Topological Characterization and Predictive Modeling of Graph Energy in Ionic Covalent Organic Frameworks

1  Introduction

Covalent organic frameworks are crystalline materials with a porous structure, formed by organic molecular building blocks made up of light elements, which are interconnected through covalent bonds [1,2]. COFs can form crystalline structures with permanent porosity, enabling organic reactions to occur without altering their properties [3]. The structural adaptability of COFs enables their use in a wide range of applications, including gas separation, drug discovery, heterogeneous catalysis, and energy conversion [4,5]. A defining feature of COFs is their uniquely predictable skeleton structures, which distinguish them from other porous polymeric materials [6]. The design of COFs with structural arrangements and tailored pore sizes relies on the strategic choice of linkers and molecular building units [7]. The polymerization process combines covalent bonds with noncovalent interactions, resulting in well-organized, extended crystalline frameworks. The crystallization of COFs successfully resolves the persistent challenge of crystallizing covalent solids. This is achieved by carefully balancing thermodynamic and kinetic factors during the reversible formation of covalent bonds, which is vital for producing extended crystalline structures [8]. Concurrently, advances in ion-regulated design, particularly the incorporation of multivalent ions [9,10], zwitterionic groups [11], and spatial ion-binding architectures [12,13], have significantly enhanced the performance of ICOFs in terms of selective ion adsorption, ion-exchange capacity, and long-range ionic conductivity, especially under conditions of high ionic competition and dynamic operating environments [14].

The structure of a covalent organic framework is influenced by the size, arrangement, and connectivity of the linkers. COFs with boronate, spiroborate, imine, hydrazone, triazine, and benzyl nitrile linkages have been synthesized through polycondensation reactions between organic precursors [15]. Among these linkers, the spiroborates, derived from boronic acid, are ionic compounds known for their exceptional hydrolytic resistance and durability in aqueous, methanolic, and alkaline conditions [16]. Their ionic properties make spiroborate-linked COFs promising candidates for ion-conductive materials. These linkages are easily formed by condensing polyols with alkali tetraborate or boric acid, or through transesterification between borates and polyols under thermodynamically controlled conditions. They have also found widespread use in the synthesis of macrocycles, facilitating a wide range of applications, including electrolytes, sensors, catalysts, and hosts for neutral molecules or ions [17–19]. Recent developments in ionic covalent organic frameworks have highlighted a wide range of synthetic strategies, particularly leveraging dynamic covalent chemistry to achieve tunable architectures and enhanced energy functionalities [20–23].

This study investigates the topological characterization of ionic covalent organic frameworks (ICOFs) via spiroborate linkages, incorporating sp3 hybridized boron anionic centers and tunable countercations [24] as depicted in Fig. 1. ICOFs are formed by their repeating ionic units and crystalline structures, providing high ionic conductivity [25–27]. By merging the properties of porous ionic polymers (PiPs) with COFs, they maintain atomic periodicity and porosity while introducing oppositely charged groups via electrostatic interactions or ion exchange [28]. Their adaptable structures, organized channels, and abundant charged sites make them highly promising for applications such as purification, ion transport, sensing, catalysis, and energy retention [29–32]. With permanent porosity and customizable architectures, ICOFs are highly suitable for selective trace element extraction, efficient ion transport, and sustained ionic flow, positioning them as ideal candidates for energy devices and material innovations [33].

Figure 1: The building block of ICOF is composed of carbon atoms depicted in grey, boron atoms in yellow, and oxygen atoms in red

Topological descriptors serve as a means to characterize the molecular graphs of ICOF, with degree-based descriptors reflecting atom valency through vertex degrees at bond ends [34,35], and potentially being used to quantify ICOFs through QSPR and QSAR analyses [36–39]. Degree-based topological indices (TIs) are key graph invariants derived from various graph properties [40] which are fundamentally grounded in the vertex degree, representing the number of edges connected to a vertex. To refine this traditional degree-based TIs, neighborhood degree-based TIs, which sum the degrees of a vertex’s adjacent vertices, are being developed as a more effective measure [41–44]. These descriptors are widely used in areas such as molecular modeling and applied materials research, delivering valuable insights across a broad range of disciplines [45–49]. In the context of ICOFs, these descriptors provide a rigorous framework for understanding structure–property relationships that influence critical functionalities such as ion transport, adsorption, and energy storage [50–54]. In information theory, Shannon’s entropy plays a crucial role in quantifying system uncertainty. Graph entropy measures, particularly those derived from degree descriptors, are widely used in graph theory to assess the complexity and diversity of molecular graphs [55–57]. Its versatility across fields such as biology and chemistry makes it an essential tool for analyzing complex systems [58–60]. In parallel, topological descriptors and entropy-based analyses have gained traction in quantifying structure–property relationships in porous and crystalline frameworks [61–64]. This evolving literature forms the foundation for our graph-theoretic approach, which complements recent machine-learning-based models and contributes to a deeper understanding of structure-driven energy properties in ICOFs.

