordingly, a modified reverse-degree parameter is employed to reveal different degree combinations and to represent vertices with lower connectivity. The reverse degree is defined as

Here, denotes the maximum degree in , and the reversing parameter takes integer values from 1 to . For , the degree values repeat cyclically. The reverse-degree parameter defined in Equation 1 follows existing formulations in the literature, where reverse-degree concepts are employed to capture complementary connectivity patterns in complex networks. Such a definition has been effectively used in recent studies to improve degree-based indices (Ahmad et al., 2023; Kalaam and Greeni, 2024; Kalaam and Greeni, 2025; Rao et al., 2024; Youssef et al., 2024). In this work, the same formulation is adopted in the context of TP-COFs to appropriately reflect their connectivity variations. The degree-based topological indices are defined as . In order to incorporate the reverse-degree parameters, we define the associated topological indices as

Furthermore, to integrate the effect of reverse-degree modification with the structural influence of pores, we employ the face-reverse-degree indices for each face , defined aswhere denotes the set of boundary edges of the face , and represents the set of all faces of . The set includes both internal faces , which are completely bounded cycles, and the external face , which outlines the outer boundary of the structure. During the boundary traversal of a face, pendant edges are counted twice (Arockiaraj et al., 2026b). The total number of faces is given by the cardinality . Aggregating the face contributions derived from Equation 3, the scaled face-reverse-degree index of is defined as

To emphasize the structural influence of these reverse-degree modifications and their pore-level extension, we consider the first Zagreb , second Zagreb , hyper-Zagreb , forgotten , arithmetic , bi-Zagreb , tri-Zagreb , bi-Zagreb-arithmetic , tri-Zagreb-harmonic , and tri-Zagreb-arithmetic , which are defined respectively as

Among these topological indices, the first Zagreb index was originally introduced in relation to the total -electron energy of molecular graphs, while the forgotten index was later shown to provide improved structural sensitivity in subsequent investigations (Furtula and Gutman, 2015; Gutman and Trinajstić, 1972). In recent studies, these indices have been extended through reverse-degree and scaled face-based formulations to better represent structural variations (Arockiaraj et al., 2026a; Kalaam and Greeni, 2025). These extensions have been shown to provide additional structural information compared to existing indices and support their applicability in structural analysis and potential QSPR studies (Ahmad et al., 2023; Aq et al., 2026; Ravi, 2024).

3 Results and discussion

In TP-COFs, the structural units can be arranged in various configurations, such as linear, rectangular, hexagonal, parallelogram, and bi-trapezium, depending on the topology and connectivity of the building units. These configurations influence the overall geometry, porosity, and functional characteristics of the materials. In the present study, the bi-trapezium configuration is used to model the arrangement of structural units as graph structures. The representative bi-trapezium configurations considered in this work are illustrated in Figures 3, 4. These figures depict the structural arrangement of the TP-COFs, where vertices represent atoms and edges represent bonds. In particular, the figures highlight variations in connectivity and the arrangement of faces, which form the basis for computing the reverse-degree and scaled face reverse-degree topological indices. This configuration includes two important structural forms, namely linear and hexagonal, as subcases, and therefore provides a general framework suitable for representing a wide range of geometric variations observed in TP-COFs. Let and represent the bi-trapezium graph structures corresponding to the two TP-COFs, namely TP-COF-DAB and TP-COF-BZ, respectively. Here, and denote the structural parameters that determine the extent of the bi-trapezium arrangement. The vertex and edge set counts of these COFs are given, respectively, by and .

FIGURE 3

FIGURE 4

3.1 Reverse degree topological indices

To streamline the computation of the reverse-degree topological indices, we first partition the bonds into bond classes using the bond-partition technique, which groups edges according to the degrees of their end vertices. Mathematically, this is defined asLet denote the set of bond classes of the TP-COFs. Then, the frameworks and share the same bond classes, . However, their cardinalities differ, and the corresponding values are presented in Table 1.

From Table 1, the degree set of is , with maximum degree 3. Accordingly, the reversing parameter ranges from 1 to 3, and the corresponding modifications are

Thus, the modified bond partitions for are given by . For , the partitions are and for , . The corresponding cardinalities remain unchanged as per order given in Table 1. Then, the general formula for reverse-degree topological indices, obtained by substituting into Equation 2, is given by

By substituting the topological indices into Equations 5 and 6, the corresponding closed-form expressions for the reverse-degree topological indices are obtained, and the results are presented in the form in Results 1 and 2.

