Reticular chemistry connects organic building blocks through strong covalent bonds, which have the capacity to regulate the pore sizes of frameworks by preserving their fundamental topology and varying the lengths of organic linkers, thus paving the way for the emergence of multiple classes of crystalline porous materials (Yaghi, 2016; Yaghi, 2019). Reticulated materials can be classified as metal organic frameworks (MOFs), created by the combination of organic linkers and metal atoms, and covalent organic frameworks (COFs), composed only of organic linkers (Gropp et al., 2020). COFs have drawn particular attention from researchers due to their regular pattern of organic building blocks, which allows for the creation of crystalline structures with extensive surface areas, stability, and customizable pores (El-Kaderi et al., 2007). COFs possess potential applicatiions in separation (Fan et al., 2023), luminescence (Haug et al., 2020), biomedicine (Shi et al., 2023), energy conversion (Sun et al., 2023), environmental remediation (Hou et al., 2023), seawater desalination (Jrad et al., 2023), photocatalysis (Gong et al., 2023), and electrocatalysis (Zhang et al., 2021). The COFs have predetermined structures based on their building blocks, allowing for highly ordered geometries (Huang et al., 2016; Chen et al., 2014). Their covalently crystalline structure gives them advantages over other porous materials such as molecular sieves, MOFs and zeolites (Yang et al., 2019; Jiao et al., 2019; Algieri and Drioli, 2021).

Covalent bonds within COFs can arise from a diverse range of functional groups. The methods for forming these bonds can be broadly classified into several categories, including boroxine-linked, boronate ester-linked, triazine-linked, imine-linked, hydrazone-linked, β -ketoenamine-linked, azine-linked, imide-linked, carbon-carbon linked, and others (Wang et al., 2020; Geng et al., 2020). Boronate ester based COFs represent a category of crystalline, porous polymers characterized by layer-stacked structures, which are formed through reversible covalent interactions between boronic acid and catechol. The initial reported methods for COF formation involved the self-condensation of boronic acids into boroxine rings and the co-condensation of boronic acids with catechols to form boronic esters. This bond type stands out as one of the most frequently observed COF formation, with COF-5 being an early example falling within this category (Cot̂é et al., 2005; Li et al., 2018). Since then the varieties of COFs featuring boron, have gained significant attention primarily due to their exceptional thermal stability (Kuhn et al., 2008).

The COFs considered in this study are made up of polycyclic aromatic hydrocarbons (PAHs) with contorted hexabenzocoronene (c-HBC) serving as the core component for constructing the two Marta-COFs. The c-HBC adopts a doubly-concave structure, which sets it apart from the planar hexabenzocoronene. Its formation occurs when the aromatic core of HBC is distorted away from planarity due to steric congestion in its proximal carbon atoms. Structurally, c-HBC is the building block composed of six benzene rings attached to the periphery of a coronene molecule (Sepúlveda et al., 2017; Kim et al., 2022). The c-HBC unit, as shown in Figure 1, when copolymerized with pyrene-2,7-diboronic acid (PDBA), results in the formation of a highly crystalline two-dimensional COF known as Marta-COF-1 (Abadía et al., 2019). Notably, the c-HBC nodes and the PDBA display substantial π -areas and extensive π -stacking within the resulting COF. In response to this, a comparable COF labeled Marta-COF-2 has been created, synthesized and investigated with the substitution of pyrene-2,7-diboronic acid by benzene-1,4-diboronic acid (BDBA) (Abadía et al., 2021). As depicted in Figure 2, graphical diagram representations of the two COF frameworks, Marta-COF-1 and Marta-COF-2, are illustrated to highlight their distinctive arrangements.

The two variations of highly crystalline Marta COFs can be evaluated through quantitative parameters called the topological descriptors that convert various structural attributes of the frameworks into measurable quantities. These quantifying functionals are essential for representing the molecular frameworks and are useful for QSPR and QSAR analyses (Jafari et al., 2024; Patil et al., 2024; Nath et al., 2023; Hayat et al., 2023). The incorporation of topological descriptors and graph-derived metrics in QSAR/QSPR studies has been extensively used in the domain of computational and material sciences. This amalgamation has provided robust tools for predicting structural behaviors and designing new materials with desired properties or functionalities. These approaches enable researchers to explore numerous applications, including drug discovery, material optimization, and the development of materials that can be tailored for specific applications or objectives (Balasubramanian and Saxena, 2021; Balasubramanian, 2022; Arockiaraj et al., 2023a; Hasani and Ghods, 2024; Abubakar et al., 2024; Meharban et al., 2024; Ullah et al., 2024; Shanmukha et al., 2023a; Gnanaraja et al., 2023; Zhang et al., 2023; Hassan et al., 2024).

The graph entropy measure enables the evaluation of the inherent complexity and diversity of COFs. This measure provides valuable insights into the arrangement and functioning of COF structures by associating fundamental graph components with appropriate weights (Junias and Clement, 2023; Arockiaraj et al., 2024a; Chu et al., 2023; Roy et al., 2023; Zhao et al., 2023; Lal et al., 2024). The applications of graph entropy continue to expand its relevance and significance across diverse domains due to its adaptable nature that surpasses disciplinary boundaries and facilitates the analysis of complex systems (Arockiaraj et al., 2023b; Junias et al., 2024; Huang et al., 2024). In recent years, there has been significant interest in the computation of topological expressions and entropies for COFs (Yang et al., 2024; Arockiaraj et al., 2023c; Augustine and Roy, 2022; Shanmukha et al., 2023b; Arockiaraj et al., 2024b). In this study, we provide hybrid topological characterizations and entropies for two variations of Marta COFs and conduct a comparative analysis of the bond-wise entropy of these frameworks. Furthermore, we construct regression models to predict the graph energy of these frameworks based on the calculated topological indices.

We consider the Marta-COF as a molecular graph where the sets V ( Marta - COF ) and E ( Marta - COF ) represent the atoms and bonds respectively. Our mathematical computation involves deriving topological descriptors and entropies, incorporating hybrid descriptors based on vertex degree and degree-sum parameters. The number of bonds incident to a vertex p ∈ V ( Marta - COF ) is denoted as d Marta - COF ( p ) which represents the degree of a vertex p . Additionally, the total sum of the degrees of all neighbors of vertex p is denoted as s Marta - COF ( p ) which is defined as the degree-sum of vertex p . That is, s Marta - COF ( p ) = ∑ q ∈ N Marta - COF ( p ) d Marta - COF ( q ) in which we used N Marta - COF ( p ) = { q ∈ V ( Marta - COF ) | p q ∈ E ( Marta - COF ) } . Let d ( r , x ) = | { m n ∈ E ( Marta - COF ) : r = d Marta - COF ( m ) and x = d Marta - COF ( n ) } | and s ( r , x ) = | { m n ∈ E ( Marta - COF ) : r = s Marta - COF ( m ) and x = s Marta - COF ( n ) } | . The total number of edges within Marta-COFs is classified into distinct edge classes based on symmetrical representations related to d ( r , x ) and s ( r , x ) . These edge classes are labeled as D (Marta-COF) and S (Marta-COF), respectively.

We now define the additive and multiplicative versions of topological descriptors related to the degree and degree-sum parameters of Marta-COF, involving the index function ξ , as follows (Hakeem et al., 2023; Paul et al., 2023; Arockiaraj et al., 2022; Mondal et al., 2022; Arockiaraj et al., 2023d; Ramane et al., 2021; Yu et al., 2023; Zaman et al., 2023; Hassan et al., 2024): ξ d M a r t a − C O F = ∑ d r , x ∈ D Marta - COF d r , x ξ r , x ξ d * M a r t a − C O F = ∏ d r , x ∈ D Marta - COF d r , x ξ r , x ξ s M a r t a − C O F = ∑ s r , x ∈ S Marta - COF s r , x ξ r , x ξ s * M a r t a − C O F = ∏ s r , x ∈ S Marta - COF s r , x ξ r , x

When the index function ξ is raised to its own power, the resulting versions of topological descriptors can take the following forms: ξ d p M a r t a − C O F = ∑ d r , x ∈ D Marta - COF d r , x ξ r , x ξ r , x ξ d p * M a r t a − C O F = ∏ d r , x ∈ D Marta - COF d r , x ξ r , x ξ r , x ξ s p M a r t a − C O F = ∑ s r , x ∈ S Marta - COF s r , x ξ r , x ξ r , x ξ s p * M a r t a − C O F = ∏ s r , x ∈ S Marta - COF s r , x ξ r , x ξ r , x

The index functions ξ ( r , x ) are considered in our study, as stated below (Arockiaraj et al., 2024a; Arockiaraj et al., 2024b; Arockiaraj et al., 2023e; Arockiaraj et al., 2024c).

• B M ( r , x ) = r + x + r x (Bi − Zagreb)

These index functions, combined with the edge classes based on d ( r , x ) and s ( r , x ) , lead to the formation of topological descriptors. However, the representative element in the edge classes does not account for the specific types of atoms involved at their terminal points. Since three types of atoms are present in Marta covalent organic frameworks, which constitute the basis for Marta, it is important to distinguish between the atoms. Therefore, we involve weight functions that consider both the atoms and bonds, thereby enhancing the partitions based on d ( r , x ) and s ( r , x ) . The weight function for atoms will be denoted by the symbol Φ , while Γ will represent the weight function for bonds. Particularly, Φ B represents the weight assigned to atom B, while Γ BC denotes the weight function corresponding to the bond B − C. As a result, the edge classification of Marta-COFs will undergo additional refinement through the utilization of the bond weight function.

In employing Shannon’s entropy method, defining a structural information function on the bonds of Marta-COFs is necessary. In our study, we adopt the index function ξ derived from degree or degree-sum parameters of Marta-COFs corresponding to the structural information function. The entropy of Marta-COF structures using the structural information function ξ is defined on E ( Marta - COF ) = { c 1 , c 2 , … , c m } and takes the following form. I ξ M a r t a − C O F = − ∑ x = 1 m ξ c x ∑ z = 1 m ξ c z log ξ c x ∑ z = 1 m ξ c z = log ∑ x = 1 m ξ c x − 1 ∑ x = 1 m ξ c x log ∏ c = 1 m ξ c x ξ c x

In a series of papers (Arockiaraj et al., 2023c; Mushtaq et al., 2022; Raza et al., 2023), the significance and implications of substituting the multiplicative factor have been comprehensively explored concerning the scalar multiplicative index. This leads to the formulation of the modified version of entropy as presented below. I ξ M a r t a − C O F = log ξ M a r t a − C O F − 1 ξ M a r t a − C O F log ξ p * M a r t a − C O F

In this section, the two types of Marta-COFs are analyzed, and their structural properties are compared using topological descriptors and entropies. We consider the geometrical configuration of bi-trapezium (BT) shaped arrangements of Marta-COFs, which yield diverse configurations of Marta-COF layers. These Marta-COF structures are constructed using the unit cells as shown in Figure 2, which are the fundamental building blocks.

The Marta-COF-BT ( t , u ) geometric formation is achieved by arranging t units linearly to form the base and u units to form the non-parallel sides, subject to the conditions t ≥ 2 and u ≤ t . By fixing t = 2 u − 1 and t = u respectively, the hexagonal and parallelogram geometries are extracted from the BT configurations which are denoted by Marta-COF-H ( u ) and Marta-COF-P ( u , u ) . The linear chain of Marta-COFs is derived by setting t = 1 and is represented as Marta-COF-L ( u ) . The representations of hexagonal structures for two variations of Marta-COFs are depicted in Figures 3, 4.

Furthermore, the covalent organic framework Marta-COF-1-BT ( t , u ) is composed of 324 t u − 162 u 2 − 16 t + 308 u − 16 vertices and 414 t u − 207 u 2 − 21 t + 393 u − 21 edges, while Marta-COF-2-BT ( t , u ) comprises 264 t u − 132 u 2 − 6 t + 258 u − 6 vertices and 336 t u − 168 u 2 − 8 t + 328 u − 8 edges. We have computed diverse molecular descriptors of Marta COFs by calculating degree and degree-sum parameters and the distribution of bonds are shown in Tables 1, 2. The explicit mathematical expressions for these descriptors in Marta-COFs are derived by assigning unit weights to atoms and bonds.

The degree based descriptors for Marta-COF-1-BT ( t , u ) are obtained for ξ using the following equation. ξ d M a r t a - C O F - 1 - B T ( t,u ) = 4 t + 4 u + 4 Γ BO ξ 2 Φ O , Φ B + Φ C + 24 t u − 12 u 2 − 4 t + 20 u − 4 Γ BO × ξ 2 Φ O + Φ C , Φ B + Φ C + 12 t u − 6 u 2 − 2 t + 10 u − 2 Γ BC ξ 2 Φ O + Φ C , 2 Φ C + Φ B + 24 t u − 12 u 2 + 24 u Γ OC ξ Φ B + Φ C , 2 Φ C + Φ O + 150 t u − 75 u 2 − 5 t + 145 u − 5 × Γ CC ξ 3 Φ C , 3 Φ C + 24 t u − 12 u 2 + 24 u Γ CC ξ 2 Φ C + Φ O , 2 Φ C + 12 t u − 6 u 2 + 12 u × Γ CC ξ 2 Φ C + Φ O , 2 Φ C + Φ O + 120 t u − 60 u 2 − 12 t + 108 u − 12 Γ CC × ξ 2 Φ C , 3 Φ C + 48 t u − 24 u 2 − 2 t + 46 u − 2 Γ CC ξ 2 Φ C , 2 Φ C In computing the degree-sum descriptors of Marta-COF-1-BT ( t , u ) , we use ξ s M a r t a - C O F - 1 - B T ( t,u ) = 4 t + 4 u + 4 Γ BO ξ 4 Φ O , 3 Φ C + 2 Φ B + 24 t u − 12 u 2 − 4 t + 20 u − 4 × Γ BO ξ 4 Φ O + 3 Φ C , 3 Φ B + 3 Φ C + 24 t u − 12 u 2 − 4 t + 20 u − 4 × Γ OC ξ 3 Φ B + 3 Φ C , 5 Φ C + 2 Φ O + 4 t + 4 u + 4 Γ OC ξ 2 Φ B + 3 Φ C , 5 Φ C + 2 Φ O + 12 t u − 6 u 2 + 12 u Γ CC ξ 2 Φ O + 5 Φ C , 2 Φ O + 5 Φ C + 48 t u − 24 u 2 + 48 u × Γ CC ξ 8 Φ C , 9 Φ C + 24 t u − 12 u 2 + 24 u Γ CC ξ 8 Φ C , 8 Φ C + 24 t u − 12 u 2 + 24 u × Γ CC ξ 6 Φ C , 8 Φ C + 24 t u − 12 u 2 + 24 u Γ CC ξ 5 Φ C , 8 Φ C + 24 t u − 12 u 2 + 24 u × Γ CC ξ 4 Φ C , 5 Φ C + 54 t u − 27 u 2 − t + 53 u − 1 Γ CC ξ 9 Φ C , 9 Φ C + 12 t u − 6 u 2 + 12 u × Γ CC ξ 4 Φ C , 4 Φ C + 72 t u − 36 u 2 − 8 t + 64 u − 8 Γ CC ξ 6 Φ C , 7 Φ C + 12 t u − 6 u 2 − 2 t + 10 u − 2 Γ CC ξ 5 Φ C , 5 Φ C + 24 t u − 12 u 2 − 4 t + 20 u − 4 × Γ CC ξ 5 Φ C , 7 Φ C + 24 t u − 12 u 2 − 4 t + 20 u − 4 Γ CC ξ 7 Φ C , 9 Φ C + 12 t u − 6 u 2 − 2 t + 10 u − 2 Γ BC ξ 4 Φ O + 3 Φ C , 4 Φ C + 3 Φ B .

The resulting outcomes are given in the form, ξ # ( Marta - COF ) = { ξ d ( Marta - COF ) , ξ s ( Marta - COF ) } .

Result 1

The quantitative expressions for Marta-COF-1-BT ( t , u ) are given by

1. G H # ( M a r t a − C O F − 1 − B T ( t , u ) ) = t u ( 480 6 + 1758 ) − u 2 ( 240 6 + 879 ) − t ( 40 6 + 55 ) + u ( 440 6 + 1703 ) − 40 6 − 55 , t u 672 3 + 216 5 + 2 ( 312 5 + 2448 ) + 7 780 6 + 144 5 + 576 + 7578 − u 2 ( 336 3 + 108 5 + 2 ( 156 5 + 1224 ) + 7 ( 390 6 + 72 5 + 288 ) + 3789 ) − t ( 7 ( 104 6 + 96 ) − 36 5 + 229 ) + u 672 3 + 252 5 + 2 ( 312 5 + 2448 ) + 7 676 6 + 144 5 + 480 + 7349 + 36 5 − 7 ( 104 6 + 96 ) − 229