Two-dimensional metal organic frameworks (MOFs) are revolutionizing nanoscale research with their unique blend of inorganic and organic components, offering exceptional advantages like porosity and tunability. These porous crystals feature cage-like architectures formed by aromatic organic moieties and square-planar metal ions, finding diverse applications in gas catalysis, drug delivery, sensors, optoelectronics, storage, and adsorption (Kinoshita et al., 1959; Lee et al., 2009; Horcajada et al., 2008; Murray et al., 2009). Metal-organic surfaces based on metal-butylated hydroxytoluene (MBHT) exhibit promising electronic and magnetic properties, particularly for transition metals like M = {Co, Fe, Mn, Cr} (Clough et al., 2015; Chakravarty et al., 2016a). Among MBHT-derived materials, CoBHT, FeBHT, and MnBHT display planar ferromagnetic half-metallism, while CrBHT possesses a spin-frustrated kagome lattice leading to antiferromagnetic semimetallic behavior. These frameworks exhibit high sensitivity towards gas molecules like carbon monoxide, altering their electronic and magnetic properties significantly upon adsorption. A coronene molecule substituted with an − NH group, complexed with transition metals (NHC-TM), presents a promising pathway to developing novel materials for spintronic devices (Chakravarty et al., 2016b). These MOFs which feature coronene molecules bonded to transition metals in a square planar geometry, exhibit favorable formation energy, making practical synthesis feasible and enhancing control over their magnetic and electronic properties (Dong et al., 2018).

Transition metal-phthalocyanine (TM-Pc) based MOFs exhibit captivating two-dimensional structures with distinctive electronic and magnetic features. TM-Pc is derived from the transition metal-tetracyanobenzene (TM-TCNB) framework through benzene ring rotation and on-surface polymerization, involving transition metals TM = {Ti, V, Cr, Co, Ni, Cu, Zn} (Mabrouk and Hayn, 2015). TM-Pc demonstrates superior stability over TM-TCNB by approximately 7 eV per cell, revealing local energy minima in free-standing layers. Both materials, characterized by TM²⁺ states, showcase potential for spintronics (Mabrouk et al., 2018). Metal-octa amino phthalocyanine (MOAPc) shows significant promise in applications like energy storage, catalysis, and sensing due to their unique bimetallic properties. Co²⁺, Ni²⁺, and Cu²⁺ serve as both metal centers and nodes within the MOAPc lattice (Li et al., 2017). This exploration provides insights into the effects of metal substitutions, enriching the understanding of MOAPc-based MOFs for diverse applications (Park et al., 2023).

In mathematical chemistry, graph theoretical techniques are employed to study molecular structures, properties, and reactions. Topological descriptors play a crucial role in this field, serving as essential tools for analyzing complex molecular systems. These descriptors are numerical values derived from molecular structure connectivity, representing the positions of atoms and bonds within the molecule. The widespread use of distance-based indices like the Wiener index and degree-based indices such as the Zagreb indices has significantly advanced the field of topological indices (Wiener, 1947; Gutman and Trinajstić, 1972; Raza et al., 2023a; Arockiaraj et al., 2023a). Applications of degree and distance-based topological indices are creating new possibilities in drug discovery and neural network research (Zhang et al., 2022; Zhang et al., 2024; Arockiaraj et al., 2024a). In particular, the robust refinement of reverse degree indices significantly improved the correlation with the physicochemical properties of molecules and was applied to drug compounds related to coronavirus, blood cancer and cardiovascular drug compounds (Arockiaraj et al., 2023b; Arockiaraj et al., 2023c; Arockiaraj et al., 2024b). This approach develops various graph degree sequences with variable parameters, enhancing statistical models for the considered datasets. In this work, we implement the modified reverse degree method to the recently introduced hybrid topological indices (Arockiaraj et al., 2023a).

Structural entropy, introduced by Shannon, deals with unpredictability or uncertainty in datasets (Shannon, 1948; Arockiaraj et al., 2023d). It indicates micro-state diversity, reflecting various positions in a system with atoms and molecules. Higher entropy within a system implies greater disorder, signifying more potential micro-states. This principle extends to chemical structures, providing a valuable tool for analyzing their stability and structural data (Dehmer, 2008). Graph entropies link probability distributions to graph elements, such as vertices and edges, aiding in comprehensive graph analysis. These explorations aid in predicting the graph energies of MOFs using graph theoretical and statistical techniques.