Based on Results 1–2, the reverse-degree topological indices of and for are computed, and the values are presented in Tables 2–4.

3.2 Scaled face reverse degree topological indices

In the case of scaled face reverse degree topological indices, the TP-COFs are partitioned into faces. For both and , the face partitions consist of five internal faces and one external face , where the internal face partitions are illustrated in Figures 5, 6. In particular, these figures illustrate the classification of faces using different colours, where each colour corresponds to a distinct face class. Higher-dimensional structures follow a similar pattern. The cardinalities of the faces , and are the same for both TP-COFs. However, the edge composition of each face, represented by , varies. An exception occurs for the face , whose cardinality varies while its edge composition remains the same. The corresponding cardinalities and edge compositions are presented in Table 5. Moreover, the total number of faces in and are given by and , respectively. Let . Then and . On substituting and the modified reverse-degrees into Equation 4, the general formulas for scaled face reverse-degree topological indices are presented below.

FIGURE 5

FIGURE 6

By substituting the topological indices into Equations 7 and 8, the closed-form expressions for the scaled face reverse-degree-based topological indices are presented as in Results 3 and 4.

The scaled face reverse-degree topological indices of and are computed using Results 3 and 4, and the corresponding values are summarized in Tables 6–8.

3.3 Existence of isentropic structures

The term isentropic, meaning constant entropy, has been of long-standing interest in thermodynamics and has led to diverse concepts such as isentropic analysis, isentropic efficiency, and isentropic flow. In the context of topological analysis, isentropic structures are defined as systems that attain equal entropy values even in the presence of variations in topology, geometry, or composition. The recognition of such structures is significant, as it reveals underlying structural similarity, indicates equivalent information content, and enables the classification of networks into entropy-equivalent families. These measures provide an effective framework for quantifying structural complexity and information distribution in molecular and networked systems (Anand and Bianconi, 2009; Bonchev, 1983; Mowshowitz, 1968). They have been successfully applied to both organic and inorganic materials, where entropy-based approaches contribute to the analysis of framework organization and support rational material design (Jacob et al., 2023; Rahul et al., 2022). These entropy formulations are constructed through additive and multiplicative self-powered transformations of the underlying indices, thereby incorporating both frequency and magnitude information associated with bond partitions. Recent advancements have incorporated reverse-degree modifications into entropy measures. As a result reverse-degree-based entropy of takes the formwhere, is the multiplicative self-powered transformation defined as . Explicit mathematical expressions for the entropy measures can be derived from the corresponding topological and self-powered indices. However, the resulting formulations are complex and difficult to present concisely. Therefore, we focus on the numerical evaluation of the entropy values for and . The computed values are presented in Tables 9–11.

From Tables 9–11, we clearly observe that the configurations pair , , , , and so on, yield similar entropy values, leading to the existence of isentropic structures. Furthermore, since the isentropic dimensional patterns are consistent for both and frameworks, this behavior can be expressed in the general forms . Selected examples illustrating the isentropic correspondence between and are listed in Table 12. It should be noted that the term “isentropic” used in this study is based on graph-theoretic entropy and does not refer to thermodynamic entropy in the classical materials science sense. The entropy considered here is a structural descriptor derived from topological indices, quantifying the distribution of connectivity within the network. Accordingly, two TP-COF structures are termed isentropic if they exhibit equal values of this graph-based entropy, even if their topology differs. Thus, the term indicates structural equivalence in terms of information content rather than thermodynamic behavior.

4 Spectral descriptors and QSPR analysis

In this section, the spectral characteristics of the considered structures are investigated to gain insight into their structural properties and connectivity patterns. However, direct computation through density functional theory (DFT) is computationally demanding, time-consuming, and often challenging for large-scale frameworks. Therefore, we adopt a graph-theoretical approach based on the eigenvalues of the adjacency matrix. In particular, graph energy, HOMO-LUMO gap, and spectral diameter are computed using Python programming through adjacency matrices. The graph energy of a comprising vertices is determined from its spectrum obtained via the adjacency matrix representation. Let denote the eigenvalues of this matrix. The corresponding graph energy, is defined as