The Hückel molecular orbital (HMO) theory is frequently used to calculate the total π-electron energy in alternant hydrocarbon frameworks, which is a key quantum-chemical feature of conjugated molecules. The π-electron energy and the graph-based energy for alternant hydrocarbons are generally correlated, although this relationship is not always valid for all molecular graphs. By applying the same approach used for graphs composed entirely of carbon atoms, this method can be modified to include graphs with heteroatoms [65,66] through the incorporation of external validation. While the current study employs linear regression models due to their simplicity and strong predictive performance, alternative nonlinear models such as exponential regressions can also be explored to capture more complex dependencies [67]. Additionally, recent developments including the thermodynamically consistent deep energy method [68] and machine learning interatomic potentials [69] demonstrate the potential of machine learning techniques for accurate energy prediction in complex materials. These approaches offer promising avenues for future research. This study involved calculating topological descriptors and entropy values for ICOFs, comparing entropy at the bond level, and applying regression models to predict graph energy.

2  Computational Methods

The ionic covalent organic framework (ICOF), characterized by spiroborate linkages, can be represented as a molecular graph, where the vertex set V(ICOF) corresponds to the atoms, and the edge set E(ICOF) represents the bonds. The molecular graph of ICOF as shown in Fig. 1 is depicted in Fig. 2, in which all the atoms are considered as vertices and the bonds as edges.

Figure 2: The molecular graph of ICOF

In this framework, the degree of a vertex g∈ V(ICOF), expressed as dICOF(g), refers to the number of bonds connected to g. The degree-sum, represented as sICOF(g), is the total of the degrees of all atoms adjacent to g. That is, sICOF(g)=∑h∈NICOF(g)dICOF(h) where NICOF(g)={h∈V(ICOF) | gh∈E(ICOF)}. Let d(f,u)=|{gh∈E(ICOF):f= dICOF(g) and u=dICOF(h)}| and s(f,u)=|{gh∈E(ICOF): f= sICOF(g) and u=sICOF(h)}|. The edge classes, determined by d(f,u) and s(f,u), are grouped into the sets D(ICOF) and S(ICOF), respectively. The index function Υ is used to define the additive and multiplicative forms of topological descriptors, as outlined below [70–73].

Υd(ICOF)=∑d(f,u)∈D(ICOF)d(f,u) Υ(f,u)

Υd∗(ICOF)=∏d(f,u)∈D(ICOF)d(f,u) Υ(f,u)

Υs(ICOF)=∑s(f,u)∈S(ICOF)s(f,u) Υ(f,u)

Υs∗(ICOF)=∏s(f,u)∈S(ICOF)s(f,u) Υ(f,u)

The index function Υ is exponentiated to its own value to derive the self-powered forms of the topological descriptors [74–76].

Υdp(ICOF)=∑d(f,u)∈D(ICOF)d(f,u) Υ(f,u)Υ(f,u)

Υdp∗(ICOF)=∏d(f,u)∈D(ICOF)d(f,u) Υ(f,u)Υ(f,u)

Υsp(ICOF)=∑s(f,u)∈S(ICOF)s(f,u) Υ(f,u)Υ(f,u)

Υsp∗(ICOF)=∏s(f,u)∈S(ICOF)s(f,u) Υ(f,u)Υ(f,u)

The hybrid descriptors, along with the index function Υ(f,u), are provided below [77–80].

