Comparative structural analysis of certain covalent organic frameworks through entropy and degree-based topological indices

Abstract

Topological indices are graph invariants of the molecular graph that facilitate the study of structural characteristics pertaining to connectivity and shape. This paper addresses the application of degree-based topological indices coupled with Shannon entropy to analyze the structural features of covalent organic frameworks JLU44 and JLU45. Covalent Organic Frameworks (COFs) are a class of crystelline, porus organic polymers characterized by strong covalent linages between their building blocks. The study addresses the complexity and uncertainties inherent in molecular structures by pinpointing graph-theoretical metrics, effectively representing the connectivity and interaction dynamics among the molecules. The utilization of Shannon entropy derived from topological indices improves the understanding of structural information, aiding in the comparative analysis of the frameworks’ structural characteristics. This demonstrates the efficiency of entropy-based metrics in measuring molecular complexity and highlights the value of topological descriptors in furthering the investigation of chemically relevant graph structures.

Introduction

Hydrogen energy is the most promising alternative to fossil fuels, providing a clean, high density energy transporter that emits only water as a byproduct after burning¹. The need for effective hydrogen generation technology to power fuel cells, industrial operations, and transportation has increased as the world moves toward sustainable energy systems. Since sunlight and water are two of the most plentiful renewable resources on Earth, photocatalytic water splitting technology is thought to be the most promising of the many generating methods since it directly transforms solar energy into chemical energy^2,3.

Photocatalytic hydrogen evolution is the process of creating hydrogen gas with water, light mostly sunlight and a photocatalyst. By absorbing photons to create electron hole pairs, which then move to the surface to propel the reduction of water into hydrogen gas, the photocatalyst plays a crucial part in this process. Researchers have concentrated on creating materials with great structural stability, effective charge separation, and broad light absorption in order to optimize efficiency. Covalent Organic Frameworks (COFs), a novel type of crystalline, porous polymers for photocatalysis, have drawn a lot of interest recently^4–8. They are perfect candidates for hydrogen evolution because to their huge surface areas, extended -conjugation, and highly adjustable architectures^9–11.In particular, there has been considerable promise in the creation of completely conjugated, benzobisoxazole-bridged COFs. Wang et al. constructed OZBT-COF and investigated its use in photocatalytic organic transformations based on its recyclability and stability¹². Zong et al. used nitrogen site engineering to optimize pyrene-based Py-COF variants to improve hydrogen production¹³. Du et al. created a cocatalyst-free -ketoenamine Tp-Py-COF to generate sacrificial hydrogen from water¹⁴. Li et al. developed Re(I)-decorated dibenzochrysene COF hybrids to reduce CO to CO¹⁵. The two most notable instances of this class are COF-JLU44 and COF-JLU45,introduced by Si Ma et al. two novel benzobisoxazole-bridged, sp carbon linked 2D-COFs, which differ mainly in their aromatic core units, pyrene for COF-JLU45 and triazine for COF-JLU44^16–18.

Despite their success in experiments, a more detailed, mathematical understanding of the structural distinctions between these frameworks is crucial. The inherent complexity and information content of these periodic networks are frequently not adequately captured by qualitative definitions of“conjugation”and“porosity.” In order to close this gap, the complicated molecular graphs of COF-JLU44 and COF-JLU45 are transformed into distinct numerical descriptors using degree-based topological indices^19–23. Additionally, we use these indices to determine the frameworks’ Shannon entropy. The idea of Shannon entropy was initially laid out by Claude Shannon in his seminal work which was published in 1948²⁴. Entropy provides a theoretical foundation for forecasting features like graph energy and thermal stability in chemical graph theory by acting as a potent metric to measure structural disorder, complexity, and the distribution of vertex degrees^25–29. The graph entropy based on vertex orbit was first presented by Rashevsky in 1955³⁰. This was utilized by Mowshowitz for a few graph operations³¹. Dehmer investigated the inequalities derived from the graph entropies and vertex probability³². Ramin Kazemi used degree-based topological indices to calculate entropy on weighted graphs³³.

The aim of this study is to analyze the chemical structures of COF-JLU44 and COF-JLU45. It is started by calculating their degree-based edge partitions. Subsequently, various degree-based topological indices are computed for these molecules. Later, the Shannon entropies of the structures are determined, enabling a comparative analysis between the two frameworks. Graph entropy provides a tool to calculate the uncertainty inherent in the structural configurations that helps to effectively model connectivity information based on edge degrees and interactions. Through the examination of these entropy metrics, the study seeks to identify differences in structural information content that may correspond to variations in chemical and physical properties in which the best network can be found between the two. This approach not only helps in assessing the complexity of these covalent organic frameworks but also offers deeper insights into how topological indices influence entropy in chemically significant graphs.

Materials and methods

The study analyzes the structural topology and organic linkers of the covalent organic frameworks JLU44 and JLU45. JLU44 has triazine units, whereas JLU45 contains pyrene units in its core structure. Both frameworks are generated using Knoevenagel polycondensation, which produces long-range structured crystalline structures with surface areas and photoelectric characteristics. The inclusion of pyrene and platinum allows the COF JLU-45 to absorb hydrogen across a wide range. JLU-45 acts as a metal-free photocatalyst, relying solely on sunlight to perform the chemical reaction that produces hydrogen fuel, making hydrogen fuel production more sustainable and reliable.

Let be a simple undirected molecular graph, and represent the vertex set and edge set respectively. Let and be a vertices such that then, and are said to be neighbours if there exists an edge connecting them. The degree of vertex is said to be the total number of neighbours of . It is denoted by ^34–36. Figures 1 and 2 represent the molecular graph of COF-JLU44 and COF-JLU45 respectively. Through out the paper, is regarded as a size metric indicative of the growth of the COF structures. refers to the growth of extended sp-conjugated framework, resulting in larger domain sizes, longer - delocalization pathways, and higher structural complexity, which is relevant to photocatalytic function.

Fig. 1.

Molecular graph of COF-JLU44.

Fig. 2.

Molecular graph of COF-JLU45.

In this study the degree-based edge partition method is used and is done by calculating the degrees of vertices and considering the adjacent edges. Let and be degrees of vertices and respectively such that then,

1

Edge partitioning involves classifying edges based on the degree of their end vertices^37–40. Later, we classify them into sets based on edges with the same degree composition and calculate the cardinality of each set. Tables 1 and 2 will provide the edge partitioning based on COF-JLU44 and COF-JLU45.

Table 1.

Edge partition of COF-JLU44.

Edges
(2, 2)
(2, 3)
(3, 3)

Table 2.

Edge partition of COF-JLU45.

Edges
(2, 2)
(2, 3)
(3, 3)

Degree-based topological indices are basic numerical invariants that describe molecular graph connectivity patterns. Table 3 lists the degree-based topological indices that were employed in this investigation^41–47. These descriptions serve as a mathematical link between the frameworks’ physical manifestations and their abstract graph-theoretic form.

Table 3.

Degree-based topological indices.

Topological index	Mathematical expression
Randić
Reciprocal Randić
Reduced Reciprocal Randić
First Zagreb
Second Zagreb
Reduced Second Zagreb
Hyper Zagreb
Augmented Zagreb
Harmonic
Sum Connectivity
Geometric Arithmetic
Inverse Sum
Albertson
Symmetric Division
Atom Bond Connectivity
Forgotten

In this study, entropy calculations are performed using Shannon entropy. Shannon entropy, a statistical measure of the uncertainty or information content included in a discrete probability distribution, forms the foundation for the quantification of structural complexity in this work. The functional

defines the entropy E(P) of a probability space

This abstract measure is transferred onto the molecular framework in chemical graph theory by defining the probability space using degree-based topological indices^24–26. A weight is allocated to each edge in the graph of the covalent organic framework based on a particular topological descriptor, like the Randić or Zagreb index, which represents the bond environment and local connectivity. We obtain the probability for each edge by normalizing these weights against the graph’s total index value,

This satisfies the normalization criterion . Thus,

is the expression for the Shannon entropy based on a topological index. A higher degree of topological symmetry and uniformity is indicated by a lower entropy value, whereas greater structural heterogeneity is indicated by a higher entropy value^48–54. This equation enables a qualitative interpretation of structural organization. We can quantitatively differentiate the small differences in the carbon-linked networks of COF-JLU44 and COF-JLU45 by using this formalism, which connects mathematical“uncertainty”to the materials’ periodic order and physical robustness.

Results

In this section, generalized formulations of 15 degree-based topological indices for the molecular graphs of the COF-JLU44 and COF-JLU45 are computed using the edge partitions given in Tables 1 and 2 which we express as the following Theorems.

Theorem 1

Let denote molecular graph of COF-JLU44 with dimension . Then,

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

Proof

Let be COF-JLU44’s molecular graph. The partitioning of the edge set of with respect to the vertex degree is employed in the formulation of degree-based topological indices to determine the following.

Theorem 2

Let denote molecular graph of COF-JLU45 with dimension . Then,

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

Proof

Let be COF-JLU45’s molecular graph. The partitioning of the edge set of with respect to the vertex degree is employed in the formulation of degree-based topological indices to determine the following.