Furthermore, the HOMO-LUMO gap and spectral diameter are computed from the ordered eigenvalue set. Specifically, the HOMO-LUMO gap is evaluated as the difference between the smallest positive eigenvalue and the largest negative eigenvalue, whereas the spectral diameter is defined as the difference between the maximum and minimum eigenvalues of the spectrum. The computed spectral descriptors presented in Table 13 demonstrate distinct behavioural patterns across the investigated structures. In particular, the HOMO–LUMO gap remains constant for each framework, attaining fixed values of for and for irrespective of the dimensional parameters . This invariance indicates structural control of the spectral gap within each class. A similar near-constant trend is observed for the spectral diameter, which exhibits only marginal variation with increasing structural size. In contrast, graph energy displays a clear monotonic growth as the parameters increase, reflecting its sensitivity to structural expansion and complexity. Owing to this discriminative behaviour, graph energy is selected as the principal descriptor for subsequent QSPR modelling.

Both univariate and multivariate regression analyses are employed for effective model construction. The univariate regression model is expressed as where denotes the slope and represents the intercept. The multivariate regression model with two independent variables is given by where and are the regression coefficients associated with the indices and , respectively, and is the intercept. The selection of indices for both univariate and multivariate modelling is initially guided by correlation analysis. For the univariate case, the correlation results are illustrated through heatmaps shown in Figures 7, 8. Based on the best-performing index, univariate regression models are constructed. Similarly, for the multivariate case, all possible pairs of indices are correlated with graph energy using Algorithm 1 for multivariate regression analysis. The most significant index pairs are subsequently employed to develop multivariate regression models. The results of both univariate and multivariate regression analyses are presented in Table 14, where the coefficient of determination () and root mean square error (RMSE) are reported to nine and six decimal places, respectively.

FIGURE 7

FIGURE 8

Table 14 clearly shows that for both and , the association between graph energy and the topological indices follows an almost identical pattern in its correlation behaviour with the indices. In the univariate analysis, the same index consistently exhibits the strongest correlation with energy for both structures. A slight variation is observed only in the case of face reverse-degree-based indices when moving to the multivariate setting. This overall consistency highlights the strong structural similarity between and with respect to their spectral characteristics. In the univariate models, the index dominates the predictive performance. However, in the multivariate framework, the models based on and indices emerge as the most effective. The multivariate regression models exhibit comparatively low RMSE values, indicating improved predictive accuracy. Among these, the reverse degree-based indices perform particularly well. Specifically, for , the reverse degree descriptor with modification parameter provides the best performance, whereas for , the descriptor with performs better. The resulting regression equations corresponding to the best-performing models, along with the standard error (.) and -value, are given below.

Substituting the index values into these reverse-degree regression equations yields

The low and high -value indicate the good fit of the model. We examine the two best-performing multivariate models, statistical significance and stability using the p-values, leave-one-out cross-validation coefficient , and the variance inflation factor (VIF). Both models yielded , confirming statistical significance at the 1 level. The cross-validation results produced , indicating perfect internal predictive agreement within the dataset. However, the VIF values are very high , suggesting the presence of multicollinearity between the and predictors. The elevated VIF values indicate a strong linear association between the and descriptors, suggesting potential redundancy. This behaviour is not unexpected, as both descriptors are degree-based and derived from closely related degree distributions. Consequently, partial linear dependence may arise, leading to possible coefficient instability in the multivariate regression model. Therefore, while the combined model provides superior fitting performance, caution is required when interpreting the individual regression coefficients, as multicollinearity may inflate statistical measures.

To confirm that the observed perfect predictive ability is not merely a consequence of linear dependence between and , external validation was performed. The regression model was applied to higher-dimensional structures that were not involved in the model fitting process, and the results are presented in Table 15. The sustained predictive accuracy observed in this independent dataset confirms that the excellent performance is not an artefact of multicollinearity but instead reflects a stable and structurally consistent relationship between the selected indices and the energy. To demonstrate the performance of the multivariate models, first-degree polynomial surface fitting were carried out, and the resulting visualizations are depicted in Figures 9, 10.

FIGURE 9

FIGURE 10

Figures 9, 10 clearly illustrate the effectiveness of the polynomial surface fitting for the considered dataset. The fitted surface closely aligns with the observed data points, indicating a strong agreement between the predicted and computed values. The smooth planar nature of the surface suggests a stable and linear relationship between the spectral descriptor and the selected topological descriptors. This confirms the suitability of the proposed multivariate model for predicting graph energy.

