Abstract
Topological indices are numerical numbers that represent the topology of a molecule and are calculated from the graphical depiction of the molecule. The importance of topological indices is due to their use as descriptors in QSPR/QSAR modeling. QSPRs (quantitative structure-property relationships) and QSARs (quantitative structure-activity relationships) are mathematical correlations between a specified molecular property or biological activity and one or more physicochemical and/or molecular structural properties. In this paper, we give explicit expressions of some degree-based topological indices of two classes of metal-organic frameworks (MOFs), namely, butylated hydroxytoluene- (BHT-) based metal-organic (, Fe, Mn, Cr) (MBHT) frameworks and (TPyP = -tetrakis(4-pyridyl)porphyrin and = Fe and Co) MOFs.
1. Introduction
Metal-organic frameworks (MOFs) are defined by their regular array of metal cores and organic linkers in a three-dimensional framework. In MOFs, all metal cores are connected to organic linkers to construct networks which can contain a range of guest molecules. The first MOF was reported in 1959 by Kinoshita et al. [1]. MOFs receive attention due to the use of reticular chemistry for their design and synthesis [2]. After that, thousands of MOFs have been synthesized and broaden the scope of their potential applications. MOFs have shown applications in the area of gas catalysis [3–5], delivery of drugs [6–8], sensing [9, 10], separation [11, 12], storage [13–15], and adsorption [16–20]. There are many open sites in MOFs that are capable of capturing industrial flue gases, such as , , NO, CO, and [21–23]. These flue gases have the harmful effect on the environment. For instance, the emission of CO2 from the burning of fossil fuel is the major issue for climate change and greenhouse effect [24]. and are the main reason for the induction of acid rain as well as smog [25], and CO and NO are asphyxiants for humans [26]. Therefore, there is a need to develop an efficient strategy to limit the release of these hazardous gases to improve the worsening environmental quality. MOFs have shown a great potential to trap in their porous structure. MOF-74 has the ability to capture at room temperature and low pressure. For more details on the ability of MOFs to capture flue gas molecules, refer [27–29].
In mathematical chemistry and in chemical graph theory, the structural formula of a chemical compound is represented by a molecular graph where the vertices are represented by the atoms, and the edges are represented by the bonds between the vertices. Let be a molecular graph, where and denote the vertex set and edge set, respectively. Two vertices and are adjacent if they are end vertices of a common edge . The set of neighbors of a vertex is denoted by and is defined as . The degree of a vertex is symbolized by and is the cardinality of the set . Let denote the sum of degrees of the neighbor of the vertex , that is, . For undefined terminologies related to graph theory, we refer the reader to [30].
Molecular descriptors play an important role in the quantitative description of the molecular structure in finding appropriate predictive models. Molecular descriptors are terms that characterize a particular aspect of a molecule [31] and can be categorized into global and local according to the way the molecular structure is characterized. The topological indices (TIs) are among the most useful molecular descriptors known today [32–34]. These descriptors are numerical values related to chemical structure to correlate chemical structure with different physical properties, chemical reactivity, or biological activity [31]. First, topological index was introduced by Weiner [35], named Weiner index, and is defined as the number of carbon-carbon between all pairs of carbon atoms in an alkane. After 25 years, the introduction of connectivity indices and their application motivated the study of such descriptors [31]. Most of the known topological indices are global molecular descriptors. It means these indices characterize the molecule as a whole. For example, it can describe its branching or shape of the entire structure. The first degree-based topological index was introduced by Randic [36] in 1975. It is denoted by and is defined as
Randic observed that there is a very good correlation between Randic index and certain physical/chemical properties of alkanes: boiling points, enthalpies of formation, chromatographic retention times, surface areas, and parameters in the Antoine equation for vapor pressure. Later, in 1998, Bollobás and Erdȯs [37] generalized this index by replacing with any real number , which is called the general Randic index:
The first and second Zagreb indices were introduced by Gutman et al. [38] and were applied to the branching problem in 1972. The first and second Zagreb indices are denoted and defined as
Recently, Shirdel et al. [39] proposed the hyper-Zagreb index:
The Zagreb indices and their variants have been used to study molecular complexity [40–44], chirality [45], ZE-isomerism [46], and heterosystems [47] whilst the overall Zagreb indices exhibit potential applicability for deriving multilinear regression models. Various researchers also use the Zagreb indices in their QSPR and QSAR studies [48–53].
In 2009, Zhou and Trinajstic [54] introduced the sum connectivity index. It was observed that the sum connectivity index correlates well with the electron energy of hydrocarbons. It is denoted and defined as
Recently, Zhou and Trinajstic [55] extended this concept to the general sum connectivity index. The general sum connectivity index is defined as
Atom-bond connectivity (ABC) index was introduced by Estrada et al. [56] in 1998 which is denoted as
The ABC index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes [56, 57].
Recently, the well-known connectivity topological index is geometric-arithmetic index which was introduced by Vukičević and Furtula in [58]. For a graph , the index is denoted and defined as
It has been demonstrated on the example of octane isomers that index is well correlated with a variety of physicochemical properties.
