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M-polynomial-based topological indices of metal-organic networks

  • Agha Kashif , Sumaira Aftab , Muhammad Javaid EMAIL logo and Hafiz Muhammad Awais
Published/Copyright: June 16, 2021
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Main Group Metal Chemistry
From the journal Main Group Metal Chemistry Volume 44 Issue 1

Abstract

Topological index (TI) is a numerical invariant that helps to understand the natural relationship of the physicochemical properties of a compound in its primary structure. George Polya introduced the idea of counting polynomials in chemical graph theory and Winer made the use of TI in chemical compounds working on the paraffin's boiling point. The literature of the topological indices and counting polynomials of different graphs has grown extremely since that time. Metal-organic network (MON) is a group of different chemical compounds that consist of metal ions and organic ligands to represent unique morphology, excellent chemical stability, large pore volume, and very high surface area. Working on structures, characteristics, and synthesis of various MONs show the importance of these networks with useful applications, such as sensing of different gases, assessment of chemicals, environmental hazard, heterogeneous catalysis, gas and energy storage devices of excellent material, conducting solids, super-capacitors and catalysis for the purification, and separation of different gases. The above-mentioned properties and physical stability of these MONs become a most discussed topic nowadays. In this paper, we calculate the M-polynomials and various TIs based on these polynomials for two different MONs. A comparison among the aforesaid topological indices is also included to represent the better one.

Keywords: M-polynomial; chemical compounds; topological indices; metal-organic network

1 Introduction

Molecular hydrogen is an odorless, tasteless, colorless, nontoxic, nonmetallic, and nominal molecule in the world with very high flammability scale. On the other hand, it is a next-generation and environmental friendly source of fuel (Petit and Bandosz, 2009). As molecular hydrogen, it is useful in different fuel cells to power engine that is very beneficial for environment. However, hydrogen gas is odorless which makes a little bit leaking identification nearly not possible for humans. The current guiding laws established by the US Energy Department, put the attention on the detecting speed of the tool which must be detect 1 per cent of volume of smell-less molecular hydrogen in air in just 60 seconds.

Koo et al. (2017) prepared a fast molecular hydrogen detecting instrument containing metal nodes and organic ligands recognized as MON with the assistance of palladium (Pd) nanowires. This instrument must sense of hydrogen leaking intensity less than one per cent in only seven seconds. Further, except sensing as well as detecting, the MON shows very useful physico-chemical properties such as changing organic ligands (Yin et al., 2015), grafting active groups (Hwang et al., 2008), post synthetic ligand, preparing composites with different substances (Ahmed and Jhung, 2014), impregnating suitable active material (Thornton et al., 2009), and ion exchange (Kim et al., 2012). Seetharaj et al. (2019) presented the relation among the temperature, solvent, architectures, pH and molar ratio of the metal-organic networks. The MON is also utilized for the separation, purification (Lin et al., 2019), precursor for the formation of different nano-structure (Yap et al., 2017) and energy storage in batteries (Xu et al., 2017). In 2019, Wasson defined the concept of linker competition with the help of MON for topological insights (Wasson et al., 2019).

Graph theory discovered the important tools in the field of chemical graph theory which is utilized to study the several kinds of organic compounds and studied their characteristics. The computed TI of the molecular graph is the numeric value that characteristics chemical reactivates, physical properties and biological activities (Gonzalez-Diaz et al., 2007) of the organic compounds such as boiling, melting, and flash points; heat of formation; surface tension, pressure, retention time in chromatographic, density, heat of evaporation, temperature, and partition coefficient (Liu et al., 2019a, 2019b). Harry Wiener studied a distance-based TI in chemical graph theory for the paraffin's boiling point (Wiener, 1947). The most important TIs are degree-based TIs that are derived with the help of degree of nodes of the molecular structure and that is also mentioned in the survey regarding topological indices. In 1972, Gutman and Trinajstić described a degree-based descriptor famous as the first and second Zagreb indices to calculate the total π-electron energy of the chemical compounds (Gutman and Trinajstić, 1972).

Furthermore, topological indices show a vital role in the field of quantitative structures activity relationships or quantitative structures property relationships to describe any structure with a chemical and biological activity or property. The aforesaid relation is denoted as P = f(M) and P is a property or activity and M is a molecular structure (Devillers et al., 1997). The topological indices of various structures such as carbon nanotubes, oxide, icosahedral honeycomb, hexagonal, octahedral, benzenoids, silicate, titania nanotube, hexagonal, and fullerenes are discussed in Javaid et al. (2016). For further details see Chu et al. (2020) and Zhang et al. (2019).

In this paper, we compute the M-polynomials for two different metal-organic networks MON1(n) and MON2(n) with increasing layers of both organic ligands and metal vertices and with the assistance of these M-polynomials – the various TIs such as first and second Zagreb indices, second modified Zagreb index, general and reciprocal general Randić indices, symmetric division deg, inverse sum, harmonic, and the augmented Zagreb indices are computed. In the end, a comparison among the obtained topological indices with the assistance of graphical presentation is also involved. The rest of the paper is settled as: Section 2 includes the different techniques and definitions that are utilized in various outcomes. Sections 3 and 4 have the major computations in order to M-polynomials that are used in certain TIs related to MON1(n) and MON2(n) and Section 5 contains the comparison and conclusions.

2 Preliminaries

A molecular graph 𝔾 = (V(𝔾), E(𝔾)), V(𝔾) = {x1, x2, x3 ..., xn} and E(𝔾) are nodes (metals or organic ligands) and edge-set (bonds among the different atoms) of 𝔾, respectively. The |V(𝔾)| = v and |E(𝔾)| = e is the order and size of 𝔾, respectively. The number of edges which are incident on a vertex is known as degree of (ϱ(x)). In a simple and connected graph, a path is occurring within two nodes and the distance within two nodes x and y, represented as ϱ(x, y), in a graph 𝔾. In the current discussion, a graph is simple and connected, having no loops and multiple edges.

2.1 First and second Zagreb indices

Let 𝔾 be a molecular graph, then its first and second Zagreb indices are:

M1(𝔾)=∑xy∈E (𝔾)[ϱ(x)+ϱ(y)],M2(𝔾)=∑xy∈E (𝔾)[ϱ(x)×ϱ(y)].

2.2 General Randić index

If R is the set of real numbers and 𝔾 be a connected graph, so the general Randić index is given as:

Rα (𝔾)=∑xy∈E (𝔾)[ϱ(x)×ϱ(y)]∝.

2.3 Symmetric division deg index

For a molecular graph 𝔾, the symmetric division deg index is:

SDD(𝔾)=∑xy∈E (𝔾)[min (ϱ(x),ϱ(y))max (ϱ(x),ϱ(y)+max (ϱ(x),ϱ(y))min (ϱ(x),ϱ(y)]

2.4 Harmonic index

For a molecular graph 𝔾, the harmonic index is:

H(𝔾)=∑xy∈E (𝔾)[2ϱ(x)+ϱ(y)].

2.5 Inverse sum index

For a molecular graph 𝔾, the inverse sum index is:

IS(𝔾)=∑xy∈E (𝔾)[ϱ(x)ϱ(y)ϱ(x)+ϱ(y)].

2.6 Augmented Zagreb index

For a molecular graph 𝔾, the augmented Zagreb index is:

AZI(𝔾)=∑xy∈E (𝔾)[ϱ(x) ϱ(y)ϱ(x)+ϱ(y)−2]3.

2.7 M-polynomial

Let 𝔾 be a molecular graph and mi,j(𝔾); i, j ≥ 1 be the number of edges e = xy of 𝔾 in such a way {ϱ(x), ϱ(y) } = {i, j}. The M-polynomial of G is:

M(𝔾,a,b)=∑i≤j[mi,j(𝔾)aibj).

In Tables 1 and 2, the relation between the above TIs and M-polynomial is defined.

Table 1

Relation between M-polynomial and TIs

Indices f(a,b) Derivation from M (𝔾, a,b)
M1 a + b (Da + Db) (M (𝔾, a, b)) | a=1=b
M2 ab (Da Db) (M (𝔾, a, b)) | a=1=b
MM2 1ab (Sa Sb) (M (𝔾, a, b)) |a=1=b
R∝ (ab)∝, ∝ ∈ N (D∝a D∝b) (M (𝔾, a, b)) |a=1=b
RR∝ 1(ab)∝, (S∝a S∝b) (M (𝔾, a, b)) |a=1=b
SDD (a2+b2ab) (Da Sb + Db Sa) (M (𝔾, a, b)) |a=1=b
Table 2

Relation between M-polynomial and TIs

Indices f(a, b) Derivation from M (𝔾, a, b)
H 2a+b 2Sa J(M (𝔾, a, b)) |a=1
IS aba+b Sa Q2 J Da Db (M (𝔾, a, b)) |a=1=b
AZI (aba+b−2)3 S3a J D3a D3b(M (𝔾, a, b)) |a=1=b

In Table 1, MM2 is the second modified Zagreb index, RR∝ is reciprocal general Randić index, Da=φ(f(a,b))φ(a) , Db=φ(f(a,b))φ(b) , Sa=∫0af(t,b))t , Sb=∫0bf(a,t))tdt . Also J(f(a, b)) = (f(a, a)), and Q∝ (f (a, b)) = a∝ (f(a,b)), where ∝ ≠ 0. For further detail discussion, we refer to Javaid and Jung (2017).

In this section, we describe the MON which is a composition of metal nodes as well as organic ligands as presented in Figure 1. The smaller nodes are organic ligands and the bigger nodes are metals that are zinc-based (Hong et al., 2020). In the metal-organic network, all the edges are bonds among the different organic ligands and the metals nodes.

Figure 1 Basic metal-organic network.
Figure 1

Basic metal-organic network.

Now we developed the metal-organic networks as shown in Figures 2a,b. The MON1(n) is created by introducing the new bonds between the bigger nodes (metals) of the two primary metal-organic networks in such a way two bigger nodes of the upper layer of a primary metal-organic network are linked with a node of the lower layer of the basic MON1(n). The MON1(n) is shown in Figure 2a, for dimension n = 2. Also, the MON2(n) by introducing the new bonds among the smaller nodes of the two basic metal-organic networks in such a way two smaller nodes of an upper layer of a primary metal-organic network are linked with a smaller node of a lower layer of the second basic metal-organic network. The MON2(n) is shown in Figure 2b, where n = 2. For the aforesaid metal-organic networks, we have |V(MON1(n))| = |V(MON2(n))| = 48n and |E(MON1(n))| = |E(MON2(n))| = 72n – 12, where n ≥ 2. For more, see Awais et al. (2019a, 2019b, 2020a, 2020b).

Figure 2 (a) First metal-organic network (MON1 (n) for n = 2). (b) Second metal-organic network (MON2 (n) for n = 2).
Figure 2

(a) First metal-organic network (MON1 (n) for n = 2). (b) Second metal-organic network (MON2 (n) for n = 2).

3 Results for first metal-organic network MON1(n)

In this section, we computed the different results regarding aforesaid TIs on the MON1(n). Before to develop the core computations, we discuss the node-set and edge-set of MON1(n) with the help of degrees of nodes.

We have four different kinds of nodes in MON1(n) i.e. 2, 3, 4, and 6. Consequently:

V1={x∈ V(MON1(n))| ϱ(x)=2},V2={x∈ V(MON1(n))| ϱ(x)=3},V3={x∈ V(MON1(n))| ϱ(x)=4},V4={x∈ V(MON1(n))| ϱ(x)=6},

where |V1| = 30n, |V2| = 12, |V3| = 12n – 6, and |V4 | = 6n – 6. Consequently:

|V (MON1 (n))|=u=|V1|+|V2|+|V3|+|V4|=48n.

Therefore, we have four different edges that are degree-based of end nodes in MON1(n) that are {2, 3}, {2, 6}, {2, 4}, and {4, 6}. Thus, we have:

E1=E{2,3}={xy ∈ E (MON1(n))| ϱ(x)=2,ϱ(y)=3},E2=E{2,6}={xy ∈ E (MON1(n))| ϱ(x)=2,ϱ(y)=6},E3=E{2,4}={xy ∈ E (MON1(n))| ϱ(x)=2,ϱ(y)=4},E4=E{4,6}={xy ∈ E (MON1(n))| ϱ(x)=4,ϱ(y)=6},

where |E{2,3}| = 36, |E{2,6}| = 24n – 24, |E{2,4}| = 36n – 12, and |E{4,6}| = 12n – 12. Therefore, |E(MON1(n)) | = e = |E1| + |E2| + |E3| + |E4| = 72n – 12.

We also define the node-set partition V (MON1 (n)) and edge-set E (MON2 (n)) of MON1 (n) with addition of degrees of neighbor nodes. So, for xy ∈ E (MON1 (n)), we got Table 3.

Table 3

Partitions of edges of MON1(n)

E{ ϱ (x), ϱ (y)} E{2,3} E{2,6} E{2,4} E{4,6}
|E{ ϱ (x), ϱ (y)}| 36 24(n − 1) 12(3n − 1) 12(n − 1)

Theorem 3.1

Let 𝔾 ≅ MON1(n) be the first metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(𝔾,a,b)=36a2b3+24(n−1)a2b6   + 12(3n−1)a2b4+12(n−1)a4b6

Proof

By using the formulae and Table 1, we computed the results as follows:

M(𝔾,a,b)=∑i≤j(mi,j𝔾aibj)=∑2≤3(m2,3𝔾a2b3)+∑2≤6(m2,6𝔾a2b6)    +∑2≤4(m2,4𝔾a2b4)+∑4≤6(m4,6𝔾a4b6)=|E1|a2b3+|E2|a2b6+|E3|a2b4+|E4|a4b6=36a2b3+24(n−1)a2b6   + 12(3n−1)a2b4+12(n−1)a4b6

Theorem 3.2

Let 𝔾 ≅ MON1(n) be the first metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(G,a,b)=36a2b3+24(n−1)a2b6   + 12(3n−1)a2b4+12(n−1)a4b6

Then, the first Zagreb index (M1(𝔾)), the second Zagreb index (M2(𝔾)), the second modified Zagreb index (MM2(𝔾)), general Randić index (R∝ (𝔾)), and reciprocal General Randić index (RR∝ (𝔾)), where α ∈ N and symmetric division degree index (SSD(𝔾)) computed from the M-polynomial, are as follows:

  1. M1 (𝔾) = 528n − 204,

  2. M2 (𝔾) = 864n − 456,

  3. MM2 (𝔾) = 7n + 2,

  4. Rα (𝔾) = (6)α36 + (12)α24(n − 1) + (8)α12 (3n − 1) + (24)α12(n − 1),

  5. RRα(𝔾)=36(6)∝+24(12)∝(n−1)+12(8)∝(3n−1)+12(24)∝(n−1), ,

  6. SDD(𝔾) = 196n − 58.

Proof

Let f(a, b) = M(𝔾, a, b) be the M-polynomial of the first metal-organic network. We have:

f(a,b)=36a2b3+24(n−1)a2b6+12(3n−1)a2b4   + 12(n−1)a4b6

The needed partial derivatives and integrals are computed as:

D a ( f ( a , b ) ) = 72 ab 3 + 48 ( n − 1 ) ab 6 + 24 ( 3 n − 1 ) ab 4 +   48 ( n − 1 ) a 3 b 6 D b ( f ( a , b ) ) = 108 a 2 b 2 + 144 ( n − 1 ) a 2 b 5 +   48 ( 3 n − 1 ) a 2 b 3 + 72 ( n − 1 ) a 4 b 5 , D a ( D b ( f ( a , b ) ) ) = 216 ab 2 + 288 ( n − 1 ) ab 5 + 96 ( 3 n − 1 ) ab 3 +   288 ( n − 1 ) a 3 b 5 S a ( f ( a , b ) ) = 18 a 2 b 3 + 12 ( n − 1 ) a 2 b 6 +   6 ( 3 n − 1 ) a 2 b 4 + 3 ( n − 1 ) a 4 b 6 S b ( f ( a , b ) ) = 12 a 2 b 3 + 4 ( n − 1 ) a 2 b 6 + 3 ( 3 n − 1 ) a 2 b 4 +   2 ( n − 1 ) a 4 b 6 S a ( S b ( f ( a , b ) ) ) = 6 a 2 b 3 + 2 ( n − 1 ) a 2 b 6 +   3 2 ( 3 n − 1 ) a 2 b 4 + ( n − 1 ) a 4 b 6 , D b S a ( f ( a , b ) ) = 54 a 2 b 2 + 72 ( n − 1 ) a 2 b 5 +   24 ( 3 n − 1 ) a 2 b 3 + 18 ( n − 1 ) a 4 b 5 D b S b ( f ( a , b ) ) = 24 ab 3 + 8 ( n − 1 ) ab 6 + 6 ( 3 n − 1 ) ab 4 +   8 ( n − 1 ) a 3 b 6 ( D a ∝ D ∝ b ) ( f ( a , b ) ) ) = ( 6 ) ∝   36 ab 2 + ( 12 )   ∝ 24 ( n − 1 ) ab 5 +   ( 8 )   ∝ 12 ( 3 n − 1 ) ab 3 +   ( 24 )   ∝ 12 ( n − 1 ) a 3 b 5 ( S ∝ a S ∝ b ) ( f ( a , b ) ) ) = 36 ( 6 ) ∝ a 2 b 3 + 24 ( 12 ) ∝ ( n − 1 ) a 2 b 6 +   12 ( 8 ) ∝ ( 3 n − 1 ) a 2 b 4 + 12 ( 24 ) ∝ ( n − 1 )   a 4 b 6 ,

Now, we obtained:

D a ( f ( a , b ) ) |     a = 1 = b = 72 + 48 ( n − 1 ) + 24 ( 3 n − 1 ) +   48 ( n − 1 ) , D b ( f ( a , b ) ) |     a = 1 = b = 108 + 144 ( n − 1 ) +   48 ( 3 n − 1 ) + 72 ( n − 1 ) , D a ( D b ( f ( a , b ) ) ) |     a = 1 = b = 216 + 288 ( n − 1 ) +   96 ( 3 n − 1 ) + 288 ( n − 1 ) , S a ( f ( a , b ) ) |     a = 1 = b = 18 + 12 ( n − 1 ) + 6 ( 3 n − 1 ) +   3 ( n − 1 ) , S b ( f ( a , b ) ) |     a = 1 = b = 12 + 4 ( n − 1 ) + 3 ( 3 n − 1 ) +   2 ( n − 1 ) , S a ( S b ( f ( a , b ) ) ) |     a = 1 = b = 6 + 2 ( n − 1 ) + 3 2 ( 3 n − 1 ) +   2 4 ( n − 1 ) , D b S a ( f ( a , b ) ) |     a = 1 = b = 54 + 72 ( n − 1 ) + 24 ( 3 n − 1 ) +   18 ( n − 1 ) , D a S b ( f ( a , b ) ) |     a = 1 = b = 24 + 8 ( n − 1 ) + 6 ( 3 n − 1 ) +   8 ( n − 1 ) , ( D a ∝ S ∝ b ) ( f ( a , b ) ) |     a = 1 = b = ( 6 )   ∝ 36 + ( 12 )   ∝ 24 ( n − 1 ) +   ( 8 )   ∝ 12 ( 3 n − 1 ) +   ( 24 )   ∝ 12 ( n − 1 ) , ( S ∝ a S ∝ b ) ( f ( a , b ) ) |     a = 1 = b = 36 ( 6 ) ∝ + 24 ( 12 ) ∝ ( n − 1 ) + 12 ( 8 ) ∝ ( 3 n − 1 ) +   12 ( 24 ) ∝ ( n − 1 ) ,

Consequently:

  1. M1(𝔾)=(Da+Db)(f(a,b))|  a=1=b=Da(f(𝔾,a,b))|  a=1=b+Db(f(𝔾,a,b))|  a=1=b=528n−204

  2. M2(𝔾)=(Da Db)(f(a,b))|  a=1=b=Da(Db(f(𝔾,a,b)))|  a=1=b=864n−456

  3. MM2(𝔾)=(Sa Sb)(f(a,b))|  a=1=b=Sa(Sb(f(𝔾,a,b)))|  a=1=b=6+2(n−1)+ 32(3n−1)+24(n−1)=7n+2

  4. R∝(𝔾)=(D∝a D∝b)(f(a,b))|  a=1=b=(6) ∝36+ (12) ∝24(n−1)+(8) ∝12(3n−1)+(24) ∝12(n−1)

  5. RR∝(𝔾)=(D∝a D∝b)(f(a,b))|  a=1=b=36(6)∝+ 24(12)∝(n−1)+12(8)∝(3n−1)+12(24)∝(n−1),

  6. SDD(𝔾)=(DaSb+DbSa)(f(a,b))|  a=1=b=(DaSb)(f(a,b))|  a=1=b+(DbSa)(f(a,b))|  a=1=b=(Da(Sb(f(a,b)))|  a=1=b+ (Db(Sa(f(a,b)))|  a=1=b=196n−58⊳

Theorem 3.3

Let 𝔾 ≅ MON1(n) be the first metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(𝔾,a,b)=36a2b3+24(n−1)a2b6   + 12(3n−1)a2b4+12(n−1)a4b6

Then, harmonic H(𝔾), inverse sum IS(𝔾), and augmented Zagreb indices AZI(𝔾) are:

  1. H(𝔾)=(1025)n+2,

  2. IS(𝔾)=(5645)n−(1885),

  3. AZI(𝔾) = 804n − 324.

Proof

Let f(a, b) = M(𝔾, a, b) be the M-polynomial of the first metal-organic network. We have:

f(a,b)=36a2b3+24(n−1)a2b6   + 12(3n−1)a2b4+12(n−1)a4b6

The needed partial derivatives and integrals are computed as:

J ( f ( a , b ) ) = 36 a 5 + 24 ( n − 1 ) a 8 +   12 ( 3 n − 1 ) a 6 + 12 ( n − 1 ) a 10 S a ( J ( f ( a , b ) ) ) = 36 5 a 5 + 3 ( n − 1 ) a 8 +   2 ( 3 n − 1 ) a 6 + 6 5 ( n − 1 ) a 10 J ( D a ( D b ( f ( a , b ) ) ) ) = 216 a 3 + 288 ( n − 1 ) a 6 +   96 ( 3 n − 1 ) a 4 + 288 ( n − 1 ) a 8 Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) = 216 a 5 + 288 ( n − 1 ) a 8 +   96 ( 3 n − 1 ) a 6 + 288 ( n − 1 ) a 10 S a Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) = 216 5 a 5 + 36 ( n − 1 ) a 8 +   16 ( 3 n − 1 ) a 6 +   144 5 ( n − 1 ) a 10 D 3 a   D 3 b   ( f ( a , b ) ) = 6 3 ( 36 ) ab 2 + 12 3 ( 24 ) ( n − 1 ) ab 5 +   8 3 ( 12 ) ( 3 n − 1 ) ab 3 +   24 3 ( 12 ) ( n − 1 ) a 3 b 5 , J   D 3 a   D 3 b   ( f ( a , b ) ) = 6 3 ( 36 ) a 3 + 12 3 ( 24 ) ( n − 1 ) a 6 +   8 3 ( 12 ) ( 3 n − 1 ) a 4 +   24 3 ( 12 ) ( n − 1 ) a 8 , S a 3   J   D 3 a   D 3 b   ( f ( a , b ) ) = 2 3 ( 36 ) a 3 + 2 3 ( 24 ) ( n − 1 ) a 6 +   2 3 ( 12 ) ( 3 n − 1 ) a 4 +   3 3 ( 12 ) ( n − 1 ) a 8 .

Now we obtained:

S a ( J ( f ( a , b ) ) ) |     a = 1 = b = 36 5 + 3 ( n − 1 ) +   2 ( 3 n − 1 ) +   6 5 ( n − 1 ) S a Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) |     a = 1 = b = 216 5 + 36 ( n − 1 ) +   16 ( 3 n − 1 ) +   144 5 ( n − 1 ) S 3 a   JD 3 a D 3 b   ( f ( a , b ) ) |     a = 1 = b = ( 6 3 ) 3 ( 36 ) +   ( 12 6 ) 3 ( 24 ) ( n − 1 ) +   ( 8 4 ) 3 ( 12 ) ( 3 n − 1 ) +   ( 24 8 ) 3 ( 12 ) ( n − 1 ) .

Consequently:

  1. H(𝔾)=2 Sa(J(f(a,b)))|  a=1=b=2[365+3(n−1))+ 2(3n−1)+65(n−1)]=(1025)n+2

  2. IS(𝔾)=SaQ2(J(Da(Db)(f(a,b)))))|  a=1=b=2165+ 36(n−1)+16(3n−1)+1445(n−1)=(5645)n−1885

  3. AZI(𝔾)=S3a JDa3 Db3(f(a,b))|  a=1=b=(63)3(36)+ (126)3 (24)(n−1)+(84)3 (12)(3n−1)+ (248)3 (12)(n−1)=804n−324⊳

4 Results for second metal-organic network

In this section, we show the consequences of TIs for the MON2(n). So, we define the partitions of node-set and edge-set of MON2(n) with respect to degree of nodes. We have three different kinds of nodes in MON2(n) with respect to degree of 2, 3, and 4. Thus, we have:

V1={x∈ V(MON2(n))| ϱ (x)=2},V2={x∈ V(MON2(n))| ϱ (x)=3},V3={x∈ V(MON2(n))| ϱ (x)=4},

where |V1| = 12n + 18, |V2| = 24n – 12, and |V3| = 12n – 6. Consequently:

|V (MON2 (n))|=u=|V1|+|V2|+|V3|=48n.

Additionally, we have different kinds of edges that is based on the degree of end nodes in MON2(n) that are {2, 3}, {2, 4}, {3, 3}, {3, 4}, and {4, 4}.

Hence, we have:

E1=E{2,3}={xy∈ E(MON2(n))| ϱ (x)=2,ϱ (y)=3},E2=E{2,4}={xy∈ E(MON2(n))| ϱ (x)=2,ϱ (y)=4},E3=E{3,3}={xy∈ E(MON2(n))| ϱ (x)=3,ϱ (y)=3},E4=E{3,4}={xy∈ E(MON2(n))| ϱ (x)=3,ϱ (y)=4},E5=E{4,4}={xy∈ E(MON2(n))| ϱ (x)=4,ϱ (y)=4},

where |E{2,3}| = 12(n + 2), |E{2,4}| = 12(n + 1), |E{3,3}| = 24(n – 1), |E{3,4}| = 12(n – 1), and |E{4,4}| = 12(n – 1). Consequently, |E(MON2(n)) | = e = |E{2,3}| + |E{2,4}| + |E{3,3}| + |E{3,4}| + |E{4,4}| = 72n – 12.

Therefore, we define the partition of node-set V(MON2(n)) and edge-set E(MON2(n)) of MON2(n) with respect to addition of degree of neighborhood nodes.

Thus, for xy∈ E (MON2 (n)), we get Table 4.

Table 4

Partitions of edges of MON2(n)

E{ϱ x), ϱ (y)} E{2,3} E{2,4} E{3,3} E{3,4} E{4,4}
|E{ϱ (x), ϱ (y)}| 12(n+2) 12(n + 1) 24(n − 1) 12(n − 1) 12(n−1)

Theorem 4.1

Let 𝔾 ≅ MON2(n) be the second metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(𝔾,a,b)=12(n+2)a2b3+12(n+1)a2b4   + 24(n−1)a3b3+12(n−1)a3b4   + 12(n−1)a4b4

Proof

By using the formulae and Table 2, we computed the results as follows:

M ( 𝔾 , a , b ) = ∑ i ≤ j ( m i , j 𝔾 a i b j ) = ∑ 2 ≤ 3 ( m 2 , 3 𝔾 a 2 b 3 ) + ∑ 2 ≤ 4 ( m 2 , 4 𝔾 a 2 b 4 )       +   ∑ 3 ≤ 3 ( m 3 , 3 𝔾 a 3 b 3 ) + ∑ 3 ≤ 4 ( m 3 , 4 𝔾 a 3 b 4 )       +   ∑ 4 ≤ 4 ( m 4 , 4 𝔾 a 4 b 4 ) = | E 1 | a 2 b 3 + | E 2 | a 2 b 4 + | E 3 | a 3 b 3       +   | E 4 | a 3 b 4 + | E 5 | a 4 b 4 = 12 ( n + 2 ) a 2 b 3 + 12 ( n − 1 ) a 2 b 4       +   24 ( n − 1 ) a 3 b 3 + 12 ( n − 1 ) a 3 b 4       +   12 ( n − 1 ) a 4 b 4

Theorem 4.2

Let 𝔾 ≅ MON2(n) be the second metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(𝔾,a,b)=12(n+2)a2b3+12(n+1)a2b4   + 24(n−1)a3b3+12(n−1)a3b4   + 12(n−1)a4b4

Then, the first Zagreb index (M1(𝔾)), the second Zagreb index (M2(𝔾)), the second modified Zagreb index (MM2(𝔾)), general Randić index (R∝(𝔾)), and reciprocal General Randić index (RR∝(𝔾)), where α ∈ N and symmetric division degree index (SSD(𝔾)) computed from the M-polynomial, are as follows:

  1. M1 (𝔾) = 456n − 132,

  2. M2 (𝔾) = 720n − 312,

  3. MM2(𝔾)=(9512)n+132, ,

  4. Rα (𝔾) = (6)α12(n + 2) + (8)α12(n + 1) + (9)α24(n − 1) + (12)α12(n − 1) + (16)α12(n − 1)

  5. RRα(𝔾)=126∝(n+2)+12(8)∝(n+1)+24(9)∝(n−1)+ 12(12)∝(n−1)+12(16)∝(n−1),

  6. SDD(𝔾) = 153n − 15.

Proof

Let f (a, b) = M (𝔾, a, b) be the M-polynomial of the second metal-organic network. We have:

f(a,b)=12(n+2)a2b3+12(n+1)a2b4   + 24(n−1)a3b3+12(n−1)a3b4   + 12(n−1)a4b4

The needed partial derivatives and integrals are computed as:

D a ( f ( a , b ) ) = 24 ( n + 2 ) ab 3 + 24 ( n + 1 ) ab 4 +   72 ( n − 1 ) a 2 b 3 + 36 ( n − 1 ) a 2 b 4 +   48 ( n − 1 ) a 3 b 4 D b ( f ( a , b ) ) = 36 ( n + 2 ) a 2 b 2 + 48 ( n + 1 ) a 2 b 3 +   72 ( n − 1 ) a 3 b 2 + 48 ( n − 1 ) a 3 b 3 +   48 ( n − 1 ) a 4 b 3 , D a ( D b ( f ( a , b ) ) ) = 72 ( n + 2 ) ab 2 + 96 ( n + 1 ) ab 3 +   216 ( n − 1 ) a 2 b 2 + 144 ( n − 1 ) a 2 b 3 +   192 ( n − 1 ) a 3 b 3 , S a ( f ( a , b ) ) = 6 ( n + 2 ) a 2 b 3 + 6 ( n + 1 ) a 2 b 4 +   8 ( n − 1 ) a 3 b 3 + 4 ( n − 1 ) a 3 b 4 +   3 ( n − 1 ) a 4 b 4 , S b ( f ( a , b ) ) = 4 ( n + 2 ) a 2 b 3 + 3 ( n + 1 ) a 2 b 4 +   8 ( n − 1 ) a 3 b 3 + 3 ( n − 1 ) a 3 b 4 +   3 ( n − 1 ) a 3 b 4 , S a ( S b ( f ( a , b ) ) ) = 2 ( n + 2 ) a 2 b 3 + 3 2 ( n + 1 ) a 2 b 4 +   8 3 ( n − 1 ) a 3 b 3 + ( n − 1 ) a 3 b 4 +   3 4 ( n − 1 ) a 4 b 4 , D b S a ( f ( a , b ) ) = 18 ( n + 2 ) a 2 b 2 + 24 ( n + 1 ) ab 4 +   24 ( n − 1 ) a 3 b 2 + 16 ( n − 1 ) a 3 b 3 +   12 ( n − 1 ) a 4 b 3 , D a S b ( f ( a , b ) ) = 8 ( n + 2 ) a 2 b 3 + 6 ( n + 1 ) ab 4 +   24 ( n − 1 ) a 2 b 3 + 9 ( n − 1 ) a 2 b 4 +   12 ( n − 1 ) a 3 b 4 , ( D ∝ a   D ∝ b )   ( f ( a , b ) ) ) = ( 6 ) ∝   12 ( n + 2 ) ab 2 +   ( 8 )   ∝ 12 ( n + 1 ) ab 3 +   ( 9 )   ∝ 24 ( n − 1 ) a 2 b 2 +   ( 12 )   ∝ 12 ( n − 1 ) a 2 b 3 +   ( 16 )   ∝ 12 ( n − 1 ) a 3 b 3 , ( S ∝ a   S ∝ b )   ( f ( a , b ) ) ) = 12 ( 6 ) ∝ ( n + 2 ) a 2 b 3 + 12 ( 8 ) ∝ ( n + 1 )   a 2 b 4 +   24 ( 9 ) ∝ ( n − 1 ) a 3 b 3 + 12 ( 12 ) ∝ ( n − 1 )   a 3 b 4 +   12 ( 16 ) ∝ ( n − 1 )   a 4 b 4 ,

Now, we obtained:

D a ( f ( a , b ) ) |     a = 1 = b = 24 ( n + 2 ) + 24 ( n + 1 ) +   72 ( n − 1 ) + 36 ( n − 1 ) + 48 ( n − 1 ) , D b ( f ( a , b ) ) |     a = 1 = b = 36 ( n + 2 ) + 48 ( n + 1 ) +   72 ( n − 1 ) + 48 ( n − 1 ) + 48 ( n − 1 ) , D a ( D b ( f ( a , b ) ) ) |     a = 1 = b = 72 ( n + 2 ) + 96 ( n + 1 ) +   216 ( n − 1 ) + 144 ( n − 1 ) +   192 ( n − 1 ) , S a ( f ( a , b ) ) |     a = 1 = b = 6 ( n + 2 ) + 6 ( n + 1 ) +   8 ( n − 1 ) + 4 ( n − 1 ) + 3 ( n − 1 ) , S b ( f ( a , b ) ) |     a = 1 = b = 4 ( n + 2 ) + 3 ( n + 1 ) +   8 ( n − 1 ) + 3 ( n − 1 ) + 3 ( n − 1 ) , S a ( S b ( f ( a , b ) ) ) |     a = 1 = b = 2 ( n + 2 ) + 3 2 ( n + 1 ) +   8 3 ( n − 1 ) + ( n − 1 ) +   3 4 ( n − 1 ) , D b S a ( f ( a , b ) ) |     a = 1 = b = 18 ( n + 2 ) + 24 ( n + 1 ) +   24 ( n − 1 ) + 16 ( n − 1 ) +   12 ( n − 1 ) , D a S b ( f ( a , b ) ) |     a = 1 = b = 8 ( n + 2 ) + 6 ( n + 1 ) +   24 ( n − 1 ) + 9 ( n − 1 ) +   12 ( n − 1 ) , ( D ∝ a S ∝ b ) ( f ( a , b ) ) |     a = 1 = b = ( 6 ) ∝   12 ( n + 2 ) +   ( 8 )   ∝ 12 ( n + 1 ) +   ( 9 )   ∝ 24 ( n − 1 ) +   ( 12 )   ∝ 12 ( n − 1 ) +   ( 16 )   ∝ 12 ( n − 1 ) , ( S ∝ a S ∝ b ) ( f ( a , b ) ) |     a = 1 = b = 12 ( 6 ) ∝ ( n + 2 ) + 12 ( 8 ) ∝ ( n + 1 ) +   24 ( 9 ) ∝ ( n − 1 ) + 12 ( 12 ) ∝ ( n − 1 ) +   12 ( 16 ) ∝ ( n − 1 ) ,

Consequently:

  1. M1(𝔾)=(Da+Db(f(a,b))|  a=1=b=Da(f(𝔾,a,b))|  a=1=b+Db(f(𝔾,a,b))|  a=1=b=456n−132

  2. M2(𝔾)=(Da Db)(f(a,b))|=(Da (Db(f(𝔾,a,b)))|  a=1=b=720n−312

  3. MM2(𝔾)=(Sa Sb) (f(a,b))|  a=1=b=Sa(Sb(f(𝔾),a,b))|  a=1=b=2(n+2)+ 32(n+1)+83(n−1)+(n−1)+34(n−1)=(9512)n+1312,

  4. R∝(𝔾)=(D∝a D∝b) (f(a,b))|  a=1=b=(6) ∝12(n+2)+(8) ∝12(n+1)+ (9) ∝24(n−1)+(12) ∝12(n−1)+ (16) ∝12(n−1),

  5. RR∝(𝔾)=(S∝a S∝b) (f(a,b))|  a=1=b=12(6)∝(n+2)+12(8)∝(n+1)+24(9)∝(n−1)+ 12(12)∝(n−1)+12(16)∝(n−1),

  6. SDD(𝔾)=(Da Sb+D Sa) (f(a,b))|  a=1=b=(Da Sb)(f(a,b))|  a=1=b+(Db Sa)(f(a,b))|  a=1=b=(Da(Sb(f(a,b)))|  a=1=b+(Db(Sa(f(a,b)))|  a=1=b=153n−15⊳

Theorem 4.3

Let 𝔾 ≅ MON2(n) be the second metal-organic network, where n ≥ 2. The M-polynomial of 𝔾 is:

M(𝔾,a,b)=12(n+2)a2b3+12(n+1)a2b4   + 24(n−1)a3b3+12(n−1)a3b4   + 12(n−1)a4b4

Then, harmonic H(𝔾), inverse sum IS(𝔾), and augmented Zagreb indices AZI(𝔾) are:

  1. H(𝔾)=2Sa(J(f(a,b)))|  a=1=b=(81335)n−2935,

  2. IS(𝔾)=SaQ2(J(Da(Db(f(a,b)))))|  a=1=b=(388435)n−(125235),

  3. AZI(𝔾)=S3a JD3a D3b(f(a,b))|  a=1=b=12(8)(n+2)+12(8)(n+1)+ 24(72964)(n−1)+12(1728125)(n−1)+12(51227)(n−1)=(77293679000)n−(34093679000).

Proof

Let f (a, b) = M (𝔾, a, b) be the M-polynomial of the second metal-organic network. We have:

f(a,b)=12(n+2)a2b3+12(n+1)a2b4   + 24(n−1)a3b3+12(n−1)a3b4   + 12(n−1)a4b4.

The needed partial derivatives and integrals are computed as:

J ( f ( a , b ) ) = 12 ( n + 2 ) a 5 + 12 ( n + 1 ) a 6 +   24 ( n − 1 ) a 6 + 12 ( n − 1 ) a 7 +   12 ( n − 1 ) a 8 , S a ( J ( f ( a , b ) ) ) = 12 5 ( n + 2 ) a 5 + 2 ( n + 1 ) a 6 +   4 ( n − 1 ) a 6 + 12 7 ( n − 1 ) a 7 +   3 2 ( n − 1 ) a 8 , J ( D a ( D b ( f ( a , b ) ) ) ) = 72 ( n + 2 ) a 3 + 96 ( n + 1 ) a 4 +   216 ( n − 1 ) a 4 +   144 ( n − 1 ) a 5 +   192 ( n − 1 ) a 6 Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) = 72 ( n + 2 ) a 3 + 96 ( n + 1 ) a 6 +   216 ( n − 1 ) a 6 +   144 ( n − 1 ) a 7 +   192 ( n − 1 ) a 8 , S a Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) = 72 5 ( n + 2 ) a 5 + 16 ( n − 1 ) a 6 +   36 ( n − 1 ) a 6 + 144 7 ( n − 1 ) a 7 +   24 ( n − 1 ) a 8 , D 3 a ( D 3 b ( f ( a , b ) ) = 6 3 ( 12 ) ( n + 2 ) ab 2 +   8 3 ( 12 ) ( n + 1 ) ab 3 +   9 3 ( 24 ) ( n − 1 ) a 2 b 2 +   12 3 ( 12 ) ( n − 1 ) a 2 b 3 +   16 3 ( 12 ) ( n − 1 ) a 3 b 3 , J   D 3 a ( D 3 b ( f ( a , b ) ) ) = 6 3 ( 12 ) ( n + 2 ) a 3 +   8 3 ( 12 ) ( n + 1 ) a 4 +   9 3 ( 24 ) ( n − 1 ) a 4 +   12 3 ( 12 ) ( n − 1 ) a 5 +   16 3 ( 12 ) ( n − 1 ) a 6 , S a 3   J   D 3 a ( D 3 b ( f ( a , b ) ) ) = ( 6 3 ) 3 ( 12 ) ( n + 2 ) a 3 +   ( 8 3 ) 3 12 ( n + 1 ) a 4 +   ( 9 4 ) 3 24 ( n − 1 ) a 4 +   ( 12 5 ) 3 12 ( n − 1 ) a 5 +   ( 16 6 ) 3 12 ( n − 1 ) a 6 .

Now we obtained:

S a ( J ( f ( a , b ) ) ) |     a = 1 = b = 12 5 ( n + 2 ) + 2 ( n + 1 ) +   4 ( n − 1 ) + ( 12 7 ) ( n − 1 ) +   ( 3 2 ) ( n − 1 ) , S a Q 2 ( J ( D a ( D b ( f ( a , b ) ) ) ) ) |     a = 1 = b = 72 5 ( n + 2 ) + 16 ( n − 1 ) +   36 ( n − 1 ) + 144 7 ( n − 1 ) +   24 ( n − 1 ) , S 3 a   J ( D 3 a   D 3 b ( f ( a , b ) ) ) |     a = 1 = b = 2 3 ( 12 ) ( n + 2 ) +   ( 2 ) 3 12 ( n + 1 ) +   ( 9 4 ) 3 24 ( n − 1 ) +   ( 12 5 ) 3 12 ( n − 1 ) +   ( 8 3 ) 3 12 ( n − 1 ) .

Consequently:

  1. H(𝔾)=(81335)n−2935,

  2. IS(𝔾)=(388435)n−(125235),

  3. AZI(𝔾)=(77293679000)n−(34093679000).

5 Conclusions

In this section, a comparison is included among the different topological indices with the assistance of graphical presentation and their numerical values of the MON1(n) and MON2(n) are presented in Figures 3–8 and Tables 5–10.

Figure 3 Graphical comparison of M1(G), M2(G), and MM2(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).
Figure 3

Graphical comparison of M1(G), M2(G), and MM2(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).

Figure 4 Graphical comparison of Ra(G), RRa(G)), and SSD(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).
Figure 4

Graphical comparison of Ra(G), RRa(G)), and SSD(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).

Figure 5 Graphical comparison of H(G), IS(G)), and AZI(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).
Figure 5

Graphical comparison of H(G), IS(G)), and AZI(G) are labeled in blue, cyan, and gold graphs, respectively, for MON1(n).

Figure 6 Graphical comparison of M1(G), M2(G), and MM2(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).
Figure 6

Graphical comparison of M1(G), M2(G), and MM2(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).

Figure 7 Graphical comparison of Ra(G), RRa(G), and SSD(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).
Figure 7

Graphical comparison of Ra(G), RRa(G), and SSD(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).

Figure 8 Graphical comparison of H(G), IS(G), and AZI(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).
Figure 8

Graphical comparison of H(G), IS(G), and AZI(G) are labeled in blue, cyan, and gold graphs, respectively, for MON2(n).

Table 5

Comparison among M1(G), M2(G), and MM2(G) of MON1(n)

n M1(G) M2(G) M2(G)
2 852 1272 16
3 1380 2136 23
4 1908 3000 30
5 2436 3864 37
6 2964 4728 44
7 3492 5592 51
8 4020 6456 58
9 4548 7320 65
10 5076 8184 72
Table 6

Comparison among Ra(G), RRa(G), and SSD(G) of MON1(n), where α = 1

n Ra(G) RRa(G) SSD(G)
2 1272 16 334
3 2136 23 530
4 3000 30 726
5 3864 37 922
6 4728 44 1118
7 5592 51 1314
8 6456 58 1510
9 7320 65 1706
10 8184 72 1902
Table 7

Comparison among H(G), IS(G), and AZI(G) of MON1(n)

n H(G) IS(G) AZI(G)
2 42.80 188 1284
3 63.20 300.8 2088
4 83.60 413.6 2892
5 104 526.4 3696
6 124.40 639.2 4500
7 144.80 752 5304
8 165.20 864.8 6108
9 185.60 977.6 6912
10 206 1090.4 7716
Table 8

Comparison among M1(G), M2(G), and MM2(G) of MON2(n)

n M1(G) M2(G) MM2(G)
2 780 1128 16.9166
3 1236 1848 24.8333
4 1692 2568 32.7500
5 2148 3288 40.6666
6 2604 4008 48.5833
7 3060 4728 56.5000
8 3516 5448 64.4166
9 3972 6168 72.3333
10 4428 6888 80.2500
Table 9

Comparison among Ra(G), RRa(G), and SSD(G) of MON2(n), where α = 1

n Ra(G) RRa(G) SSD(G)
2 1128 17 291
3 1848 25 444
4 2568 33 597
5 3288 41 750
6 4008 49 903
7 4728 57 1056
8 5448 64 1209
9 6168 72 1326
10 6888 80 1515
Table 10

Comparison among H(G), IS(G), and AZI(G) of MON2(n)

N H(G) IS(G) AZI(G)
2 45.6285 186.1714 1338.8186
3 68.8571 297.1428 2197.6371
4 92.0857 408.1142 3056.45567
5 115.3142 519.0857 3915.2742
6 138.5428 630.0571 4774.09278
7 161.7714 741.0285 5632.91133
8 185 852 6491.7299
9 208.2285 962.9714 7350.5484
10 231.4571 1073.9428 8209.367

In this paper, we computed the M-polynomials of first and second metal-organic networks. Moreover, with the assistance of aforesaid M-polynomials, we calculated the various TIs such as first and second Zagreb indices, second modified Zagreb index, general and reciprocal general Randić indices, symmetric division deg, inverse sum, harmonic, and the augmented Zagreb indices of these two networks. Figures 4–6 present that M2(G), Rα(G), and AZI(G) are better for both networks discussed in this paper. These M-polynomials and calculated TIs can assist us to understand the various chemical reactivities, physical features, and biological activities of the first and second metal-organic networks. The obtained conclusions can provide a meaningful decision in the pharmaceuticals industries.

Acknowledgement

The authors are indebted to the anonymous referees for their valuable comments to improve the original version of this paper.

  1. Funding information:

    Authors state no funding involved.

  2. Author contributions:

    Agha Kashif: writing – review and editing, resources, formal analysis, visualization, methodology; Sumaira Aftab: writing – original draft, methodology; Muhammad Javaid: resources, formal analysis, visualization, methodology. Hafiz Muhammad Awais: writing – original draft, visualization, methodology

  3. Conflict of interest:

    One of the authors (Muhammad Javaid) is a Guest Editor of the Main Group Metal Chemistry's Special Issue “Topological descriptors of chemical networks: Theoretical studies” in which this article is published.

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Received: 2020-12-08
Accepted: 2021-02-21
Published Online: 2021-06-16

© 2021 Agha Kashif et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  3. Fabrication, characterization, and photocatalytic performance of ternary cadmium chalcogenides CdIn2S4 and Cd7.23Zn2.77S10-ZnS thin films
  4. New cyclic and spirocyclic aminosilanes
  5. Determination of essential and non-essential element contents of drinking water and baby water for infant’s nutrition
  6. Syntheses and molecular structures of some di(amidino)monosilanes
  7. New horizons for the synthesis of nanoparticles: Germanium nanoparticles from metastable GeBr-solutions
  8. Synthesis and antimalarial activity of some triphenyltin(IV) aminobenzoate compounds against Plasmodium falciparum
  9. Coordination behavior of PPh2{C6H4[CH2N(CH2CH2)2O]-2}: Case study CdCl2[PPh2{C6H4[CH2N(CH2CH2)2O]-2}]
  10. Rapid Communications
  11. Synthesis and structure of 2,8-dimethyl-10,10-dichlorophenoxatellurine
  12. Synthesis and crystal structure of one new Pb(II) complex constructed by 1,10-phenanthroline and two carboxylate ligands
  13. Organotin(IV) selenate derivatives – Crystal structure of [{(Ph3Sn)2SeO4} ⋅ CH3OH]n
  14. Synthesis and structural characterization of a novel 2D supramolecular lead coordination polymer with phenanthroline derivate and adipic acid
  15. X-ray crystal structures of carbonate and hydroxide bridged MnII/MgII heterobimetallic complexes formed by reduction of CO2 or H2O by a Mn0–MgII bonded compound
  16. Review Articles
  17. Variable combinations of organophosphines in PtP3X derivatives: Structural aspects
  18. Tetradentate organophosphines in Pt(η4–A4L) (A = P4, P3Si, P2X2 (X2 = N2, S2, C2), PX3 (X3 = N3, N2O)): Structural aspects
  19. Special Issue: Topological descriptors of chemical networks: Theoretical studies (Guest Editors: Muhammad Imran and Muhammad Javaid)
  20. Editorial: Topological investigations of chemical networks
  21. Forgotten coindex of some non-toxic dendrimers structure used in targeted drug delivery
  22. Eccentricity based topological indices of face centered cubic lattice FCC(n)
  23. Irregular topological indices of certain metal organic frameworks
  24. Molecular topological invariants of certain chemical networks
  25. Szeged-type indices of subdivision vertex-edge join (SVE-join)
  26. Fractional metric dimension of metal-organic frameworks
  27. On topological descriptors of ceria oxide and their applications
  28. On analysis of thermodynamic properties of cuboctahedral bi-metallic structure
  29. M-polynomial-based topological indices of metal-organic networks
  30. Two modified Zagreb indices for random structures
  31. Some degree-based topological indices of caboxy-terminated dendritic macromolecule
  32. Hosoya properties of the commuting graph associated with the group of symmetries
  33. On topological properties of hierarchical hypercube network based on Ve and Ev degree
  34. On vertex PI index of certain triangular tessellation networks
  35. Some topological properties of uniform subdivision of Sierpiński graphs
https://doi.org/10.1515/mgmc-2021-0018
Keywords for this article
M-polynomial; chemical compounds; topological indices; metal-organic network
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. A novel approach for the production of zinc borate (4ZnO·B2O3·H2O) using a single-step hydrothermal method
  3. Fabrication, characterization, and photocatalytic performance of ternary cadmium chalcogenides CdIn2S4 and Cd7.23Zn2.77S10-ZnS thin films
  4. New cyclic and spirocyclic aminosilanes
  5. Determination of essential and non-essential element contents of drinking water and baby water for infant’s nutrition
  6. Syntheses and molecular structures of some di(amidino)monosilanes
  7. New horizons for the synthesis of nanoparticles: Germanium nanoparticles from metastable GeBr-solutions
  8. Synthesis and antimalarial activity of some triphenyltin(IV) aminobenzoate compounds against Plasmodium falciparum
  9. Coordination behavior of PPh2{C6H4[CH2N(CH2CH2)2O]-2}: Case study CdCl2[PPh2{C6H4[CH2N(CH2CH2)2O]-2}]
  10. Rapid Communications
  11. Synthesis and structure of 2,8-dimethyl-10,10-dichlorophenoxatellurine
  12. Synthesis and crystal structure of one new Pb(II) complex constructed by 1,10-phenanthroline and two carboxylate ligands
  13. Organotin(IV) selenate derivatives – Crystal structure of [{(Ph3Sn)2SeO4} ⋅ CH3OH]n
  14. Synthesis and structural characterization of a novel 2D supramolecular lead coordination polymer with phenanthroline derivate and adipic acid
  15. X-ray crystal structures of carbonate and hydroxide bridged MnII/MgII heterobimetallic complexes formed by reduction of CO2 or H2O by a Mn0–MgII bonded compound
  16. Review Articles
  17. Variable combinations of organophosphines in PtP3X derivatives: Structural aspects
  18. Tetradentate organophosphines in Pt(η4–A4L) (A = P4, P3Si, P2X2 (X2 = N2, S2, C2), PX3 (X3 = N3, N2O)): Structural aspects
  19. Special Issue: Topological descriptors of chemical networks: Theoretical studies (Guest Editors: Muhammad Imran and Muhammad Javaid)
  20. Editorial: Topological investigations of chemical networks
  21. Forgotten coindex of some non-toxic dendrimers structure used in targeted drug delivery
  22. Eccentricity based topological indices of face centered cubic lattice FCC(n)
  23. Irregular topological indices of certain metal organic frameworks
  24. Molecular topological invariants of certain chemical networks
  25. