Topological properties of metal-organic frameworks

1 Introduction

Hydrogen gas has considered as a next-generation and eco-friendly source of energy which are very benefices for environment. On the other hand, it is highly combustible gas that can burst out even with a minor trigger. The modern guiding principle given by the energy sector of the United State put the emphasis on the speed proficiency of the instrument that should sense 1% by volume of fragrance-free and colorless hydrogen in environment in fewer than 60 s. An instrument (Koo et al., 2017) prepared an extreme-fast hydrogen detector composed of metal and organic ligands known as MOF with Palladium (Pd) nanowire that can be aware of hydrogen gas stages lower than 1% in fewer than 7 s. Furthermore, this instrument can notice hundreds of portions for each million stages of hydrogen gas in less than 60 s at room temperature.

Moreover, other than sensing and detecting, the MOFs represent very important physical and chemical characteristics such as grafting (Hwang et al., 2008), changing organic ligands (Yin et al., 2015), impregnating suitable active materials (Thornton et al., 2009), post synthetic ligand and ion exchange (Kim et al., 2012), environmental pollutants (Tong et al., 2005) and making composites with suitable materials (Ahmed and Jhung, 2014). Seetharaj et al. (2019) gave the relation between solvents, pH, molar ratio, temperature and the architectures of the MOFs. The metal-organic frameworks are used; for energy storage in batteries (Xu et al., 2017), super capacitors (Wang et al., 2017), separation and purification (Lin et al., 2019) of the chemical compounds. MOFs are also used as precursors for the preparation of various nanostructures (Yap et al., 2017).

Molecular graph theory created a significant usage in the field of chemistry which is used to calculate the different types of organic compounds and discussed their behavior. A topological index (TI) is a numerical value that shows physical, chemical and biological activities (Gonzalez-Diaz et al., 2007) of the fundamental organic compounds such as heat of evaporation and formation; melting, boiling and flash point; temperature, pressure, partition coefficient, tension and density of liquid and retention times in chromatographic (Liu et al., 2019).

First of all, Wiener created the practice of the indices in organic complexes with the reference of boiling point of paraffin (Wiener, 1947). He presented many symbols which are related to the numerical form of a graph as the addition of length in the middle of each pair of carbon atoms, the positions of bonds between carbon to carbon atoms that late known by the title of Wiener index. Gutman and Trinajsti (1972) described a couple of degree-based molecular descriptors known as the first and second Zagreb indices to compute the total energy of π-electron of the conjugated molecules. Later on, Randic used indices as the branching indices (Randic, 1975). Recently, Wasson et al. (2008) gave the idea of linker competition within a metal-organic framework for topological insights.

Furthermore, TIs play an important role in the studies of quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) to relate the molecular structure with a biological activity or a property. This relation of dependence of a molecular structure can be expressed mathematically as P = f (M), where P is activity or property (value of a chemical or a biological measurement) and M is a molecular structure. For further study, we refer to Devillers et al. (1997). We have several types of indices but degree-based TIs have countless significance and show the necessary part in graphs and for the most part in chemistry. Among the degree-based TIs, the most studied indices are Zagreb indices. The Zagreb indices of the various molecular frameworks such as Oxide, hexagonal, octahedral, icosahedral honeycomb, fullerenes, nanotubes, carbon nanotubes, benzenoid are computed in Javaid et al. (2016).

In this paper, we extended two different metal-organic frameworks MOF₁ (t) and MOF₂ (t) with respect to the number of increasing layers t with metal and organic ligands as well. We also compute the generalized Zagreb index and generalized Zagreb connection index of these frameworks. Moreover, the various indices and connection indices are obtained by using the aforesaid generalized versions. At the end, a comparison is also included between the indices and connection indices with the help of numerical values and their 3D plots.

The remaining portion is prepared as, Section 2 involves the basic concepts and techniques that are commonly utilized in the core conclusions. In Section 3 and 4, we calculate the generalized Zagreb index and generalized Zagreb connection index of the metal-organic frameworks MOF₁ (t) and MOF₂ (t). Moreover, using these generalized Zagreb index and connection index, we compute various other Zagreb indices and Zagreb connection indices. In Section 5, we compare the obtained results and draw the conclusion.