4.1 Y-scrambling

To further validate the robustness and predictive reliability of the proposed QSPR models, a Y-scrambling test is performed. In this procedure, the dependent variable values are randomly shuffled while keeping the independent variables unchanged, and new regression models are constructed for each permutation. The RMSE obtained from each scrambled model is then compared with that of the original model. In this study, the dependent variable is randomly permuted 500 times. For each permutation, the model is recalibrated and the corresponding RMSE is computed. The distribution of RMSE values obtained from the scrambled models is subsequently compared with the RMSE of the original model to assess the presence of chance correlations. The comparative results of the calibrated RMSE () and the recalibrated RMSE () are graphically illustrated in Figure 11.

FIGURE 11

From Figure 11, a substantial difference between and is observed. The significantly lower compared to the values confirms that the developed QSPR models are not influenced by random correlation and demonstrates their statistical robustness.

5 Conclusion

In this work, we carried out a comprehensive structural and spectral investigation of triple-pore covalent organic frameworks using reverse-degree, and scaled face reverse-degree indices. The study provides a systematic mathematical characterization of these frameworks and establishes clear relationships between their structure and spectral behavior. The entropy analysis revealed that the considered TP-COF structures exhibit isentropic behavior, indicating structural regularity and uniform information distribution. From the spectral perspective, graph energy increases monotonically with structural growth, while the HOMO–LUMO gap remains invariant and the spectral diameter shows only marginal variation. The invariance of HOMO–LUMO indicates structural control of the gap within the series in the graph-theoretical model, whereas the monotonic growth of graph energy highlights its suitability as a descriptor for QSPR modelling. Regression analysis further strengthened these observations. Both univariate and multivariate models were constructed to examine the predictive capability of the proposed indices. The multivariate models, particularly those involving the first and second Zagreb indices, produced extremely low RMSE (0.000006) with a leave-one-out cross-validation coefficient close to one. The statistical significance was confirmed through very low p-values, and external validation demonstrated high predictive accuracy. Furthermore, the regression models were validated using a Y-scrambling test. These results provide a mathematical characterization of TP-COFs and establish their structural, spectral, and predictive properties within a unified framework.

References

• 1

  AbrahamJ.ArockiarajM.JencyJ.KavithaS. R. J.BalasubramanianK. (2022). Graph entropies, enumeration of circuits, walks and topological properties of three classes of isoreticular metal organic frameworks. J. Math. Chem.60, 695–732. 10.1007/s10910-021-01321-8

  • CrossRef
  • Google Scholar

• 2

  AbuzeidH. R.EL-MahdyA. F. M.KuoS.-W. (2021). Covalent organic frameworks: design principles, synthetic strategies, and diverse applications. Giant6, 100054. 10.1016/j.giant.2021.100054

  • CrossRef
  • Google Scholar

• 3

  AhmadA.KoamA. N. A.AzeemM. (2023). Reverse-degree-based topological indices of fullerene cage networks. Mol. Phys.121 (14), e2212533. 10.1080/00268976.2023.2212533

  • CrossRef
  • Google Scholar

• 4

  AnandK.BianconiG. (2009). Entropy measures for networks: toward an information theory of complex topologies. Phys. Rev. E Stat. Nonlin. Soft Matter Phys.80, 045102. 10.1103/PhysRevE.80.045102

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 5

  AqibM.MalikM. A.AliY.JuttS.ImranM. (2026). Predicting optimal drugs for fin rot by reverse topological QSPR analysis. Comput. Biol. Chem.121, 108847. 10.1016/j.compbiolchem.2025.108847

  • CrossRef
  • Google Scholar

• 6

  ArockiarajM.Jeni GodlinJ. J.RadhaS.AzizT.Al-HarbiM. (2025a). Comparative study of degree, neighborhood and reverse degree based indices for drugs used in lung cancer treatment through QSPR analysis. Sci. Rep.15, 3639. 10.1038/s41598-025-88044-x

  • CrossRef
  • Google Scholar

• 7

  ArockiarajM.MaaranA.DossC. I. A. (2025b). Topological characterization and predictive modeling of graph energy in ionic covalent organic frameworks. Comput. Mater. Contin.85 (1), 637–655. 10.32604/cmc.2025.065674

