Modified reverse degree descriptors for combined topological and entropy characterizations of 2D metal organic frameworks: applications in graph energy prediction

Topological descriptors are widely utilized as graph theoretical measures for evaluating the physicochemical properties of organic frameworks by examining their molecular structures. Our current research validates the usage of topological descriptors in studying frameworks such as metal-butylated hydroxytoluene, NH-substituted coronene transition metal, transition metal-phthalocyanine, and conductive metal-octa amino phthalocyanine. These metal organic frameworks are crucial in nanoscale research for their porosity, adaptability, and conductivity, making them essential for advanced materials and modern technology. In this study, we provide the topological and entropy characterizations of these frameworks by employing robust reverse degree based descriptors, which offer insightful information on structural complexities. This structural information is applied to predict the graph energy of the considered metal organic frameworks using statistical regression models.

1 Introduction

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 $7\text{eV}$ 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.

2 Methodology

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 $\mathrm{\Delta }(G)$. 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 $\left(M_k\mathfrak{R}\left(d_v\right)\right)$, incorporating a variable parameter $k$ $(k\ge 1)$ and defined as

$M_k\mathfrak{R}\left(d_v\right)=\left\{\begin{array}{cc}\mathrm{\Delta }\left(G\right)-d_v+k\hfill & :k\le d_v\hfill \\ \mathrm{\Delta }\left(G\right)-d_v+k\left(mod\mathrm{\Delta }\left(G\right)\right)\hfill & :k>d_v\hfill \end{array}\right.$

Therefore, the general form of topological descriptors $(TD)$ for the modified reverse degree classification of metal organic frameworks is defined as follows:

$M_k\mathfrak{R}TD\left(G\right)={\displaystyle \sum _{uv\in E\left(G\right)}}{M}_k\mathfrak{R}TD\left(d_u,d_v\right)={\displaystyle \sum _{uv\in E\left(G\right)}}TD\left(M_k\mathfrak{R}\left(d_u\right),M_k\mathfrak{R}\left(d_v\right)\right)$

Here, $TD(M_k\mathfrak{R}(d_u),M_k\mathfrak{R}(d_v))$ denotes the topological descriptors function, ensuring symmetrical mutuality as given below.

$TD\left(M_k\mathfrak{R}\left(d_u\right),M_k\mathfrak{R}\left(d_v\right)\right)=TD\left(M_k\mathfrak{R}\left(d_v\right),M_k\mathfrak{R}\left(d_u\right)\right)$

Suppose the edge set of $G$ is partitioned into equivalence classes $E(G)=\{E_1\cup E_2\cup \cdots \cup E_n\}$, 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 $uv\in E_i$.

$M_k\mathfrak{R}TD\left(E_i\right)=|E_i|\cdot TD\left(M_k\mathfrak{R}\left(d_u\right),M_k\mathfrak{R}\left(d_v\right)\right)$

Therefore, the overall modified reverse degree descriptors for graph $G$ is calculated by summing the individual contributions from each class $E_i$.

$M_k\mathfrak{R}TD\left(G\right)={\displaystyle \sum _{i=1}^n}|E_i|\cdot TD\left(M_k\mathfrak{R}\left(d_u\right),M_k\mathfrak{R}\left(d_v\right)\right)$

The modified reverse degree based topological descriptor functions considered in this study are stated below.

• Modified reverse first Zagreb descriptor:

$M_k\mathfrak{R}{M}_1\left(d_u,d_v\right)=M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)(1)$

    • Modified reverse second Zagreb descriptor:

$M_k\mathfrak{R}{M}_2\left(d_u,d_v\right)=M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)(2)$

    • Modified reverse forgotten descriptor:

$M_k\mathfrak{R}F\left(d_u,d_v\right)={\left(M_k\mathfrak{R}\left(d_u\right)\right)}^2+{\left(M_k\mathfrak{R}\left(d_v\right)\right)}^2(3)$

    • Modified reverse hyper-Zagreb descriptor:

$M_k\mathfrak{R}HZ\left(d_u,d_v\right)={\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)\right)}^2(4)$

    • Modified reverse third redefined Zagreb descriptor:

$M_k\mathfrak{R}Re{Z}_3\left(d_u,d_v\right)=\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)\right)\times \left(M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)\right)(5)$

    • Modified reverse bi-Zagreb descriptor:

$M_k\mathfrak{R}BM\left(d_u,d_v\right)=\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)+\left(M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)\right)\right)(6)$

    • Modified reverse tri-Zagreb descriptor:

$M_k\mathfrak{R}BM\left(d_u,d_v\right)=\left({\left(M_k\mathfrak{R}\left(d_u\right)\right)}^2+{\left(M_k\mathfrak{R}\left(d_v\right)\right)}^2+\left(M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)\right)\right)(7)$

    • Modified reverse bi-Zagreb harmonic descriptor:

$M_k\mathfrak{R}BMH\left(d_u,d_v\right)=\frac{\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)+\left(M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)\right)\right)\times \left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)\right)}{2}(8)$