2 Preliminaries

In the molecular graph Γ = (Q (Γ), E (Γ)), where Q (Γ) = {q₁, q₂,...,q_n} and E (Γ) are set of vertices and edges respectively such that the edges are considered as a bond between the metals and the organic ligands or between the two organic ligands. The |Q (Γ)| = q is the order and |E (Γ)| = e is the size of Γ. A graph is connected if there exists a path for any couple of nodes. The length (number of edges) of the shortest path between two nodes p and q is called their distance which is denoted by d (p,q). For a vertex q ϵ Q (Γ) and a positive integer k, the open k-neighborhood of q in the graph Γ is denoted by Nk (q) and defined as Nk (q) = {qϵ Q (Γ): d (p,q) = k}. The k-distance degree of a vertex q ϵ Γ is d_k (q) = |Nk (q)|. In particular, for k = 1, d₁ (q) = d (q) is the degree of vertex qϵ V (Γ) and for k = 2, d₂ (q) = τ is called connection number of the vertex qϵ V (Γ). The k-distance generalized Zagreb index is defined as follows:

2.1 Definition

For r, s ϵ Z⁺ and k ≥ 1, the k-distance generalized Zagreb index Mr,skΓis:

Mr,sk(Γ)=∑pq∈E(Γ)(dk(p)rdk(q)s+dk(q)rdk(p)s).

For k = 1, it is called generalized Zagreb index Mr,s1(Γ)and for k = 2, it is known as generalized Zagreb connection index Mr, sc(Γ)which are presented as follows:

Mr,s1(Γ)=∑pq∈E(Γ)(d(p)rd(q)s+d(q)rd(p)s), and

Mr, sc(Γ)=∑pq∈E(Γ)(τ(p)rτ(q)s+τ(q)rτ(p)s).

In Table 1, the Zagreb indices such as first Zagreb index M₁ (Γ), second Zagreb index M₂ (Γ), forgotten index F (Γ), Redefined index ReZM (Γ), general first Zagreb index M   1α(Γ),general Randic index R_α (Γ) and symmetric division deg index SDD (Γ)depending on the degree of the vertices of the graph are given in Table 1.

Table 1

Zagreb indices.

M1(Γ)=∑pq∈E(Γ)[d(p)+d(q)]

M2(Γ)=∑pq∈E(Γ)[d(p)×d(q)]

F(Γ)=∑pq∈E(Γ)[d(p)2+d(q)2].

ReZM(Γ)=∑pq∈E(Γ)d(p)d(q)[d(p)+d(q)]

M1α(Γ)=∑p∈Q(Γ)[d(p)]∞

Rα(Γ)=∑pq∈E(Γ)[d(p)×d(q)]∞

SDD(Γ)=∑pq∈E(Γ)[d(p)d(q)+d(q)d(p)]

In Table 2, the relation between these Zagreb indices and the generalized Zagreb indices are presented.

Table 2

Zagreb indices in terms of generalized Zagreb indices.

M1(Γ)=M1,01(Γ)

M2(Γ)=12M1,11(Γ)

F(Γ)=M2,01(Γ)

ReZM(Γ)=M2,11(Γ)

M1α(Γ)=M∞−1,01(Γ)

Rα(Γ)=12M∞,∞1(Γ)

SDD(Γ)=M1,−11(Γ)

In Table 3, the Zagreb indices such as first Zagreb index M1  c(Γ),second Zagreb index M2  c(Γ),forgotten index F^c (Γ), Redefined index ReZM^c (Γ), general first Zagreb index M^αc₁ (Γ), general Randic index R^c_α (Γ) and symmetric division deg index SDD^c (Γ) depending on the degree of the vertices of the graph are given in Table 3.

Table 3

Zagreb connection indices.

M1c(Γ)=∑pq∈E(Γ)[τ(p)+τ(q)].

M2c(Γ)=∑pq∈E(Γ)[τ(p)×τ(q)].

Fc(Γ)=∑pq∈E(Γ)[τ(p)2+τ(q)2].

ReZMc(Γ)=∑pq∈E(Γ)τ(p)τ(q)[τ(p)+τ(q)].

M1αc(Γ)=∑p∈Q(Γ)[τ(p)]∞.

Rαc(Γ)=∑pq∈E(Γ)[τ(p)×τ(q)]∞.

SDDc(Γ)=∑pq∈E(Γ)[τ(p)τ(q)+τ(q)τ(p)].

In Table 4, the relation between these Zagreb connection indices and the generalized Zagreb connection indices are presented.

Table 4

Zagreb connection indices in terms of generalized Zagreb connection indices.

M1c(Γ)=M1,0c(Γ)

M2c(Γ)=12M1,1c(Γ)

Fc(Γ)=M2,0c(Γ)

ReZMc(Γ)=M2,1c(Γ)

M1αc(Γ)=M∞−1,0c(Γ)

Rαc(Γ)=12M∞,∞c(Γ)

SDDc(Γ)=M1,−1c(Γ)

The Zagreb indices such as first Zagreb index, second Zagreb index, forgotten index, redefined Zagreb index, general first Zagreb index, general Randic index, symmetric division deg index and generalized Zagreb index are defined by Gutman and Trinajsti (1972), Furtula and Gutman (2015), Ranjini et al. (2013), Hu et al. (2005), Randic (1975), Vukicevic and Gasperov (2010), and Azari and Iranmanesh (2011), respectively and for further study of Zagreb connection indices, see Tang et al. (2019).