Numerical computation of topological indces

In this section, we use the analytical expressions developed in Theorems 1 and 2 to calculate different degree-based topological indices in order to assess the structural evolution of COF-JLU44 and COF-JLU45. We offer a quantitative trajectory of the frameworks’ connection by increasing the growth parameter n from 1 to 10. The outcomes are thoroughly described in Tables 4 and 5 and graphically examined in Figs. 3 and 4.

Table 4.

Numerical values for degree-based topological indices of COF-JLU44 for to 10.

n	1	2	3	4	5	6	7	8	9	10
R	100.5857	484.9286	1153.0286	2104.8857	3340.4999	4859.8714	6662.9999	8749.8856	11120.5284	13774.9284
RR	579.5143	2825.5714	6738.1713	12317.3141	19562.9997	28475.2281	39053.9994	51299.3135	65211.1704	80789.5701
RRR	333.5879	1631.9394	3895.0545	7122.9333	11315.5757	16472.9818	22595.1515	29682.0848	37733.7818	46750.2424
	1176	5736	13680	25008	39720	57816	79296	104160	132408	164040
	1416	6936	16560	30288	48120	70056	96096	126240	160488	198840
	480	2364	5652	10344	16440	23940	32844	43152	54864	67980
HM	5832	28584	68256	124848	198360	288792	396144	520416	661608	819720
AZ	2001.375	9718.875	23152.5	42302.25	67168.125	97750.125	134048.25	176062.5	223792.875	277239.375
H	99.2	478	1136.4	2074.4	3292	4789.2	6566	8622.4	10958.4	13574
SC	108.9298	526.6492	1253.1581	2288.4565	3632.5445	5285.4220	7247.0890	9517.5456	12096.7916	14984.8272
GA	236.6057	1147.0286	2731.2685	4989.3256	7921.1999	11526.8913	15806.3998	20759.7254	26386.8682	32687.8281
IS	285.6	1392	3319.2	6067.2	9636	14025.6	19236	25267.2	32119.2	39792
SD	508	2468	5880	10744	17060	24828	34048	44720	56844	70420
ABC	168.7351	818.2195	1948.4532	3559.4364	5651.1688	8223.6506	11276.8818	14810.8623	18825.5922	23321.0714
F	3000	14712	35136	64272	102120	148680	203952	267936	340632	422040

Table 5.

Numerical values for degree-based topological indices of COF-JLU45 for to 10.

n	1	2	3	4	5	6	7	8	9	10
R	86.9224	458.0735	1113.453	2053.0612	3276.8979	4784.9632	6577.2571	8653.7795	11014.5305	13659.51
RR	525.5347	2780.4408	6764.7183	12478.3672	19921.3875	29093.7792	39995.5423	52626.6768	66987.1828	83077.0601
RRR	309.0193	1635.6063	3979.761	7341.4834	11720.7734	17117.6312	23532.0565	30964.0496	39413.6104	48880.7388
	1064	5632	13704	25280	40360	58944	81032	106624	135720	168320
	1324	7008	17052	31456	50220	73344	100828	132672	168876	209440
	472	2496	6072	11200	17880	26112	35896	47232	60120	74560
HM	5424	28736	69936	129024	206000	300864	413616	544256	692784	859200
AZ	1845.1875	9719.5	23622.9375	43555.5	69517.1875	101508	139527.9375	183577	233655.1875	289762.5
H	85.8667	452.2667	1099.2	2026.6667	3234.6667	4723.2	6492.2667	8541.8667	10872	13482.6667
SC	95.2063	502.286	1221.2392	2252.0658	3594.7659	5249.3395	7215.7865	9494.107	12084.3009	14986.3683
GA	209.4139	1105.7763	2689.0873	4959.3469	7916.555	11560.7117	15891.8169	20909.8707	26614.8731	33006.824
IS	259.6	1372.8	3339.6	6160	9834	14361.6	19742.8	25977.6	33066	41008
SD	445.3333	2357.3333	5736	10581.3333	16893.3333	24672	33917.3333	44629.3333	56808	70453.3333
ABC	148.1273	782.901	1904.3212	3512.3879	5607.101	8188.4606	11256.4667	14811.1192	18852.4182	23380.3636
F	2776	14720	35832	66112	105560	154176	211960	278912	355032	440320

Fig. 3.

Graphical comparision of Topological indices of COF-JLU44.

Fig. 4.

Graphical comparision of Topological indices of COF-JLU45.

Entropy analysis

Numerical computation

The computation of Shannon entropy is demonstrated below by computing the entropy of COF-JLU44 using the Randić index.

Randić index entropy for COF-JLU44:

Let denote COF-JLU44. The Randić index entropy is computed as follows:

The generic entropy equations for each of the COFs would be too long to offer as Theorems. Apply the above approach to determine degree-based entropies for each topological index.

The structural entropy of COF-JLU44 and COF-JLU45 is assessed in this section by varying the unit cell parameters n between 1 and 10. The numerical findings, which are shown graphically in Figs. 5 and 6 and described in depth in Tables 6 and 7, demonstrate a systematic evolution of structural information as the framework grows. To calculate these data and create a graphical representation, we utilized MATLAB. This comparative study offers a quantitative window into the topological uncertainty present in the carbon-linked networks and goes beyond simple calculation. The importance of this approach is that it unravels the fluctuations of symmetry and connection density when the material moves from a molecular cluster to a bulk periodic structures. In the context of graph theory, this “uncertainty” is a stand-in for the amount of information needed to characterize the system’s connectivity; a framework with higher entropy usually has a more varied distribution of bond environments, which can affect the localized electronic density and exciton migration pathways during photocatalytic hydrogen evolution.

Fig. 5.

Graphical comparision of entropies of COF-JLU44.

Fig. 6.

Graphical comparision of entropies of COF-JLU45.

Table 6.

Numerical values for degree-based entropies of COF-JLU44.

n	1	2	3	4	5	6	7	8	9	10
ER	5.474642	7.053961	7.921763	8.524395	8.986694	9.36187	9.677628	9.950247	10.190114	10.404262
ERR	5.474707	7.05403	7.92183	8.524462	8.986759	9.361935	9.677693	9.950312	10.190178	10.404326
ERRR	5.463333	7.043306	7.91128	8.513992	8.976336	9.351542	9.667321	9.939956	10.179834	10.393992
	5.474543	7.053938	7.921758	8.524398	8.9867	9.36188	9.67764	9.95026	10.190128	10.404277
	5.45696	7.037288	7.905357	8.508113	8.970482	9.345705	9.661496	9.934139	10.174024	10.388186
	5.411324	6.994037	7.862734	8.465779	8.928315	9.303646	9.619512	9.892212	10.13214	10.346337
EHM	5.456989	7.037578	7.905716	8.508503	8.970891	9.346125	9.661924	9.934574	10.174463	10.38863
EAZ	5.473883	7.052688	7.92035	8.522917	8.985178	9.36033	9.676071	9.948677	10.188533	10.402674
EH	5.474146	7.053542	7.921365	8.524007	8.986311	9.361492	9.677253	9.949874	10.189741	10.403891
ESC	5.479041	7.058125	7.925861	8.528462	8.990742	9.365907	9.681657	9.95427	10.194131	10.408275
EGA	5.480595	7.059576	7.927283	8.529871	8.992143	9.367303	9.683049	9.955659	10.195519	10.409662
EIS	5.474786	7.05404	7.921822	8.524445	8.986737	9.36191	9.677666	9.950283	10.190148	10.404294
ESD	5.479981	7.05899	7.926705	8.529296	8.991571	9.366732	9.682479	9.95509	10.19495	10.40909
EABC	5.480488	7.059462	7.927168	8.529755	8.992027	9.367187	9.682933	9.955543	10.195402	10.409545
EF	5.456702	7.037549	7.905753	8.508572	8.970977	9.346224	9.662031	9.934686	10.17458	10.38875

Table 7.

Numerical values for degree-based entropies of COF-JLU45.

n	1	2	3	4	5	6	7	8	9	10
ER	5.348457	7.013468	7.902361	8.514541	8.982283	9.360979	9.679198	9.953635	10.194899	10.410155
ERR	5.348477	7.013486	7.902378	8.514556	8.982299	9.360993	9.679212	9.953649	10.194913	10.41017
ERRR	5.332997	6.998951	7.888063	8.500339	8.968137	9.346868	9.665112	9.939567	10.180845	10.396112
	5.34867	7.013702	7.9026	8.51478	8.982524	9.36122	9.679439	9.953877	10.195141	10.410398
	5.32438	6.990842	7.880073	8.492402	8.96023	9.338979	9.657237	9.931702	10.172988	10.388261
	5.264111	6.933794	7.823777	8.436441	8.904458	9.28333	9.601672	9.8762	10.117534	10.332845
EHM	5.325884	6.992413	7.881659	8.493995	8.961827	9.340579	9.658838	9.933305	10.174592	10.389866
EAZ	5.344918	7.009689	7.898527	8.510681	8.97841	9.357096	9.675309	9.949741	10.191002	10.406255
EH	5.348254	7.013298	7.9022	8.514382	8.982127	9.360824	9.679044	9.953482	10.194747	10.410003
ESC	5.354526	7.019161	7.907966	8.520106	8.987826	9.366507	9.684716	9.959146	10.200404	10.415656
EGA	5.356536	7.021035	7.909808	8.521933	8.989645	9.368321	9.686526	9.960953	10.20221	10.41746
EIS	5.348182	7.01317	7.902057	8.514234	8.981974	9.360669	9.678887	9.953323	10.194587	10.409843
ESD	5.355829	7.020347	7.909125	8.521252	8.988966	9.367642	9.685848	9.960275	10.201532	10.416783
EABC	5.356308	7.020813	7.909588	8.521713	8.989426	9.368102	9.686307	9.960734	10.201991	10.417241
EF	5.326939	6.993537	7.8828	8.495144	8.962979	9.341734	9.659995	9.934463	10.175751	10.391026

Comparitive discussion

The structural properties of covalent organic frameworks JLU44 and JLU45 were analysed by calculating the topological Shannon entropy values for the two systems based on their graph theoretic interpretations. In this context, entropy is used as a measure of topological heterogeneity in the molecular graph and is treated as a structural complexity metric rather than a thermodynamic property. High entropy values are related to high diversity in vertex connectivity, asymmetry, or non-uniform structure. Entropy values were computed for various topological indices upto ten-dimensional configurations. In all instances, JLU44 exhibited slightly higher entropy values than JLU45. For example, in the Randić index-based entropy, the mean value for JLU44 exceeded that of JLU45 by approximately 0.1 units, with similar differences exhibited for the remaining indices. Although the differences are relatively small in magnitude, they are uniform across several descriptors, which reveals a consistent difference in the structure between the two frameworks. The consistently higher entropy value in COF-JLU44 could be an indication of relatively more heterogeneous connectivity structure in the graph representation of COF-JLU44. Conversely, the lower entropy values for COF-JLU45 suggest a more uniform and regular topological arrangement. It must be noted that these interpretations are purely based on graph models and therefore refer to relative differences in structural organization rather than physical properties as measured by experiments. Although characteristics like enhanced pore diversity or flexible local environments may be associated with greater structural heterogeneity, these associations are primarily theoretical in the absence of experimental support. By contrast, a lower entropy value does not necessarily indicate a higher degree of thermodynamic stability, rigidity, or crystallinity. Therefore, the above findings should be viewed as a qualitative guide for topological structure rather than a predictive tool for material properties. From the above analysis, it can be concluded that COF-JLU44 has a slightly more complex structural topology than COF-JLU45, while COF-JLU45 has a relatively regular structure. The analysis shows that the topological metric of entropy can provide a useful guide for comparing the structural topology in covalent organic frameworks and can serve as a theoretical parameter for designing these COF architectures.

Conclusion

This study examines the degree-based topological indices of COF-JLU44 and COF-JLU45, two covalent organic frameworks with unique structural characteristics. Shannon entropy was evaluated in order to gain a deeper quantitative understanding of the structural complexity, regularity, and potential stability of these two frameworks, building on the degree-based topological indices. Based on estimated entropy values, COF-JLU44 suggests a framework with increased configurational flexibility and structural diversity. In contrast, COF-JLU45 exhibits lower entropy values, indicating a more regular symmetric topological structure. This is typically associated with increased thermodynamic stability, stiffness, and maybe improved crystallinity. Overall, the comparative entropy study reveals that JLU45 has a more organized and thermodynamically resilient architecture, but JLU44 is notable for its structural adaptability and diversity. This work demonstrates the effectiveness of entropy-based metrics, which are obtained from degree-based topological indices, as useful instruments for differentiating between functionality and stability in the topological design of COFs. In order to comprehend the structural information, this analysis is restricted to the degree-based topological indices and the shannon entropy obtained from a concept of vetrex degree. The COFs’ spectural or distance effects on the structure have not been investigated. Thus, spectral graph descriptors (e.g., graph Laplacian eigenvalues), distance-based topological indices, and quantum-chemical parameters (e.g., band gaps, charge distribution, and frontier molecular orbitals) could be included in future research. Integrating these descriptors with entropy and degree-based measurements might improve QSPR models and offer a multiscale insight into structure property relationships in COFs. These can aid in the advancement of the function of information theoretic and graph theoretical techniques.

Author contributions

FS: Conceptualization; writing-original draft; methodology RS: Supervision; validation; conceptualization; writing- review and editing JK: Writing-original draft; software; investigation GKB: Visualization; software.

Funding

Open access funding provided by Vellore Institute of Technology. Not applicable.

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.