•   M1(f,u)=f+u (First Zagreb)

•   M2(f,u)=fu (Second Zagreb)

•   HM(f,u)=(f+u)2 (Hyper Zagreb)

•   AZ(f,u)=(fuf+u−2)3 (Augmented Zagreb)

•   BM(f,u)=f+u+fu (Bi-Zagreb)

•   TM(f,u)=f2+u2+fu (Tri-Zagreb)

•   GBM(f,u)=fuf+u+fu (Geometric − Bi-Zagreb)

•   GTM(f,u)=fuf2+u2+fu (Geometric − Tri-Zagreb)

•   BMG(f,u)=(f+u+fu)fu (Bi-Zagreb − Geometric)

•   TMG(f,u)=f2+u2+fufu (Tri-Zagreb − Geometric)

•   TMA(f,u)=2(f2+u2+fu)f+u (Tri-Zagreb − Arithmetic)

•   BMA(f,u)=2(f+u+fu)f+u (Bi-Zagreb − Arithmetic)

As we see from the definitions of topological descriptors, the classification of edge classes does not consider the specific types of atoms at their terminal points. However, it is essential to distinguish between the three types of atoms that form the basis of ionic covalent organic frameworks. To enhance the partitions based on d(f,u) and s(f,u), we introduce weight functions that incorporate both atom and bond contributions. The function Φ represents atom weights, while Γ denotes bond weights. To be more specific, ΦB refers to the weight assigned to atom B, whereas ΓBC represents the weight function for the bond between atoms B and C.

Shannon’s entropy method is used to define a structural information function for the bonds in ICOF. A high entropy value signifies greater complexity or disorder, while a low entropy value indicates more order and regularity. The following form is used to define the entropy of the ICOF structure using Υ as the structural information function on E(ICOF)={c1,c2,…,cm} as shown below.

IΥ(ICOF)=−∑k=1mΥ(ck)∑x=1mΥ(cx)log⁡(Υ(ck)∑x=1mΥ(cx))=log⁡(∑k=1mΥ(ck))−1∑k=1mΥ(ck)log⁡(∏c=1mΥ(ck)Υ(ck))

The modified entropy approach, which replaces the multiplicative form with a scalar multiplicative form, is outlined below [57,81].

IΥ(ICOF)=log⁡(Υ(ICOF))−1Υ(ICOF)log⁡(Υp∗(ICOF))

The adjacency matrix of an ICOF determines its spectrum through its eigenvalues, which are denoted as λ1,λ2,λ3,…,λn, arranged in descending order. Collectively, these eigenvalues form the eigen-spectrum of the ICOF. As given below, the graph energy [82], represented as Eπ(ICOF) and expressed in β-units, is obtained by adding the non-negative values of the ICOF’s eigenvalues.

Eπ(ICOF)=∑i=1n|λi|

3  Results and Discussion

The structural configuration of ionic covalent organic frameworks, with an emphasis on comparing their topological features using entropy information is explored in this section. The COFs featuring spiroborate linkages are depicted in a parallelogram peripheral shape, achieved by arranging the ionic COF units in a p×r grid-like pattern, where p,r≥1, as illustrated in Fig. 3. The ICOF is composed of 78pr−5p−5r vertices and 96pr−8p−8r edges, respectively.

Figure 3: ICOF in parallelogram of dimensions (3,5)

The bond classes of ICOF are distributed over the set {B−O,O−C,C−C} with their corresponding bond degree classes being {(1,4),(2,2),(2,3),(2,4),(3,3),(3,4)} while degree-sum bond classes are {(4,8),(5,5),(5,6),(5,9),(6,6),(6,8),(6,10),(8,8),(8,9),(9,10)}. The bond classification of ICOFs is identified, and their numerical values are shown in Tables 1 and 2.

The degree and degree-sum-based descriptors for ICOF((p,r)) are calculated using the following equations with respect to Υ.

Υd(ICOF(p,r))=(8pr−4p−4r)ΓBOΥ(4ΦO,ΦB+ΦC)+(8pr−4p−4r)ΓOCΥ(2ΦC,ΦB+ΦC)+(4p+4r)ΓCCΥ(ΦC,4ΦC)+(12pr)ΓCCΥ(2ΦC,2ΦC)+(40pr)ΓCCΥ(2ΦC,3ΦC)+(8pr−4p−4r)ΓCCΥ(2ΦC,4ΦC)+(12pr)ΓCCΥ(3ΦC,3ΦC)+(8pr)ΓCCΥ(3ΦC,4ΦC)

Υs(ICOF(p,r))=(8pr−4p−4r)ΓBOΥ(4ΦB+2ΦC,8ΦO)+(8pr−4p−4r)ΓOCΥ(4ΦC+2ΦO,4ΦB+2ΦC)+(4p+4r) ΓCCΥ(4ΦC,8ΦC)+(12pr)ΓCCΥ(5ΦC,5ΦC)+(16pr)ΓCCΥ(5ΦC,6ΦC)+(8pr)ΓCCΥ(5ΦC,9ΦC)+(8pr)ΓCCΥ(6ΦC,6ΦC)+(8pr)ΓCCΥ(6ΦC,8ΦC)+(8pr−4p−4r)ΓCCΥ(6ΦC,10ΦC)+(4pr)ΓCCΥ(8ΦC,8ΦC)+(8pr+4p+4r)ΓCCΥ(8ΦC,9ΦC)+(8pr−4p−4r)ΓCCΥ(9ΦC,10ΦC)

By setting the atom and bond weights to unity, the above two equations can be simplified in the following form.