Now, we talk about the MOF which is composition of metals and organic ligands as shown in Figure 1. The bigger node is a metal which is zeolite imidazole (Zinc-based) and the smaller node is an organic ligand. The edges are considered as a bond between the metals and the organic ligands as well as between the two organic ligands. A framework is merely a limited associated graph which has several edges and it is connected if there exists at least one edge in the middle of any couple of nodes. For further study of this MOF, we refer to Koo et al. (2017). Now, we construct two MOFs from the basic MOF of Figure 1 with respect to the number of increasing levels or dimensions consisting on metals and organic ligands as well, where each new level or dimension increases two layers in the previous level or dimension. The first metal-organic framework is obtained by creating the bonds between the metals of the two basic MOFs such that two metals of upper layer of a basic MOF are joined with a metal of lower layer of the second basic MOF. For dimension t = 2, the first metal-organic framework (MOF₁ (t)) is shown in Figure 2a. Similarly, we construct the second metal-organic framework by creating the bonds between the organic ligands of the two basic MOFs such that two organic ligands of upper layer of a basic MOF are joined with an organic ligand of lower layer of the second basic MOF. For dimension t = 2, the second metal-organic framework MOF₂ (t) is shown in Figure 2b. Moreover, for both the newly constructed MOFs, we have |Q (MOF₁ (t))| = |Q (MOF₂ (t))| = 48t and |E (MOF₁ (t))| = |E (MOF₂ (t))| = 72t − 12, where t ≥ 2.

Figure 1

Basic metal-organic framework.

Figure 2

(a) First metal-organic framework (MOF1 (t) for t = 2); (b) Second metal-organic framework (MOF2 (t) for t = 2).

3 Topological indices for the first metal-organic framework

In this area, we figure out the topological indices for the first metal-organic framework. In Theorem 3.1, we find the topological index that is based on the degrees of the nodes that is generalized Zagreb index and in Corollary 3.2, we find the first and second Zagreb index, forgotten topological index, redefined Zagreb index, general first Zagreb index, general Randic and symmetric division deg index with the help of generalized Zagreb index of the first metal-organic framework.

Now, we describe the division of the set of nodes Q (MOF₁ (t)) and the set of edges E (MOF₁ (t)) of MOF₁ (t) with respect to degree of nodes. We have four kinds of nodes in MOF₁ (t) that are of degree 2, degree 3, degree 4 and degree 6. Thus, we have Q₁ = {qϵ Q(MOF₁(t))| d(q) = 2}, Q₂ = {qϵ Q(MOF₁(t))| d(q) = 3}, Q₃ = {qϵ Q(MOF₁(t))| d(q) = 4}, Q₄ = {qϵ Q(MOF₁(t))| d(q) = 6}, where |Q₁| = 30t, |Q₂| = 12, |Q₃| = 12t − 6 and |Q₄| = 6t − 6. Consequently, |Q (MOF₁ (t))| = q = |Q₁| + |Q₂| + |Q₃| + |Q₄| = 48t.

We have four kinds of edges that is based on the degrees of end nodes in MOF₁ (t) that are {2, 3}, {2, 6}, {2, 4} and {4, 6}. Thus, we have E_{2,3} = {pqϵ E (MOF₁ (t))| d(p) = 2, d(q) = 3}, E_{2,6} = {pqϵ E (MOF₁ (t))| d(p) = 2, d(q) = 6}, E_{2,4} = {pqϵ E (MOF₁ (t))| d(p) = 2, d(q) = 4}, E_{4,6} = {pqϵ E (MOF₁(t))| d(p) = 4, d(q) = 6},

where |E_{2,3}| = 36, |E_{2,6}| = 24t − 24, |E_{2,4}| = 36t − 12 and |E_{4,6}| = 12t − 12. Consequently, |E(MOF₁(t))| = e = |E₁| + |E₂| + |E₃| + |E₄| = 72t − 12.

Likewise for connection numbers, we describe the partitions of the set of nodes Q (MOF₁ (t)) and the set of edges E (MOF₂ (t)) of MOF₁ (t) with respect to the addition of the degrees of neighbors at 2-distance of the nodes of MOF₁(t). So, at 2-distance pqϵ E (MOF₁ (t)), we get Tables 5 and 6.