  • CrossRef
  • Google Scholar

• 8

  ArockiarajM.JencyJ.ShaliniA. J.BalasubramanianK. (2025c). Degree-based topological insights and graph entropies of kagome lattice covalent organic frameworks. Chem. Pap.79, 135–157. 10.1007/s11696-024-03743-5

  • CrossRef
  • Google Scholar

• 9

  ArockiarajM.RajendranT.BalasubramanianK. (2026a). Face-degree and reverse-degree topological indices for quantitative structure property predictions of benzenoid hydrocarbons. Chem. Pap.80, 377–402. 10.1007/s11696-025-04399-5

  • CrossRef
  • Google Scholar

• 10

  ArockiarajM.RajendranT.ShaliniA. J.BalasubramanianK. (2026b). Graph theoretical characterization and QSPR modeling of aluminum and magnesium hydroxide networks through integrated face degree topological indices. Eur. Phys. J. E49, 14. 10.1140/epje/s10189-025-00554-8

  • CrossRef
  • Google Scholar

• 11

  AugustineT.RoyS. (2022). Topological study on triazine-based covalent-organic frameworks. Symmetry14 (8), 1590. 10.3390/sym14081590

  • CrossRef
  • Google Scholar

• 12

  BaiL.PhuaS. Z. F.LimW. Q.JanaA.LuoZ.ThamH. P.et al (2016). Nanoscale covalent organic frameworks as smart carriers for drug delivery. Chem. Commun.52 (22), 4128–4131. 10.1039/c6cc00853d

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 13

  BhambriH.KhullarS.MandalS. K. (2022). Nitrogen-rich covalent organic frameworks: a promising class of sensory materials. Mater. Adv.3 (1), 19–124. 10.1039/d1ma00506e

  • CrossRef
  • Google Scholar

• 14

  BonchevD. G. (1983). Information theoretic indices for characterization of chemical structures. Chichester, England: John Wiley Sons.

  • Google Scholar

• 15

  CarringtonM. E.RampalN.MaddenD. G.O’NolanD.CasatiN. P. M.DivitiniG.et al (2022). Sol-gel processing of a covalent organic framework for the generation of hierarchically porous monolithic adsorbents. Chem8 (11), 2961–2977. 10.1016/j.chempr.2022.07.013

  • CrossRef
  • Google Scholar

• 16

  ChenF.ZhengH.YusranY.LiH.QiuS.FangQ. (2025). Exploring high-connectivity three-dimensional covalent organic frameworks: topologies, structures, and emerging applications. Chem. Soc. Rev.54 (1), 484–514. 10.1039/d4cs00703d

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 17

  CôtéA. P.BeninA. I.OckwigN. W.O’KeeffeM.MatzgerA. J.YaghiO. M. (2005). Porous, crystalline, covalent organic frameworks. Science310 (5751), 1166–1170. 10.1126/science.1120411

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 18

  DalapatiS.JinE.AddicoatM.HeineT.JiangD. (2016). Highly emissive covalent organic frameworks. J. Am. Chem. Soc.138 (18), 5797–5800. 10.1021/jacs.6b02700

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 19

  DingS.-Y.DongM.WangY.-W.ChenY.-T.WangH.-Z.SuC.-Y.et al (2016). Thioether-based fluorescent covalent organic framework for selective detection and facile removal of mercury(II). J. Am. Chem. Soc.138 (9), 3031–3037. 10.1021/jacs.5b10754

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 20

  DuY.YangH.WhiteleyJ. M.WanS.JinY.LeeS.-H.et al (2016). Ionic covalent organic frameworks with spiroborate linkage. Angew. Chem. Int. Ed. Engl.55 (5), 1737–1741. 10.1002/anie.201509014

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 21

  FurtulaB.GutmanI. (2015). A forgotten topological index. J. Math. Chem.53, 1184–1190. 10.1007/s10910-015-0480-z

  • CrossRef
  • Google Scholar

• 22

  GaoQ.LiX.NingG.-H.XuH.-S.LiuC.TianB.et al (2018). Covalent organic framework with frustrated bonding network for enhanced carbon dioxide storage. Chem. Mater.30 (5), 1762–1768. 10.1021/acs.chemmater.8b00117