• Modified reverse tri-Zagreb harmonic descriptor:

$M_k\mathfrak{R}TMH\left(d_u,d_v\right)=\frac{\left({\left(M_k\mathfrak{R}\left(d_u\right)\right)}^2+{\left(M_k\mathfrak{R}\left(d_v\right)\right)}^2+\left(M_k\mathfrak{R}\left(d_u\right)\times M_k\mathfrak{R}\left(d_v\right)\right)\right)\times M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)}{2}(9)$

3 Computation of modified reverse degree metrics

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$(m,n)$, $|V(G)|=29mn+23(m+n)+17$ and $|E(G)|=40mn+30(m+n)+20$; for MOAPc$(m,n)$, $|V(G)|=51mn+52(m+n)+53$ and $|E(G)|=68mn+68m+68n+68$; for MBHT$(m,n)$, $|V(G)|=27mn+28(m+n)+1$ and $|E(G)|=36mn+36m+36n$; and for NHC-TM$(m,n)$, $|V(G)|=34mn+35(m+n)+36$ and $|E(G)|=46mn+46m+46n+46$. From these MOFs, we observe that they have a maximum degree of 4. Hence, the modified reverse degrees of the vertices are presented below.

$M_1\mathfrak{R}\left(d_v\right)=\left\{\begin{array}{cc}4\hfill & :d_v=1\hfill \\ 3\hfill & :d_v=2\hfill \\ 2\hfill & :d_v=3\hfill \\ 1\hfill & :d_v=4\hfill \end{array}\right.$

$M_2\mathfrak{R}\left(d_v\right)=\left\{\begin{array}{cc}1\hfill & :d_v=1\hfill \\ 4\hfill & :d_v=2\hfill \\ 3\hfill & :d_v=3\hfill \\ 2\hfill & :d_v=4\hfill \end{array}\right.$

$M_3\mathfrak{R}\left(d_v\right)=\left\{\begin{array}{cc}2\hfill & :d_v=1\hfill \\ 1\hfill & :d_v=2\hfill \\ 4\hfill & :d_v=3\hfill \\ 3\hfill & :d_v=4\hfill \end{array}\right.$

Figure 1

Figure 1. MOFs of dimensions (2,3) (A) TM-Pc and (B) MBHT.

Figure 2

Figure 2. MOFs of dimensions (2,3) (A) NHC-TM and (B) MOAPc.

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,

$\begin{array}{l}{M}_1\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=\left(4\left(m+n+2\right)\right)\cdot \left(3+3\right)\\ +\left(16mn+12+\left(m+n\right)+8\right)\\ \cdot \left(3+2\right)+\left(20mn+10\left(m+n\right)\right)\\ \cdot \left(2+2\right)+\left(4\left(mn+m+n+1\right)\right)\\ \cdot \left(2+1\right)\\ M_1\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=172mn+136\left(m+n\right)+100\end{array}$

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,

$\begin{array}{l}{M}_2\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=\left(4\left(m+n+2\right)\right)\cdot \left(4+4\right)\\ +\left(16mn+12+\left(m+n\right)+8\right)\\ \cdot \left(4+3\right)+\left(20mn+10\left(m+n\right)\right)\\ \cdot \left(3+3\right)+\left(4\left(mn+m+n+1\right)\right)\\ \cdot \left(3+2\right)\\ M_2\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=252mn+196\left(m+n\right)+140\end{array}$

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,

$\begin{array}{l}{M}_3\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=\left(4\left(m+n+2\right)\right)\cdot \left(1+1\right)\\ +\left(16mn+12+\left(m+n\right)+8\right)\\ \cdot \left(1+4\right)+\left(20mn+10\left(m+n\right)\right)\\ \cdot \left(4+4\right)+\left(4\left(mn+m+n+1\right)\right)\\ \cdot \left(4+3\right)\\ M_3\mathfrak{R}{M}_1\left(\text{TM}-\text{Pc}\left(m,n\right)\right)=268mn+176\left(m+n\right)+84\end{array}$

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 $k=\mathrm{1,2},$ and $3$.

Table 1

Table 1. Edge partitions of MOFs based on vertex degrees.

Table 2

Table 2. Modified degree descriptors for TM-Pc MOF with reversing parameters $k$ =1, 2, 3.

Table 3

Table 3. Modified reverse degree descriptors for MOAPc MOFs with reversing parameters $k$=1, 2, 3.

Table 4

Table 4. Modified reverse degree descriptors for MBHT MOFs with reversing parameters $k$=1, 2, 3.

Table 5

Table 5. Modified reverse degree descriptors for NHC-TM MOFs with reversing parameters $k$=1, 2, 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 $M\mathfrak{R}{M}_1$, $M\mathfrak{R}{M}_2$, $M\mathfrak{R}BM$, $M\mathfrak{R}F$, and $M\mathfrak{R}TM$ 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.

Table 6

Table 6. Modified reverse degree based topological descriptors for MOFs when $k=1$.