4 Topological indices for the second metal-organic framework

In this area, we figure out the topological indices for the second metal-organic framework. In Theorem 4.1, we find the topological index that is based on the degree of the nodes that is generalized Zagreb index and in Corollary 4.2, we find the first and second Zagreb index, forgotten topological index, redefined Zagreb index, general first Zagreb index, general Randic and symmetric division deg index with the help of generalized Zagreb index for the second metal-organic framework.

Firstly, we describe the division of the nodes set Q (MOF₂ (t)) and the set of edges E (MOF₂ (t)) of MOF₂ (t) that is based on the degree of the nodes. Also each node MOF₂ (t) is of degree 2, degree 3 and degree 4. Thus, we have Q₁ = {qϵ Q(MOF₂(t))| d(q) = 2}, Q₂ = {qϵ Q(MOF₂(t))| d(q) = 3}, Q₃ = {qϵ Q(MOF₂(t))| d(q) = 4},

where |Q₁| = 12t + 18, |Q₂| = 24t − 12, and |Q₃| = 12t − 6. Consequently, |Q (MOF₂ (t))| = q = |Q₁| + |Q₂| + |Q₃| = 48t.

Further, we have three kinds of edges based on the degrees of end nodes in MOF₂ (t) namely with degrees of end nodes {2, 3}, {2, 4}, {3, 3}, {3, 4} and {4, 4}. Thus, we have E_{2,3} = {pqϵ E(MOF₂(t))| d(p) = 2, d()q = 3}, E_{2,4} = {pqϵ E(MOF₂(t))| d(p) = 2, d(q) = 4}, E_{3,3} = {pqϵ E(MOF₂(t))| d(p) = 3, d(q) = 3}, E_{3,4} = {pqϵ E(MOF₂(t))| d(p) = 3, d(q) = 4}, E_{4,4} = {pqϵ E(MOF₂(t))| d(p) = 4, d(q) = 4},

where |E_{2,3}| = 12(t + 2), |E_{2,4}| = 12(t + 1), |E_{3,3}| = 24(t − 1), |E_{3,4}| = 12(t − 1) and |E_{4,4}| = 12(t − 1). Consequently, |E(MOF₂(t))| = e = |E_{2,3}| + |E_{2,4}| + |E_{3,3}| + |E_{3,4}| + |E_{4,4}| = 72t − 12.

Likewise for connection numbers, we can define the division of the node set Q (MOF₂ (t)) and the edge set E (MOF₂ (t)) of MOF₂ (t) that based on the addition of the degrees of neighbors at 2-distance of the nodes of MOF₂ (t). So, the connection number pqϵ E (MOF₂ (t)). For details please see Tables 7 and 8.

5 Conclusions

At last, a comparison is included between the Zagreb indices and Zagreb connection indices with the help of 3D plots and their numerical values for the first and second metal-organic frameworks are presented in Figures 3a-n and Tables 9-12.

Figure 3

(a) 3D plot of M₁ (Γ) and M^c₁ (Γ) are labeled in red and blue graphs; (b) 3D plot of M₂ (Γ) and M2cΓ are labeled in gold and green graphs; (c) 3D plot of F (Γ) and F^c(Γ) are labeled in purple and silver graphs; (d) 3D plot of ReZM (Γ) and ReZM^c (Γ) are labeled in blue and red graphs; (e) 3D plot of M^∞(Γ) and M^cα(Γ) if α = 2, are labeled in golden and green graphs; (f) 3D plot of R_α (Γ) and Rαc(Γ)are labeled in silver and red graphs; (g) 3D plot of SDD (Γ) and SDD^c (Γ) are labeled in blue and mahogany graphs; (h) 3D plot of M₁ (Γ) and M1c(Γ)are labeled in blue and red graphs; (i) 3D plot of M₂ (Γ) and M2c(Γ)are labeled in sky blue and yello graphs; (j) 3D plot of F(Γ) and F^c(Γ) are labeled in golden and green graphs; (k) 3D plot of ReZM (Γ) and ReZM^c (Γ) are labeled in blue and red graphs; (l) 3D plot of M^∝(Γ) and M^cα(Γ) if α=2, are labeled in silver and green graphs; (m) 3D plot of R_α (Γ) and Rαc(Γ)are labeled in red and blue graphs; (n) 3D plot of SDD (Γ) and SDD^c (Γ) are labeled in golden and red graphs.

Now we close our discussion with conclusion given below.

• From Figures 3a-n (3D plots), it is clear that the results of Zagreb connection indices are more better than

Zagreb indices for both the metal-organic frameworks with the increasing number of layers consisting of nodes and organic ligands.

• Also, from Table 9-12, numerical calculations of all the obtained indices and connection indices for both the metal-organic frameworks show that the Zagreb connection indices are better than Zagreb indices.