The fourth version of the atom-bond connectivity index () was introduced by Ghorbani and Hosseinzadeh [45] in 2010 and is defined as
The fifth version of the topological index is proposed by Graovac et al. [59] in 2011 which is expressed as
For more details on the computation of topological indices, we refer the readers to [60–65].
The main aim of this work is to compute the degree-based topological indices of two classes of MOFs. The computed topological indices can be used in QSPR/QSAR studies to improve the physical/chemical properties of the considered MOFs. The same technique can be used to compute the degree-based topological indices of other classes of MOFs. In the next section, we compute the above-defined topological indices of metal-organic frameworks.
2. Topological Aspects of Structure of Metal-Organic Frameworks
Wurster et al. [66] prepared MOFs and observed that these MOFs have two metal centers at two distinguished coordination environments. They are bimetallic MOFs. Depending on and , these MOFs become either hetero- or homobimetallic. The performance of these MOFs was observed in oxygen evolution reactions, which generate from water. They reported that the catalytic activity of metalloporphyrins was lower than that of heterobimetallic MOFs. The 2D structure of MOFs is depicted in Figure 1. We denote the graph of MOFs by , where and represent the number of unit cell in each row and column, respectively. The 2D structure of the molecular graph of is shown in Figure 1.
A simple calculation shows that has vertices and edges. First, the general Randic connectivity and general sum connectivity indices of were computed.
Theorem 1. The Randic connectivity index and general sum connectivity indices of the graph are
Proof. To compute the general Randic connectivity index and general sum connectivity index, we need to find the edge partition of depending on the degree of end vertices. This partition is given in Table 1. Now using the values of the edge partition in the definition of these indices, we get the result as follows:
Corollary 1. The values of Randic connectivity, first and second Zagreb, hyper-Zagreb, and sum connectivity indices can be computed from the above theorem by using the value of .In the next theorem, the ABC and GA indices of were computed.
Theorem 2. The ABC and GA indices of the graph are
Proof. By using the values of the edge partition given in Table 1 in the definition of the ABC index, this index can be determined as follows:Similarly, the GA index can be calculated asFinally, the expression of and indices are calculated in the next theorem.
Theorem 3. The and indices of graph are
Proof. To compute the values of and indices, we need to find the partition of the edge set based on the sum of degrees of the neighbors of the end vertices of each edge. This partition is presented in Table 2. Now, using the values in the definition of index, this index can be calculated asSimilarly, the value of index can be calculated as
3. Topological Aspects of CoBHT (CO) Lattice
Clough et al. [67] synthesize the 2D cobalt bis(dithioline) (CoBHT) metal-organic surface. Chakravarty et al. [10] investigated the electronic and magnetic properties of a 2D metal-organic (MBHT) framework. They observed that all these frameworks are planar, perfect Kagome lattices with a six-fold symmetry. There is a possibility that these MOFs can be used as gas sensors such as CO sensing. The optimized structure of these MOFs is shown in Figure 2.
We denote the molecular graph of 2D metalorganic superlattice by with unit cells in each row and unit cell in each column. Figure 2 depicts the molecular graph . A simple calculation shows that has vertices and edges. First, we compute the general Randic connectivity and general sum connectivity indices of .
Theorem 4. The Randic connectivity index and general sum connectivity indices of graph are
Proof. To compute the general Randic connectivity index and general sum connectivity index, we need to find the edge partition of depending on the degree of end vertices. This partition is given in Table 3. Now using the values of the edge partition in the definition of these indices, we get the result as follows:
Corollary 2. The values of Randic connectivity, first and second Zagreb, hyper-Zagreb, and sum connectivity indices can be computed from the above theorem by using the value of .In the next theorem, the ABC and GA indices of were computed.
Theorem 5. The ABC and GA indices of the graph are
Proof. By using the values of the edge partition given in Table 3 in the definition of the ABC index, this index can be determined as follows:Similarly, the GA index can be calculated asFinally, the expression of and indices are calculated in the next theorem.
Theorem 6. The and indices of the graph are
Proof. To compute the values of and indices, we need to find the partition the edge set based on the sum of degrees of the neighbors of the end vertices of each edge. This partition is presented in Table 4. Now, using the values in the definition of index, this index can be calculated asSimilarly, the value of index can be calculated as
4. Conclusion
The importance of computing the molecular descriptors can be understood by taking into account the recent advances in drug discovery technologies. Combinatorial chemistry, pharmacogenomics, and high-throughput screening enable us to obtain and evaluate thousands of compounds in a short period of time. These technologies pose new challenges for computational scientists as they require new approaches to computer-aided lead discovery and optimization in an accelerated way. The traditional quantitative structure-activity relationships (QSARs) are not only used to improve the biological activity of leads but also to improve their physicochemical, pharmacokinetic, and toxicological properties. Hence, the computed topological indices can be used in QSPR/QSAR studies to improve the physical/chemical properties of the considered MOFs.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.