Table 7

Table 7. Modified reverse degree based topological descriptors for MOFs when $k=2$.

Table 8

Table 8. Modified reverse degree based topological descriptors for MOFs when $k=3$.

Figure 3

Figure 3. Graphical comparison of numerical descriptors for all MOFs across a range of $k$ values. (A) Comparision of numerical descriptor values between MOFs when $k=1$. (B) Comparision of numerical descriptor values between MOFs when $k=2$ (C) Comparision of numerical descriptor values between MOFs when $k=3$

4 Comparative study of entropy levels

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 $(f:X\to {\mathbb{R}}^{+})$. Assuming $X=E(G)=\{E_1\cup E_2\cup \cdots \cup E_n\}$, the bond graph entropy of $G$ concerning the topological descriptors function $f\in \{M_k\mathfrak{R}TD\}$ is given by

$\begin{array}{ll}\hfill I_f\left(G\right)& =-{\displaystyle \sum _{uv\in E\left(G\right)}}\frac{f\left(u,v\right)}{{\sum }_{uv\in E\left(G\right)}f\left(u,v\right)}\log \left[\frac{f\left(u,v\right)}{{\sum }_{uv\in E\left(G\right)}f\left(u,v\right)}\right]\hfill \\ \hfill & =\log \left[{\displaystyle \sum _{i=1}^n}f\left(E_i\right)\right]-\frac{1}{{\sum }_{i=1}^{n}f\left(E_i\right)}\log \left[\prod _{i=1}^n{f{\left(u,v\right)}^{f\left(u,v\right)}}^{|E_i|}\right]\hfill \end{array}$

A recent study (Paul et al., 2023) scrutinized the aforementioned expression, involving the replacement of the term $\prod _{i=1}^n{f{\left(u,v\right)}^{f\left(u,v\right)}}^{|E_i|}$ with $\prod _{i=1}^n|E_i|\cdot f{\left(u,v\right)}^{f\left(u,v\right)}$. Consequently, our study adopts the resulting entropy formula.

$I_f\left(G\right)=\log \left[{\displaystyle \sum _{i=1}^n}f\left(E_i\right)\right]-\frac{1}{{\sum }_{i=1}^{n}f\left(E_i\right)}\log \left[\prod _{i=1}^n|E_i|\cdot f{\left(u,v\right)}^{f\left(u,v\right)}\right]$

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

$I_{M_1\mathfrak{R}{M}_1}\left(G\right)=\log \left(M_1\mathfrak{R}{M}_1\right)-\frac{1}{\left(M_1\mathfrak{R}{M}_1\right)}\log \left[\prod _{i=1}^n|E_i|\cdot {\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)\right)}^{\left(M_k\mathfrak{R}\left(d_u\right)+M_k\mathfrak{R}\left(d_v\right)\right)}\right]$

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.

$\begin{array}{cc}\hfill I_{M_1\mathfrak{R}{M}_1}\left(\text{TM}-\text{Pc}\left(m,n\right)\right)& =\log \left(172mn+136\left(m+n\right)+100\right)\hfill \\ \hfill & -\frac{1}{172mn+136\left(m+n\right)+100}\log \left[\left(4\left(m+n+2\right)\right)\times {\left(3+3\right)}^{\left(3+3\right)}\right.\hfill \\ \hfill & \times \left(16mn+12\left(m+n\right)+8\right)\times {\left(3+2\right)}^{\left(3+2\right)}\times \left(20mn+10\left(m+n\right)\right)\hfill \\ \hfill & \left.\times {\left(2+2\right)}^{\left(2+2\right)}\times \left(4\left(mn+m+n+1\right)\right)\times {\left(2+1\right)}^{\left(2+1\right)}\right]\hfill \end{array}$

Assuming $m=n=3$, we acquire

$\begin{array}{c}{I}_{M_1\mathfrak{R}{M}_1}\left(\text{TM}-\text{Pc}\left(\mathrm{3,3}\right)\right)=\log \left(2464\right)\hfill \\ -\frac{1}{2464}\log \left[1007769600000\times 110100480\right]\\ I_{M_1\mathfrak{R}{M}_1}\left(\text{TM}-\text{Pc}\left(\mathrm{3,3}\right)\right)=3.39164070349\\ -\left(0.00040584415\times 20.0451504658\right)\\ I_{M_1\mathfrak{R}{M}_1}\left(\text{TM}-\text{Pc}\left(\mathrm{3,3}\right)\right)=3.38350\end{array}$

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, ${\mathrm{C}}_4$C₈ nanosheets, honeycomb network, $\gamma$-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 $k=\mathrm{1,2,3}$. Notably, entropy consistently increases at $k=3$ compared to $k=1$ and 2, with $M_3\mathfrak{R}TMH$ 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 $(BIs)$, normalizing entropy by considering the number of edges in the frameworks (Arockiaraj et al., 2023d). The bond-wise entropy $BIs$ offers a detailed perspective on structural characteristics and stability dynamics within MOFs. For example, TM-Pc(1,3) has $I_{M_3\mathfrak{R}TMH}=4.4325$, and $|E(\text{TM}-\text{Pc}(\mathrm{1,3}))|$=260, then the bond-wise entropy is measured for TM-Pc(1,3) by the following formula.

$B{I}_{M_3\mathfrak{R}TMH}\left(\text{TM}-\text{Pc}\left(\mathrm{1,3}\right)\right)=\frac{I_{M_3\mathfrak{R}TMH}\left(\text{TM}-\text{Pc}\left(\mathrm{1,3}\right)\right)}{|E\left(\text{TM}-\text{Pc}\left(\mathrm{1,3}\right)\right)|}=\frac{4.4325}{260}=0.017048$

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.

Table 9

Table 9. Comparing entropy levels for TM-Pc$(m,m)$ based MOFs between $k=1$, $k=2$ and $k=3$.

Table 10

Table 10. Comparing entropy measures for MOAPc$(m,m)$ based MOFs between $k=1$, $k=2$ and $k=3$.

Table 11

Table 11. Comparing entropy levels for NHC-TM$(m,m)$ based MOFs between $k=1$, $k=2$ and $k=3$.

Table 12

Table 12. Comparing entropy levels for MBHT$(m,m)$ based MOFs between $k=1$, $k=2$ and $k=3$.

Table 13

Table 13. Bond-wise entropy comparison among MOFs with similar bond ranges.

Figure 4

Figure 4. Graphical comparison of bond-wise entropy levels among MOFs with different bond ranges.

5 Estimating graph energy with regression models

Graph energy significantly influences electronic properties, impacting chemical reactions, materials science, energy conservation, and optoelectronics. Moreover, graph energy directly relates to total $\pi$-electronic energy. In this study, we use integrated software newGRAPH for computing the adjacency matrix from molecular structures (Stevanović et al., 2021). Let $\mathcal{A}(G)$ represent the adjacency matrix of metal organic framework $G$ with order $m$. The eigenvalues of $\mathcal{A}(G)$, denoted ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$, constitute graphs spectrum (Gutman, 1978). The energy $E_{\pi }(G)$ of the graph $G$ is determined by the absolute sum of the eigenvalues of the spectrum.

$E_{\pi }\left(G\right)={\displaystyle \sum _{i=1}^m}|{\lambda }_i|$

In the literature, studies focused on distance, degree, degree-sum based topological descriptors, and energies for structures such as zeolite frameworks, benzenoid hydrocarbons, polyhex nanotubes, hypercubes, and porous graphene (Taherpour and Mohammadinasab, 2010; Hayat et al., 2020; Hayat et al., 2021a; Balasubramanian, 2023; Govardhan and Roy, 2023; Hayat et al., 2021b). However, in this study, we investigate the relationship between graph energy $E_{\pi }(G)$ and various modified reverse degree descriptors. High correlation coefficient values between these descriptors and $E_{\pi }(G)$ underscore the expanded importance of graph energy beyond its conventional use in Hückel molecular orbital theory (Gutman et al., 2017).

Calculating graph energy for higher-order dimensions $(m,n)$ of metal organic frameworks using newGRAPH software can be complex as it involves generating the adjacency matrix for higher dimensions. However, we develop statistical models to accurately predict energy for these higher orders by integrating all four distinct frameworks into a unified dataset. Energy values for fixed graph frameworks of $(m,n)$ are computed using newGRAPH software detailed in Table 12. We derive linear and quadratic regression equations for predicting graph energy based on topological descriptor values computed from the respective frameworks, as shown in Tables 6–8. Additionally, a comparison between the linear and quadratic regression equations is utilized to assess their predictive capacity against the regression models.

5.1 Regression models for prediction of spectral properties

In this section, we conduct a comprehensive correlation analysis of the topological descriptors and spectral characteristics of various metal organic frameworks.

The predictive regression models are proposed using the following equations.

$\text{Linear\hspace{0.17em}equation}:Y=aX+b$

where $Y$ is the spectral property to be predicted, $X$ is the respective topological descriptor, $a$ represents the slope coefficient and $b$ represents as constant coefficient of the regression line such that the standard error (SE) should be low, and the F-value should be high. Correlation coefficient $(r)$ measures the robustness of linear relationships and the goodness of fit. The values of the correlation coefficients, along with statistical metrics like $r^2$ and adjusted $r^2$, are examined and discussed.

We perform a regression analysis to identify the best predicting topological descriptor for the considered metal organic frameworks. This analysis explores the relationship between the graph energy values provided in Table 14 and the molecular descriptors listed in Tables 6–8, emphasizing descriptors that display strong positive correlations. To improve predictive accuracy, we analyze individual MOFs separately. As an example, we illustrate the regression analysis for the NHC-TM$(m,n)$ framework to first choose the best descriptor for predicting graph energy values provided in Table 14, using the NHC-TM$(m,n)$ framework descriptor values presented in Table 6 for the reversing parameter $k=1$ as outlined in Table 15.

Table 15, shows that the values of $r$, $r^2$, and adjusted $r^2$ are similar for all topological descriptors. When considering other parameters for effective prediction modeling, again the Zagreb index $(M_1\mathfrak{R}{M}_1)$ stands out due to its higher $F$ value and lower standard error $(S.E.)$, suggesting it as the optimal predictive model. We further compare the predictive ability for NHC-TM(1,4) Zagreb descriptor-based linear regression models between the reversing parameters $k=\mathrm{1,2}$ and three is presented in Table 16.

Table 16 demonstrates that the regression equation for $k=3$ predicts energy values more accurately compared to other $k$ values. Similarly, for all other mentioned metal organic frameworks (MOFs), the Zagreb descriptor $(M_3\mathfrak{R}{M}_1)$ serves as the best predictor for the graph energies. The detailed results for the mentioned frameworks are presented in Table 17.

Utilizing the regression models from Table 17, we predict the graph energies of higher-dimensional metal organic frameworks and present a comparison between the actual graph energies and the predicted energies using the optimal model and present in the Table 18.

Table 18 and Figure 5 illustrate that the predictive model provides energy values closely matching the actual graph energies computed by the newGRAPH software. This capability enables accurate prediction of graph energy values for higher-dimensional distinct MOFs.

Table 14

Table 14. Energy values for MOFs obtained from newGRAPH.

Table 15

Table 15. Regression equations for MOFs correlating energy and the topological descriptors when $k=1$.

Table 16

Table 16. Comparing graph energy for NHC-TM(1,4) with $M_k\mathfrak{R}{M}_1$ predicted values for $k=\mathrm{1,2}$, and 3.

Table 17

Table 17. Optimal regression models for metal organic frameworks.

Table 18

Table 18. Comparing graph energy of MOFs using optimal models.

Figure 5

Figure 5. Comparison of predicted energy of MOFs.

6 Conclusion

We have conducted a comprehensive analysis by calculating modified reverse degree topological descriptors for four types of MOFs. Simultaneously, we have performed a detailed assessment of entropy levels for each MOF and compared these levels with the bond-wise scaled entropy approach across all frameworks. Moreover, we have presented an optimal linear regression model for predicting the graph energy of distinct structural frameworks, aiming to reduce the computational complexity of software and produce results in polynomial time. The graph theoretical and statistical methods explored in this study can enhance machine learning applications in computational chemistry and QSAR/QSPR studies for material advancements.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

AK: Methodology, Writing–original draft, Conceptualization. AG: Formal Analysis, Investigation, Methodology, Supervision, Writing–review and editing. MA: Methodology, Supervision, Validation, Writing–review and editing.

Funding

The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The author(s) declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

Publisher’s note

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References

Abraham, J., Arockiaraj, M., Jency, J., Kavitha, S. R., and Balasubramanian, K. (2022). Graph entropies, enumeration of circuits, walks and topological properties of three classes of isoreticular metal organic frameworks. J. Math. Chem. 60 (4), 695–732. doi:10.1007/s10910-021-01321-8

CrossRef Full Text | Google Scholar

Abul Kalaam, A. R., and Berin Greeni, A. (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), 27326. doi:10.1002/qua.27326

CrossRef Full Text | Google Scholar

Al-Dayel, I., Nadeem, M. F., and Khan, M. A. (2024). Topological analysis of tetracyanobenzene metal–organic framework. Sci. Rep. 14 (1), 1789. doi:10.1038/s41598-024-52194-1

PubMed Abstract | CrossRef Full Text | Google Scholar

Arockiaraj, M., Campena, F. J., Greeni, A. B., Ghani, M. U., Gajavalli, S., Tchier, F., et al. (2024a). QSPR analysis of distance-based structural indices for drug compounds in tuberculosis treatment. Heliyon 10 (2), e23981. doi:10.1016/j.heliyon.2024.e23981

PubMed Abstract | CrossRef Full Text | Google Scholar

Arockiaraj, M., Greeni, A. B., and Kalaam, A. A. (2023b). Linear versus cubic regression models for analyzing generalized reverse degree based topological indices of certain latest corona treatment drug molecules. Int. J. Quantum Chem. 123 (16), e27136. doi:10.1002/qua.27136

CrossRef Full Text | Google Scholar

Arockiaraj, M., Greeni, A. B., and Kalaam, A. A. (2023c). Comparative analysis of reverse degree and entropy topological indices for drug molecules in blood Cancer treatment through QSPR regression models. Polycycl. Aromat. Compd., 1–18. doi:10.1080/10406638.2023.2271648

CrossRef Full Text | Google Scholar

Arockiaraj, M., Greeni, A. B., Kalaam, A. A., Aziz, T., and Alharbi, M. (2024b). Mathematical modeling for prediction of physicochemical characteristics of cardiovascular drugs via modified reverse degree topological indices. Eur. Phys. J. E 47 (8), 53. doi:10.1140/epje/s10189-024-00446-3

PubMed Abstract | CrossRef Full Text | Google Scholar

Arockiaraj, M., Jency, J., Mushtaq, S., Shalini, A. J., and Balasubramanian, K. (2023e). Covalent organic frameworks: topological characterizations, spectral patterns and graph entropies. J. Math. Chem. 61, 1633–1664. doi:10.1007/s10910-023-01477-5

CrossRef Full Text | Google Scholar

Arockiaraj, M., Paul, D., Clement, J., Tigga, S., Jacob, K., and Balasubramanian, K. (2023a). Novel molecular hybrid geometric-harmonic-Zagreb degree based descriptors and their efficacy in QSPR studies of polycyclic aromatic hydrocarbons. Environ. Res. 34, 569–589. doi:10.1080/1062936x.2023.2239149

CrossRef Full Text | Google Scholar

Arockiaraj, M., Paul, D., Ghani, M. U., Tigga, S., and Chu, Y. M. (2023d). Entropy structural characterization of zeolites BCT and DFT with bond-wise scaled comparison. Sci. Rep. 13 (1), 10874. doi:10.1038/s41598-023-37931-2

PubMed Abstract | CrossRef Full Text | Google Scholar

Azeem, M., Aslam, A., Iqbal, Z., Binyamin, M. A., and Gao, W. (2021). Topological aspects of 2D structures of trans-Pd (NH2) S lattice and a metal-organic superlattice. Arabian J. Chem. 14 (3), 102963. doi:10.1016/j.arabjc.2020.102963

CrossRef Full Text | Google Scholar

Balasubramanian, K. (2023). Topological indices, graph spectra, entropies, Laplacians, and matching polynomials of n-dimensional hypercubes. Symmetry 15 (2), 557. doi:10.3390/sym15020557

CrossRef Full Text | Google Scholar

Chakravarty, C., Mandal, B., and Sark, P. (2016a). Bis (dithioline)-based metal–organic frameworks with superior electronic and magnetic properties: spin frustration to spintronics and gas sensing. J. Phys. Chem. C 120 (49), 28307–28319. doi:10.1021/acs.jpcc.6b09416

CrossRef Full Text | Google Scholar

Chakravarty, C., Mandal, B., and Sark, P. (2016b). Coronene-based metal–organic framework: a theoretical exploration. Phys. Chem. Chem. Phys. 18 (36), 25277–25283. doi:10.1039/c6cp05495a

PubMed Abstract | CrossRef Full Text | Google Scholar

Clough, A. J., Yoo, J. W., Mecklenburg, M. H., and Marinescu, S. C. (2015). Two-dimensional metal–organic surfaces for efficient hydrogen evolution from water. J. Am. Chem. Soc. 137 (1), 118–121. doi:10.1021/ja5116937

PubMed Abstract | CrossRef Full Text | Google Scholar

Dehmer, M. (2008). Information processing in complex networks: graph entropy and information functionals. Appl. Math. Comput. 201, 82–94. doi:10.1016/j.amc.2007.12.010

CrossRef Full Text | Google Scholar

Dong, R., Zhang, Z., Tranca, D. C., Zhou, S., Wang, M., Adler, P., et al. (2018). A coronene-based semiconducting two-dimensional metal-organic framework with ferromagnetic behavior. Nat. Commun. 9 (1), 2637. doi:10.1038/s41467-018-05141-4

PubMed Abstract | CrossRef Full Text | Google Scholar

Ediz, S. (2015). Maximum chemical trees of the second reverse Zagreb index. Pac. J. Appl. Math. 7 (4), 287.

Google Scholar

Ghani, M. U., Sultan, F., Tag El Din, E. S., Khan, A. R., Liu, J. B., and Cancan, M. (2022). A paradigmatic approach to find the valency-based K-banhatti and redefined Zagreb entropy for niobium oxide and a metal–organic framework. Molecules 27 (20), 6975. doi:10.3390/molecules27206975

PubMed Abstract | CrossRef Full Text | Google Scholar

Govardhan, S., and Roy, S. (2023). Topological analysis of hexagonal and rectangular porous graphene with applications to predicting π electron energy. Eur. Phys. J. Plus 138 (7), 670. doi:10.1140/epjp/s13360-023-04307-4

CrossRef Full Text | Google Scholar

Govardhan, S., Roy, S., Prabhu, S., and Arulperumjothi, M. (2024). Topological characterization of cove-edged graphene nanoribbons with applications to NMR spectroscopies. J. Mol. Struct. 1303, 137492. doi:10.1016/j.molstruc.2024.137492

CrossRef Full Text | Google Scholar

Gutman, I. (1978). The energy of a graph, Ber. Math. Stat. Sekt. Forschungszentrum Graz 103, 1–22.

Google Scholar

Gutman, I., Radenković, S., Dordević, S., Milovanović, I. Z., and Milovanović, E. I. (2017). Extending the McClelland formula for total π -electron energy. J. Math. Chem. 55 (10), 1934–1940. doi:10.1007/s10910-017-0772-6

CrossRef Full Text | Google Scholar

Gutman, I., and Trinajstić, N. (1972). Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17 (4), 535–538. doi:10.1016/0009-2614(72)85099-1

CrossRef Full Text | Google Scholar

Hayat, S., Khan, S., and Imran, M. (2021b). Quality testing of spectrum-based distance descriptors for polycyclic aromatic hydrocarbons with applications to carbon nanotubes and nanocones. Arabian J. Chem. 14 (3), 102994. doi:10.1016/j.arabjc.2021.102994

CrossRef Full Text | Google Scholar

Hayat, S., Khan, S., Khan, A., and Imran, M. (2020). Distance-based topological descriptors for measuring the π-electronic energy of benzenoid hydrocarbons with applications to carbon nanotubes. Math. Methods Appl. Sci. doi:10.1002/mma.6668

CrossRef Full Text | Google Scholar

Hayat, S., Khan, S., Khan, A., and Imran, M. (2021a). A computer-based method to determine predictive potential of distance-spectral descriptors for measuring the π-electronic energy of benzenoid hydrocarbons with applications. IEEE Access 9, 19238–19253. doi:10.1109/access.2021.3053270

CrossRef Full Text | Google Scholar

Horcajada, P., Serre, C., Maurin, G., Ramsahye, N. A., Balas, F., Vallet-Regí, M., et al. (2008). Flexible porous metal-organic frameworks for a controlled drug delivery. J. Am. Chem. Soc. 130 (21), 6774–6780. doi:10.1021/ja710973k

PubMed Abstract | CrossRef Full Text | Google Scholar

Imran, M., Khan, A. R., Husin, M. N., Tchier, F., Ghani, M. U., and Hussain, S. (2023). Computation of entropy measures for metal-organic frameworks. Molecules 28 (12), 4726. doi:10.3390/molecules28124726

PubMed Abstract | CrossRef Full Text | Google Scholar

Kavitha, S. R. J., Abraham, J., Arockiaraj, M., Jency, J., and Balasubramanian, K. (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. A 125 (36), 8140–8158. doi:10.1021/acs.jpca.1c06264

PubMed Abstract | CrossRef Full Text | Google Scholar

Kinoshita, Y., Matsubara, I., Higuchi, T., and Saito, Y. (1959). The crystal structure of bis (adiponitrile) copper (I) nitrate. Bull. Chem. Soc. Jpn. 32 (11), 1221–1226. doi:10.1246/bcsj.32.1221

CrossRef Full Text | Google Scholar

Lee, J., Farha, O. K., Roberts, J., Scheidt, K. A., Nguyen, S. T., and Hupp, J. T. (2009). Metal–organic framework materials as catalysts. Chem. Soc. Rev. 38 (5), 1450–1459. doi:10.1039/b807080f

PubMed Abstract | CrossRef Full Text | Google Scholar

Li, W., Sun, L., Qi, J., Jarillo-Herrero, P., Dincă, M., and Li, J. (2017). High temperature ferromagnetism in π-conjugated two-dimensional metal–organic frameworks. Chem. Sci. 8 (4), 2859–2867. doi:10.1039/c6sc05080h

PubMed Abstract | CrossRef Full Text | Google Scholar

Mabrouk, M., and Hayn, R. (2015). Magnetic moment formation in metal-organic monolayers. Phys. Rev. B 92 (18), 184424. doi:10.1103/physrevb.92.184424

CrossRef Full Text | Google Scholar

Mabrouk, M., Hayn, R., Denawi, H., and Chaabane, R. B. (2018). Possibility of a ferromagnetic and conducting metal-organic network. J. Magnetism Magnetic Mater. 453, 48–52. doi:10.1016/j.jmmm.2018.01.005

CrossRef Full Text | Google Scholar

Manzoor, S., Siddiqui, M. K., and Ahmad, S. (2021). On physical analysis of degree-based entropy measures for metal–organic superlattices. Eur. Phys. J. Plus 136 (3), 287. doi:10.1140/epjp/s13360-021-01275-5

CrossRef Full Text | Google Scholar

Murray, L. J., Dincă, M., and Long, J. R. (2009). Hydrogen storage in metal–organic frameworks. Chem. Soc. Rev. 38 (5), 1294–1314. doi:10.1039/b802256a

PubMed Abstract | CrossRef Full Text | Google Scholar

Nadeem, M. F., Imran, M., Siddiqui, H. M., Azeem, M., Khalil, A., and Ali, Y. (2021). Topological aspects of metal-organic structure with the help of underlying networks. Arabian J. Chem. 14 (6), 103157. doi:10.1016/j.arabjc.2021.103157

CrossRef Full Text | Google Scholar

Park, C., Baek, J. W., Shin, E., and Kim, I. D. (2023). Two-dimensional electrically conductive metal–organic frameworks as chemiresistive sensors. ACS Nanosci. Au 3 (5), 353–374. doi:10.1021/acsnanoscienceau.3c00024

PubMed Abstract | CrossRef Full Text | Google Scholar

Paul, D., Arockiaraj, M., Jacob, K., and Clement, J. (2023). Multiplicative versus scalar multiplicative degree based descriptors in QSAR/QSPR studies and their comparative analysis in entropy measures. Eur. Phys. J. Plus 138 (4), 323. doi:10.1140/epjp/s13360-023-03920-7

CrossRef Full Text | Google Scholar

Peter, P., and Clement, J. (2024). Predictive models on potential energies of zeolite ZK-5 using bond weighted information entropy measures. J. Mol. Struct. 1307, 137945. doi:10.1016/j.molstruc.2024.137945

CrossRef Full Text | Google Scholar

Rahul, M. P., Clement, J., Junias, J. S., Arockiaraj, M., and Balasubramanian, K. (2022). Degree-based entropies of graphene, graphyne and graphdiyne Using Shannon’s Approach. J. Mol. Struct. 1260, 132797. doi:10.1016/j.molstruc.2022.132797

CrossRef Full Text | Google Scholar

Raja, N. J., and Anuradha, A. (2024). Topological entropies of single walled carbon nanotubes. J. Math. Chem. 62 (4), 809–818. doi:10.1007/s10910-023-01532-1

CrossRef Full Text | Google Scholar

Raza, Z., Akhter, S., and Shang, Y. (2023a). Expected value of first Zagreb connection index in random cyclooctatetraene chain, random polyphenyls chain, and random chain network. Front. Chem. 10, 1067874. doi:10.3389/fchem.2022.1067874

PubMed Abstract | CrossRef Full Text | Google Scholar

Raza, Z., Arockiaraj, M., Maaran, A., Kavitha, S. R., and Balasubramanian, K. (2023b). Topological entropy characterization, NMR and ESR spectral patterns of coronene-based transition metal organic frameworks. ACS omega 8 (14), 13371–13383. doi:10.1021/acsomega.3c00825

PubMed Abstract | CrossRef Full Text | Google Scholar

Shannon, C. E. (1948). A mathematical theory of communication. Bell Syst. Tech. J. 27 (3), 379–423. doi:10.1002/j.1538-7305.1948.tb01338.x

CrossRef Full Text | Google Scholar

Stevanović, L., Brankov, V., Cvetković, D., and Simić, S. (2021). newGRAPH: a fully integrated environment used for research process in graph theory. Available at: http://www.mi.sanu.ac.rs/newgraph/index.html.

Google Scholar

Taherpour, A., and Mohammadinasab, E. (2010). Topological relationship between Wiener, Padmaker-Ivan, and Szeged Indices and energy and electric moments in armchair polyhex nanotubes with the same circumference and varying lengths. Fullerenes, Nanotub. Carbon Nanostructures 18 (1), 72–86. doi:10.1080/15363830903291580

CrossRef Full Text | Google Scholar

Waheed, M., Saleem, U., Javed, A., and Jamil, M. K. (2023). Computational aspects of entropy measures for metal organic frameworks. Mol. Phys. 122, 2254418. doi:10.1080/00268976.2023.2254418

CrossRef Full Text | Google Scholar

Wiener, H. (1947). Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20. doi:10.1021/ja01193a005

PubMed Abstract | CrossRef Full Text | Google Scholar

Yang, J., Siddiqui, M. K., Bashir, A., Manzoor, S., Eldin, S. M., and Cancan, M. (2023). On physical analysis of topological co-indices for beryllium oxide via curve fitting models. J. Mol. Struct. 1278, 134933. doi:10.1016/j.molstruc.2023.134933

CrossRef Full Text | Google Scholar

Yu, Y., Khalid, A., Aamir, M., Siddiqui, M. K., Muhammad, M. H., and Bashir, Y. (2023). On some topological indices of metal-organic frameworks. Polycycl. Aromat. Compd. 43 (6), 5607–5628. doi:10.1080/10406638.2022.2105909

CrossRef Full Text | Google Scholar

Zaman, S., Jalani, M., Ullah, A., Ahmad, W., and Saeedi, G. (2023). Mathematical analysis and molecular descriptors of two novel metal–organic models with chemical applications. Sci. Rep. 13 (1), 5314. doi:10.1038/s41598-023-32347-4

PubMed Abstract | CrossRef Full Text | Google Scholar

Zhang, X., Bajwa, Z. S., Zaman, S., Munawar, S., and Li, D. (2024). The study of curve fitting models to analyze some degree-based topological indices of certain anti-cancer treatment. Chem. Pap. 78 (2), 1055–1068. doi:10.1007/s11696-023-03143-1

CrossRef Full Text | Google Scholar

Zhang, X., Idrees, N., Kanwal, S., Saif, M. J., and Saeed, F. (2022). Computing topological invariants of deep neural networks. Comput. Intell. Neurosci. 2022 (1), 1–11. doi:10.1155/2022/9051908

CrossRef Full Text | Google Scholar