Recent studies on degree-based descriptors have been instrumental in analyzing various metal organic and covalent organic frameworks, such as phthalocyanine frameworks, trans-Pd–(NH₂)S lattice, metal butylated hydroxytoluene frameworks, and FeTPyP-CO MOFs (Nadeem et al., 2021; Azeem et al., 2021; Zaman et al., 2023; Yu et al., 2023; Al-Dayel et al., 2024). Additionally, entropy-based investigations have focused on metal phthalocyanine COFs, isoreticular metal-organic frameworks, and coronene-based MOFs (Arockiaraj et al., 2023e; Abraham et al., 2022; Manzoor et al., 2021; Raza et al., 2023b; Imran et al., 2023; Waheed et al., 2023; Ghani et al., 2022; Abul Kalaam and Berin Greeni, 2024; Yang et al., 2023). This paper presents the implementation of modified reverse degree-based descriptors for four types of MOFs, the computation of entropy measures through bond-wise comparative analysis, and the development of predictive models for graph energy.

In this study, MOFs are displayed through two-dimensional graph structures. We denote such a two-dimensional structure by G with | V ( G ) | and | E ( G ) | indicating the number of vertices and edges in the graph G respectively. In this context, the term vertex degree, denoted as d v , represents the total number of vertices adjacent to vertex v . The maximum degree in graph G is denoted as Δ ( G ) . The reverse version of the degree (Ediz, 2015) and its generalized form (Arockiaraj et al., 2023b) have received significant attention in recent years. It is denoted as M k R d v , incorporating a variable parameter k ( k ≥ 1 ) and defined as M k R d v = Δ G − d v + k : k ≤ d v Δ G − d v + k m o d Δ G : k > d v Therefore, the general form of topological descriptors ( T D ) for the modified reverse degree classification of metal organic frameworks is defined as follows: M k R T D G = ∑ u v ∈ E G M k R T D d u , d v = ∑ u v ∈ E G T D M k R d u , M k R d v

Here, T D ( M k R ( d u ) , M k R ( d v ) ) denotes the topological descriptors function, ensuring symmetrical mutuality as given below. T D M k R d u , M k R d v = T D M k R d v , M k R d u Suppose the edge set of G is partitioned into equivalence classes E ( G ) = { E 1 ∪ E 2 ∪ ⋯ ∪ E n } , such that each edge in the class E i has the same modified degree parameters at the end vertices. Then, the modified reverse degree topological descriptor for the class E i is expressed in the following form with u v ∈ E i . M k R T D E i = | E i | ⋅ T D M k R d u , M k R d v

Therefore, the overall modified reverse degree descriptors for graph G is calculated by summing the individual contributions from each class E i . M k R T D G = ∑ i = 1 n | E i | ⋅ T D M k R d u , M k R d v The modified reverse degree based topological descriptor functions considered in this study are stated below.

• Modified reverse first Zagreb descriptor: M k R M 1 d u , d v = M k R d u + M k R d v (1)     • Modified reverse second Zagreb descriptor: M k R M 2 d u , d v = M k R d u × M k R d v (2)     • Modified reverse forgotten descriptor: M k R F d u , d v = M k R d u 2 + M k R d v 2 (3)     • Modified reverse hyper-Zagreb descriptor: M k R H Z d u , d v = M k R d u + M k R d v 2 (4)     • Modified reverse third redefined Zagreb descriptor: M k R R e Z 3 d u , d v = M k R d u + M k R d v × M k R d u × M k R d v (5)     • Modified reverse bi-Zagreb descriptor: M k R B M d u , d v = M k R d u + M k R d v + M k R d u × M k R d v (6)     • Modified reverse tri-Zagreb descriptor: M k R B M d u , d v = M k R d u 2 + M k R d v 2 + M k R d u × M k R d v (7)     • Modified reverse bi-Zagreb harmonic descriptor: M k R B M H d u , d v = M k R d u + M k R d v + M k R d u × M k R d v × M k R d u + M k R d v 2 (8)

• Modified reverse tri-Zagreb harmonic descriptor: M k R T M H d u , d v = M k R d u 2 + M k R d v 2 + M k R d u × M k R d v × M k R d u + M k R d v 2 (9)

Metal organic frameworks based on transition metal-phthalocyanine, conductive metal-octa amino phthalocyanine, metal-butylated hydroxytoluene, and NH-substituted coronene transition metal are respectively denoted as TM-Pc ( m , n ) , MOAPc ( m , n ) , MBHT ( m , n ) and NHC-TM ( m , n ) where m and n representing the void space in the rows and columns, as dimensions (2,3) shown in Figures 1, 2. The graph representations have the following properties: for TM-Pc ( m , n ) , | V ( G ) | = 29 m n + 23 ( m + n ) + 17 and | E ( G ) | = 40 m n + 30 ( m + n ) + 20 ; for MOAPc ( m , n ) , | V ( G ) | = 51 m n + 52 ( m + n ) + 53 and | E ( G ) | = 68 m n + 68 m + 68 n + 68 ; for MBHT ( m , n ) , | V ( G ) | = 27 m n + 28 ( m + n ) + 1 and | E ( G ) | = 36 m n + 36 m + 36 n ; and for NHC-TM ( m , n ) , | V ( G ) | = 34 m n + 35 ( m + n ) + 36 and | E ( G ) | = 46 m n + 46 m + 46 n + 46 . From these MOFs, we observe that they have a maximum degree of 4. Hence, the modified reverse degrees of the vertices are presented below. M 1 R d v = 4 : d v = 1 3 : d v = 2 2 : d v = 3 1 : d v = 4 M 2 R d v = 1 : d v = 1 4 : d v = 2 3 : d v = 3 2 : d v = 4 M 3 R d v = 2 : d v = 1 1 : d v = 2 4 : d v = 3 3 : d v = 4

To compute the bond-additive modified reverse degree-based topological descriptors, we need the edge partitions of MOFs based on the normal degrees, which are presented in Table 1. Given the complexity of computing each descriptor and variable parameter, we illustrate the first Zagreb descriptor in Equation 1 with the modified reverse variable k = 1 for the TM-Pc framework. The normal degree classes (2,2), (2,3), (3,3), and (3,4) are modified into (3,3), (3,2), (2,2), and (2,1), respectively, when the reverse variable is fixed to k = 1 . Therefore, M 1 R M 1 TM - Pc m , n = 4 m + n + 2 ⋅ 3 + 3 + 16 m n + 12 + m + n + 8 ⋅ 3 + 2 + 20 m n + 10 m + n ⋅ 2 + 2 + 4 m n + m + n + 1 ⋅ 2 + 1 M 1 R M 1 TM - Pc m , n = 172 m n + 136 m + n + 100 For k = 2 , the normal degree classes (2,2), (2,3), (3,3), and (3,4) are modified to (4,4), (4,3), (3,3), and (3,2) respectively. Therefore, M 2 R M 1 TM - Pc m , n = 4 m + n + 2 ⋅ 4 + 4 + 16 m n + 12 + m + n + 8 ⋅ 4 + 3 + 20 m n + 10 m + n ⋅ 3 + 3 + 4 m n + m + n + 1 ⋅ 3 + 2 M 2 R M 1 TM - Pc m , n = 252 m n + 196 m + n + 140 Similarly for k = 3 , the normal degree classes (2,2), (2,3), (3,3), and (3,4) are modified to (1,1), (1,4), (4,4), and (4,3) respectively. Hence, M 3 R M 1 TM - Pc m , n = 4 m + n + 2 ⋅ 1 + 1 + 16 m n + 12 + m + n + 8 ⋅ 1 + 4 + 20 m n + 10 m + n ⋅ 4 + 4 + 4 m n + m + n + 1 ⋅ 4 + 3 M 3 R M 1 TM - Pc m , n = 268 m n + 176 m + n + 84 Using the modified reverse topological descriptors detailed in Equations 1–9 and the edge partitions in Table 1, we computed the topological indices for MOFs, which are systematically presented in Tables 2–5 for reversing parameters k = 1,2 , and 3 .

We computed the numerical values for the derived topological descriptors of MOFs, which are presented in Tables 6–8 and illustrated in Figure 3. It is clear that the majority of descriptors have largest numerical values for the MOAPc framework, indicating a high level of structural coherence in these MOFs. For all frameworks, the modified reverse degree descriptors M R M 1 , M R M 2 , M R B M , M R F , and M R T M are numerically smaller compared to the other descriptors. This suggests that these descriptors should be prioritized when evaluating large frameworks to avoid potential mathematical complexity caused by the exponential growth of other descriptors.

Consider a subset X that belongs to the union of V ( G ) and E ( G ) . Let f be a function that provides structural information, defined on X and mapping to positive real numbers ( f : X → R + ) . Assuming X = E ( G ) = { E 1 ∪ E 2 ∪ ⋯ ∪ E n } , the bond graph entropy of G concerning the topological descriptors function f ∈ { M k R T D } is given by I f G = − ∑ u v ∈ E G f u , v ∑ u v ∈ E G f u , v log f u , v ∑ u v ∈ E G f u , v = log ∑ i = 1 n f E i − 1 ∑ i = 1 n f E i log ∏ i = 1 n f u , v f u , v | E i | A recent study (Paul et al., 2023) scrutinized the aforementioned expression, involving the replacement of the term ∏ i = 1 n f u , v f u , v | E i | with ∏ i = 1 n | E i | ⋅ f u , v f u , v . Consequently, our study adopts the resulting entropy formula. I f G = log ∑ i = 1 n f E i − 1 ∑ i = 1 n f E i log ∏ i = 1 n | E i | ⋅ f u , v f u , v We now illustrate the calculation of the first Zagreb entropy value for the TM-Pc structure. Let G represent the TM-Pc metal organic framework. Upon substitution into the entropy equation, we obtain I M 1 R M 1 G = log M 1 R M 1 − 1 M 1 R M 1 log ∏ i = 1 n | E i | ⋅ M k R d u + M k R d v M k R d u + M k R d v By substituting the edge partition classes specified in Table 1, we can express entropy for TM-Pc ( m , n ) based on M 1 descriptor as follows. I M 1 R M 1 TM - Pc m , n = log 172 m n + 136 m + n + 100 − 1 172 m n + 136 m + n + 100 log 4 m + n + 2 × 3 + 3 3 + 3 × 16 m n + 12 m + n + 8 × 3 + 2 3 + 2 × 20 m n + 10 m + n × 2 + 2 2 + 2 × 4 m n + m + n + 1 × 2 + 1 2 + 1

Assuming m = n = 3 , we acquire I M 1 R M 1 TM - Pc 3,3 = log 2464 − 1 2464 log 1007769600000 × 110100480 I M 1 R M 1 TM - Pc 3,3 = 3.39164070349 − 0.00040584415 × 20.0451504658 I M 1 R M 1 TM - Pc 3,3 = 3.38350 The above-outlined approach is implemented to calculate the entropy levels for MOFs based on the reverse degree indices. We would like to point out that recent literature includes a comprehensive comparative analysis across diverse chemical structures such as graphene, graphyne, graphdiyne, C 4 C₈ nanosheets, honeycomb network, γ -graphyne, kekulene structures, zigzag graphyne nanoribbons, and carbon nanosheets (Govardhan et al., 2024; Rahul et al., 2022; Raja and Anuradha, 2024; Peter and Clement, 2024; Kavitha et al., 2021) based on degree indices. Therefore, the current investigation focuses on modified reverse degree-based entropy levels to provide robust measures, thus assessing their effectiveness and offering insights for potential structure developments. A comparative analysis is presented in Tables 7–10, highlighting the impact of varying the reverse parameter k on entropy levels across different MOFs.

In evaluating the entropy levels from Tables 9–12, we explore dynamic variations across MOFs at k = 1,2,3 . Notably, entropy consistently increases at k = 3 compared to k = 1 and 2, with M 3 R T M H showing highest levels. However, for MOAPc and MBHT, the measures show minimal difference at k = 2 and k = 3 . To assess the complexity, the direct comparison is challenging due to edge variability. Thus, we incorporate scaled bond-wise entropy measures ( B I s ) , normalizing entropy by considering the number of edges in the frameworks (Arockiaraj et al., 2023d). The bond-wise entropy B I s offers a detailed perspective on structural characteristics and stability dynamics within MOFs. For example, TM-Pc(1,3) has I M 3 R T M H = 4.4325 , and | E ( TM - Pc ( 1,3 ) ) | =260, then the bond-wise entropy is measured for TM-Pc(1,3) by the following formula. B I M 3 R T M H TM - Pc 1,3 = I M 3 R T M H TM - Pc 1,3 | E TM - Pc 1,3 | = 4.4325 260 = 0.017048 Now, we consider the bond ranges with reasonably acceptable classes and compute the bond-wise entropy for the tri-Zagreb harmonic index, which are represented in Table 11.

Table 13 and Figure 4 provide a comparison of bond-wise entropies across different structures, maintaining consistent | E ( G ) | ranges among all MOFs. Bond-wise entropy, representing entropy per bond, provides intuitive comparisons for each molecular structure. Across all frameworks, TM-Pc exhibits the highest normalized entropy at all dimensions, while NHC-TM and MBHT display lower values, suggesting lesser complexity. Additionally, as the frameworks expand in dimensions ( m , n ) , the normalized entropies of TM-Pc, MOAPc, NHC-TM and MBHT converge, indicating comparable complexity levels across maximum-dimensional MOFs, regardless of bonding patterns.