Υd(ICOF(p,r))=(4p+4r)Υ(1,4)+(8pr−4p−4r)Υ(2,2)+(12pr)Υ(2,2)+(40pr)Υ(2,3)   +(16pr−8p−8r)Υ(2,4)+(12pr)Υ(3,3)+(8pr)Υ(3,4)

Υs(ICOF(p,r))=(4p+4r)Υ(4,8)+(12pr)Υ(5,5)+(16pr)Υ(5,6)+(8pr)Υ(5,9)+(16pr−4p−4r)Υ(6,6)+(16pr−4p−4r)Υ(6,8)+(8pr−4p−4r)Υ(6,10)+(4pr)Υ(8,8)+(8pr+4p+4r)Υ(8,9)+(8pr−4p−4r)Υ(9,10)

We now illustrate the computation of the index M1(ICOF) of dimension (2,2) as shown below.

M1d(ICOF(2,2))=16×(1+4)+64×(2+2)+160×(2+3)+32×(2+4)+48×(3+3)+32×(3+4)=16×(5)+64×(4)+160×(5)+32×(6)+48×(6)+32×(7)=1840

M1s(ICOF(2,2))=16×(4+8)+48×(5+5)+64×(5+6)+32×(5+9)+48×(6+6)+48×(6+8) +16×(6+10)+16×(8+8)+48×(8+9)+16×(9+10)=16×(12)+48×(10)+64×(11)+32×(14)+48×(12)+48×(14)+16×(16)+16×(16) +48×(17)+16×(19)=4704

We now derive the numerical expressions for the topological descriptors of ICOF(p,r), with the resulting expressions presented in simplified form, Υ#(ICOF) = {Υd(ICOF),Υs(ICOF)}.

Result 1. The topological descriptors for ICOF are algebraically expressed for p,r≥2.

1.   M1#(ICOF(p,r))={(504pr−44p−44r),(1304pr−128p−128r)}

2.   M2#(ICOF(p,r))={(652pr−64p−64r),(4516pr−520p−520r)}

3.   HM#(ICOF(p,r))={(2720pr−252p−252r),(18416pr−2096p−2096r)}

4.   AZ#(ICOF(p,r))={((1710559pr)/2000−(2336p)/27−(2336r)/27),((31743354508473967412051pr)/5091288592416768000−(293250748539392650537p)/377132488327168000−(293250748539392650537r)/377132488327168000)}

5.   BM#(ICOF(p,r))={(1156pr−108p−108r),(5820pr−648p−648r)}

6.   TM#(ICOF(p,r))={(2068pr−188p−188r),(13900pr−1576p−1576r)}

7.   GBM#(ICOF(p,r))={(pr(1504802+3(2394002+55440)+487179)−p(752402+7315)−r(752402+7315)/65835),(pr(5(23433822535202+3(41533116363202+2240602330120)+4329299417520)+57399700142402+109861791670403+43787771251488)−p(5(21171691126760+11203011650603)−67401163046002+27465447917603+5321430534035)−r(5(11716911267602+11203011650603)−67401163046002+27465447917603+5321430534035)/10642861068070)}

8.   GTM#(ICOF(p,r))={(2(pr(84362+3(155402+3192)+34447)−p(42182+2109)−r(42182+2109)/14763),(pr(5(3(4731187809602+219662291160)+2383052162402+427686845040)+5952139502402+11636164612803+4993045912673)−p(2909041153203−6820159846502+5(1191526081202+1098311455803)+597969570380)−r(2909041153203−6820159846502+5(1191526081202+1098311455803)+597969570380)/2690863066710)}

9.   BMG#(ICOF(p,r))={(2(pr(842+3(1102+38)+210)−p(422−3)−r(422−3)/3),(pr(8902+12403+5(4362+3(3282+304)+472)+3780)−p(3103−7752+5(2182+1523)+480)−r(3103−7752+5(2182+1523)+480)/15)}

10.   TMG#(ICOF(p,r))={(2(pr(1682+3(1902+74)+342)−p(842−27)−r(842−27))/3),(pr(21702+29603+5(10842+3(7282+784)+1208)+8460)−p(7403−19252+5(5422+3923)+1080)−r(7403−19252+5(5422+3923)+1080)/15)}

11.   TMA#(ICOF(p,r))={((6460pr)/3−128p−128r),((230462pr)/45−(4163p)/9−(4163r)/9)}

12.   BMA#(ICOF(p,r))={((4012pr)/3−96p−96r),((115934pr)/45−(2123p)/9−(2123r)/9)}

The numerical values for the dimensions of ICOFs are computed using mathematical expressions derived from prior results, as presented in Table 3, and are further validated using the pseudocode algorithm for computing degree-based topological indices, as presented in Algorithm 1. These values are essential for determining the frameworks entropy levels such that the multiplicative self-powered descriptors are available. The algebraic expressions for these descriptors are given below.

1.   Υdp∗(ICOF(p,r))=Υα(4915200p6r5−3440640p6r4+491520p6r3+4915200p5r6−6881280p5r5+1474560p5r4−3440640p4r6+1474560p4r5+491520p3r6)

2.   Υsp∗(ICOF(p,r))=Υβ(3221225472p10r9−3221225472p10r8+201326592p10r7+704643072p10r6−251658240p10r5+25165824p10r4+3221225472p9r10−6442450944p9r9+603979776p9r8+2818572288p9r7−1258291200p9r6+150994944p9r5−3221225472p8r10+603979776p8r9+4227858432p8r8−2516582400p8r7+377487360p8r6+201326592p7r10+2818572288p7r9−2516582400p7r8+503316480p7r7+704643072p6r10−1258291200p6r9+377487360p6r8−251658240p5r10+150994944p5r9+25165824p4r10)

where we used D={(1,4),(2,2),(2,3),(2,4),(3,3),(3,4)}, S={(4,8),(5,5),(5,6),(5,9),(6,6),(6,8),(6,10),(8,8),(8,9),(9,10)} such that Υα=∏(f,u)∈DΥ(f,u)Υ(f,u), Υβ=∏(f,u)∈S1Υ(f,u)Υ(f,u).

The formulas based on topological and self-powered descriptors are used to derive entropy expressions, resulting in more complex formulations. An illustrative computation of the modified Shannon entropy for ICOF(3,3), using the M1d index, is presented below.

IM1d(ICOF(3,3))=log⁡(4272)−(14272)×log⁡(1006157168640×(4)4×2(5)5×2(6)6×(7)7)=8.3401

We have now correlated the topological index and entropy values for the case where the dimensions p=r. Among the degree-based indices, the bi-Zagreb-geometric index is the most effective for modeling the logarithmic regression model of ICOF, as shown below with the correlation (r), F-value (F) and standard error (Se),

IBMGd(ICOF(p,r))=1.0072 log⁡(BMGd)−0.0757, r=1, F=50.08743672, Se=3773.59342.

Extending this approach, QSPR models could be formulated when experimental physicochemical data for ICOF become available.

As a result, entropy values of topological descriptors based on degree and degree-sum are calculated by equating their dimensions, setting p=r in their configurations to ensure that entropy values are measured on a consistent scale. Among these, the geometric-tri-Zagreb index exhibits the highest entropy, whereas the hyper-Zagreb index displays the lowest entropy. To ensure consistency in comparing entropies across ICOF configurations, the values are scaled relative to the total number of bonds in each structure, accounting for variations caused by their fixed dimensions. This scaling facilitates a direct comparison of degree and degree-sum entropies across different ICOF configurations. For ICOF(3,3), the scaled entropy based on M1 is calculated as 8.3401/816=0.0102. Table 4 shows that ICOFs consistently exhibits higher entropy values than ICOFd, as illustrated in Fig. 4. These results reveal that scaled entropies decrease asymptotically as the dimensions of p and r increase.

Figure 4: Bar diagram of scaled entropy for ICOF(3,3) based on various descriptors

4  Modeling of Graph Energy

In this section, we develop linear regression models for producing the graph energies of ICOFs using the topological indices that were determined in the preceding section. The relationship between a graph’s structural characteristics and the overall π-electron energy in molecules is examined through graph energy. McClelland’s theory for calculating Eπ is a groundbreaking contribution to the field of total π-electron energy. As an effective computational substitute for complex quantum chemistry methods, McClelland introduced the formula Eπ(G)≈0.91×2|V(G)||E(G)|, which provides a straightforward method for estimating the total π-electron energy [83,84]. While McClelland’s formula offers a fast and simple approximation of π-electron energy based solely on the number of vertices and edges in a molecular graph, it does not capture the full complexity of electronic interactions in large or highly conjugated systems. As the size and structural intricacy of ICOFs increase, the limitations of this approach become more evident. For example, as shown in Table 5, for ICOF(5,5), the actual graph energy is 2653.47942, whereas McClelland’s estimate is 2701.9, resulting in a discrepancy of approximately 48.4 units. To address this, our work incorporates regression models based on topological descriptors, which better reflect structural nuances and provide energy estimates that more closely align with spectral values [85–88]. In this paper, we compute graph energies using the well-known platform newGRAPH, a comprehensive tool designed to facilitate and advance research in graph theory.

The eigen-spectrum and graph energy for specific dimensions of ICOFs were calculated using the machine-generated package [89] in conjunction with McClelland’s graph energy as shown in Table 5. A linear regression model, Eπ(ICOF)=a(Υ(ICOF))+b, was developed to predict graph energy based on topological descriptor values. In this equation, a and b are constants and parameters related to statistics, such as correlation (r), standard error (Se), and F-value, are incorporated.

The bi-Zagreb-geometric index is the most effective for modeling linear regression models of ICOFs using degree descriptors, whereas the tri-Zagreb-arithmetic index yields the highest accuracy for degree-sum descriptors, as shown below.

Eπ(ICOF(p,r)=0.24816806778517(BMGd)+0.146261847, r=0.999999999, F=4611352006, Se=0.041570275Eπ(ICOF(p,r))=0.0214883975178587(TMAs)+2.072468999, r=0.999999825, F=37225220.86, Se=0.462677438

The geometric-tri-Zagreb and bi-Zagreb-geometric indices exhibit higher entropy values due to their strong topological structural relevance, allowing them to effectively capture the intricate connectivity and branching patterns of ICOF frameworks. This enhances their capacity to reflect the spectral and energetic characteristics of these materials.

Given that the computation of graph energy through spectral methods involves eigenvalue analysis, which becomes increasingly complex and resource-intensive for large molecular structures, we developed linear regression models based on topological descriptors as a practical alternative. These models enable efficient estimation of graph energies without the need for full spectral decomposition. As shown in Table 6, the predicted values closely approximate the actual spectral energies, with minimal error (e.g., an error of only 0.089 units for ICOF-P(6,1)). Although minor discrepancies are observed, the models offer a computationally efficient and scalable approach for estimating graph energies in complex systems. This predictive framework serves as a useful tool for large-scale applications, where exact spectral calculations may be impractical.

The graph energy of ICOFs is predicted using degree and degree-sum descriptors, based on the regression equations outlined earlier. As shown in Table 6, these predicted values are compared with the original Eπ values and depicted in Fig. 5.

Figure 5: Graph energy prediction using (a) degree and (b) degree-sum

5  Conclusion

We have discussed the topological descriptors and entropies of ICOFs with parallelogram configurations. Our analysis, using scaled entropy measures, demonstrated that degree-based entropy reveals a higher level of informational disorder than degree-sum entropy, offering a more comprehensive understanding of the structural arrangement. To further enhance computational efficiency, we developed a set of optimized linear regression models specifically designed to predict graph energy across ICOF frameworks of varying dimensions. By refining these models, we were able to significantly reduce the computational complexity while maintaining high predictive accuracy. The results not only expand our understanding of the structural diversity within ICOF frameworks but also provide valuable insights into the relationship between framework dimensions and graph energy. These findings open new avenues for future research by providing a robust framework for exploring ICOF properties and guiding the design of advanced materials in this domain.

Acknowledgement: Not applicable.

Funding Statement: The authors received no specific funding for this study.

Author Contributions: The authors confirm their contributions to the study as follows: Conceptualization, Micheal Arockiaraj and Aravindan Maaran; methodology, Aravindan Maaran and C. I. Arokiya Doss; validation, Aravindan Maaran and C. I. Arokiya Doss; investigation, Micheal Arockiaraj and Aravindan Maaran; writing—original draft preparation, Aravindan Maaran; writing—review and editing, Micheal Arockiaraj and C. I. Arokiya Doss. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: All data generated during this work are included in this paper.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest to report regarding the present study.

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