  • CrossRef
  • Google Scholar

• 23

  GengK.HeT.LiuR.DalapatiS.TanK. T.LiZ.et al (2020). Covalent organic frameworks: design, synthesis, and functions. Chem. Rev.120 (16), 8814–8933. 10.1021/acs.chemrev.9b00550

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 24

  GhaziZ. A.KhattakA. M.IqbalR.AhmadR.KhanA. A.UsmanM.et al (2018). Adsorptive removal of Cd²⁺ from aqueous solutions by a highly stable covalent triazine-based framework. New J. Chem.42 (12), 10234–10242. 10.1039/c8nj01778f

  • CrossRef
  • Google Scholar

• 25

  GutmanI.TrinajstićN. (1972). Graph theory and molecular orbitals. Total pi-electron energy of alternant hydrocarbons. Chem. Phys. Lett.17, 535–538. 10.1016/0009-2614(72)85099-1

  • CrossRef
  • Google Scholar

• 26

  HanX.-H.QiQ.-Y.ZhouZ.-B.ZhaoX. (2020). Designed synthesis of a two-dimensional covalent organic framework with three-level hierarchical porosity. Chin. J. Chem.38 (12), 1676–1680. 10.1002/cjoc.202000317

  • CrossRef
  • Google Scholar

• 27

  HussainN.HussainM.RiazA.HaidarA. (2025). Statistical modeling between entropy- and degree-based topological indices of two dimension carbon nitride monolayers. Chem. Pap.79, 8561–8573. 10.1007/s11696-025-04335-7

  • CrossRef
  • Google Scholar

• 28

  JacobK.ClementJ.ArockiarajM.PaulD.BalasubramanianK. (2023). Topological characterization and entropy measures of tetragonal zeolite merlinoites. J. Mol. Struct.1277, 134786. 10.1016/j.molstruc.2022.134786

  • CrossRef
  • Google Scholar

• 29

  JawaharK.ClementJ. (2026). Exploring entropy-energy relationships in PHI zeolite through topological indices. J. Chem. Inf. Model.66, 2820–2829. 10.1021/acs.jcim.5c03095

  • CrossRef
  • Google Scholar

• 30

  JingX.ZhangM.MuZ.ShaoP.ZhuY.LiJ.et al (2023). Gradient channel segmentation in covalent organic framework membranes with highly oriented nanochannels. J. Am. Chem. Soc.145 (38), 21077–21085. 10.1021/jacs.3c07393

  • CrossRef
  • Google Scholar

• 31

  JuniasJ. S.ClementJ. (2024). Predictive analytics of conductance and HOMO-LUMO gaps with topological descriptors of porphyrin nanosheets. Phys. Scr.99, 015208. 10.1088/1402-4896/ad0c1b

  • CrossRef
  • Google Scholar

• 32

  KalaamA. R. A.GreeniA. B. (2024). Comparative analysis of modified reverse degree topological indices for certain carbon nanosheets using entropy measures and multi criteria decision-making analysis. Int. J. Quantum Chem.124 (1), e27326. 10.1002/qua.27326

  • CrossRef
  • Google Scholar

• 33

  KalaamA. R. A.GreeniA. B. (2025). Comparative analysis of kekulene tessellation patterns using generalized reverse degree-sum descriptors combined with graph entropy and energy properties. Int. J. Quantum Chem.125 (8), e70043. 10.1002/qua.70043

  • CrossRef
  • Google Scholar

• 34

  KalaamA. R. A.GreeniA. B.ArockiarajM. (2024). Modified reverse degree descriptors for combined topological and entropy characterizations of 2D metal organic frameworks: applications in graph energy prediction. Front. Chem.12, 1470231. 10.3389/fchem.2024.1470231

  • CrossRef
  • Google Scholar

• 35

  KandambethS.BiswalB. P.ChaudhariH. D.RoutK. C.Kunjattu HS.MitraS.et al (2017). Selective molecular sieving in self-standing porous covalent-organic-framework membranes. Adv. Mater.29 (2), 1603945. 10.1002/adma.201603945

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 36

  KavithaS. R. J.AbrahamJ.ArockiarajM.JencyJ.BalasubramanianK. (2021). Topological characterization and graph entropies of tessellations of kekulene structures: existence of isentropic structures and applications to thermochemistry, nuclear magnetic resonance, and electron spin resonance. J. Phys. Chem. A125 (36), 8140–8158. 10.1021/acs.jpca.1c06264

  • CrossRef
  • Google Scholar

• 37

  KricheldorfH. R. (2006). Polypeptides and 100 years of chemistry of alpha-amino acid N-carboxyanhydrides. Angew. Chem. Int. Ed. Engl.45 (35), 5752–5784. 10.1002/anie.200600693

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 38

  KurianS.RoyS.GayathriK. B.JyothishK. (2025). Entropy comparison of benzothiadiazole-based covalent-organic frameworks. Front. Chem.13, 1704165. 10.3389/fchem.2025.1704165

  • CrossRef
  • Google Scholar

• 39

  LiX.CuiY.-Y.YangC.-X. (2021). Covalent coupling fabrication of microporous organic network bonded capillary columns for gas chromatographic separation. Talanta224, 121914. 10.1016/j.talanta.2020.121914

  • CrossRef
  • Google Scholar

• 40

  LiangR.-R.JiangS.-Y.AR.-H.ZhaoX. (2020). Two-dimensional covalent organic frameworks with hierarchical porosity. Chem. Soc. Rev.49 (12), 3920–3951. 10.1039/d0cs00049c

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 41

  LiaoH.WangH.DingH.MengX.XuH.WangB.et al (2016). A 2D porous porphyrin-based covalent organic framework for sulfur storage in lithium–sulfur batteries. J. Mater. Chem. A4 (19), 7416–7421. 10.1039/c6ta00483k

  • CrossRef
  • Google Scholar

• 42

  LiuR.JiaY.XiaY.WangS. (2026). Metalloporphyrin-based covalent organic frameworks: design, construction, and photocatalytic applications. Catalysts16 (1), 76. 10.3390/catal16010076

  • CrossRef
  • Google Scholar

• 43

  ManuelM.AngamuthuP. (2025). Mathematical modeling of H1-antihistamines: a QSPR approach using topological indices. ACS Omega10 (41), 49019–49034. 10.1021/acsomega.5c07577

  • CrossRef
  • Google Scholar

• 44

  MondalS.DasP.RazaZ.PalA.GhorbaniM. (2026). Graph spectrum of neighbourhood sombor matrix and structure-property modelling. Theor. Comput. Sci.1068, 115758. 10.1016/j.tcs.2026.115758

  • CrossRef
  • Google Scholar

• 45

  MowshowitzA. (1968). Entropy and the complexity of graphs. I. An index of the relative complexity of a graph. Bull. Math. Biophys.30, 175–204. 10.1007/BF02476948

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 46

  NaeemM.RaufA.AkhtarM. S.IqbalZ. (2024). QSPR modeling with curvilinear regression on the reverse entropy indices for the prediction of physicochemical properties of benzene derivatives. Polycycl. Aromat. Compd.44 (3), 1452–1469. 10.1080/10406638.2023.2196429

  • CrossRef
  • Google Scholar

• 47

  PaulD.ArockiarajM.EmiletD. A.GreeniA. B.KalaamA. R. A. (2025). Molecular descriptor-based QSPR analysis of physicochemical properties in neuromuscular drugs. Mod. Phys. Lett. B39 (28), 2550155. 10.1142/s0217984925501556

  • CrossRef
  • Google Scholar

• 48

  PeterP.ClementJ.ArockiarajM.JacobK. (2025). Predictive modeling of molecular interaction energies using topological and spectral entropies of zeolite AWW. Front. Chem.13, 1543588. 10.3389/fchem.2025.1543588

  • CrossRef
  • Google Scholar

• 49

  QianC.QiQ.-Y.JiangG.-F.CuiF.-Z.TianY.ZhaoX. (2017). Toward covalent organic frameworks bearing three different kinds of pores: the strategy for construction and COF-to-COF transformation via heterogeneous linker exchange. J. Am. Chem. Soc.139 (19), 6736–6743. 10.1021/jacs.7b02303

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 50

  RahulM. P.ClementJ.JuniasJ. S.ArockiarajM.BalasubramanianK. (2022). Degree-based entropies of graphene, graphyne and graphdiyne using Shannon’s approach. J. Mol. Struct.1260, 132797. 10.1016/j.molstruc.2022.132797

  • CrossRef
  • Google Scholar

• 51

  RaoY.ChenR.AhmadH.AhmadU. (2024). Reverse Zagreb indices and their application in the evaluation of physiochemical properties of anticancer/antibacterial drugs. ACS Omega9 (28), 31056–31080. 10.1021/acsomega.4c04409

  • CrossRef
  • Google Scholar

• 52

  RaviV. (2024). QSPR analysis of drugs used for treatment of hepatitis via reduced reverse degree-based topological descriptors. Phys. Scr.99, 105236. 10.1088/1402-4896/ad729d

  • CrossRef
  • Google Scholar

• 53

  RazaZ.ArockiarajM.MaaranA.ShaliniA. J. (2024). A comparative study of topological entropy characterization and graph energy prediction for marta variants of covalent organic frameworks. Front. Chem.12, 1511678. 10.3389/fchem.2024.1511678

  • CrossRef
  • Google Scholar

• 54

  TangJ.-H.HamrazA.UllahA.HamedY. S.ZamanS.BelayM. B. (2025). Chemical applicability and predictive potential of certain graphical indices for determining structure-property relationships in polycrystalline acid magenta (C₂₀H₁₇N₃Na₂O₉S₃). Sci. Rep.15, 13886. 10.1038/s41598-025-98024-w

  • CrossRef
  • Google Scholar

• 55

  TuY.YousafS.TariqS.TawfiqF. M.AslamA.TolaK. A. (2025). Computational analysis of metal organic framework and covalent organic framework using degree based topological indices with QSPR validation. Sci. Rep.15, 37311. 10.1038/s41598-025-21240-x

  • CrossRef
  • Google Scholar

• 56

  XiaoY.MaC.JinZ.WangJ.HeL.MuX.et al (2021). Functional covalent organic framework for exceptional Fe²⁺, Co²⁺ and Ni²⁺ removal: an upcycling strategy to achieve water decontamination and reutilization as smoke suppressant and flame retardant simultaneously. Chem. Eng. J.421, 127837. 10.1016/j.cej.2020.127837

  • CrossRef
  • Google Scholar

• 57

  XinJ.ZhouY.WangX.XuG.XieM.LiuL.et al (2021). Room-temperature synthesis of magnetic covalent organic frameworks for analyzing trace benzoylurea insecticide residue in tea beverages. Food Chem.347, 129075. 10.1016/j.foodchem.2021.129075

  • CrossRef
  • Google Scholar

• 58

  YaghiO. M. (2016). Reticular chemistry–construction, properties, and precision reactions of frameworks. J. Am. Chem. Soc.138 (48), 15507–15509. 10.1021/jacs.6b11821

  • Pubmed Abstract
  • CrossRef
  • Google Scholar

• 59

  YoussefM. Z.Al-DayelI.HanifM. F.SiddiquiM. K.AhmedH.TolasaF. T. (2024). On statistical evaluation of reverse degree based topological indices for iron telluride networks. Sci. Rep.14, 20533. 10.1038/s41598-024-71645-3

  • CrossRef
  • Google Scholar

• 60

  YuX.ChenZ.LiX. (2022). Reticular chemistry in action: 3D porphyrinic COFs with scu topology. Chem8 (11), 2899–2901. 10.1016/j.chempr.2022.10.013

  • CrossRef
  • Google Scholar

• 61

  YuG.SiddiquiM. K.HussainM.HussainN.SaddiqueZ.PetrosF. B. (2024). On topological indices and entropy measures of beryllonitrene network via logarithmic regression model. Sci. Rep.14, 7187. 10.1038/s41598-024-57601-1

  • CrossRef
  • Google Scholar

• 62

  YusranY.MiaoB.QiuS.FangQ. (2024). Functional covalent organic frameworks: design principles to potential applications. Acc. Mater. Res.5 (10), 1263–1278. 10.1021/accountsmr.4c00195

  • CrossRef
  • Google Scholar

• 63

  ZhangX.ArockiarajM.MaaranA.DossC. I. A. (2025). Topological information entropy and spectral energy analysis in aminal-linked covalent organic frameworks. Sci. Rep.15, 19861. 10.1038/s41598-025-01210-z

  • CrossRef
  • Google Scholar

• 64

  ZhangG.LiY.RaufA.AfzalM. A.AliP.AslamA. (2025). Estimating the prediction ability of reverse degree-based entropy indices for the physicochemical properties of lymes disease drugs. Front. Phys.13, 1536995. 10.3389/fphy.2025.1536995

  • CrossRef
  • Google Scholar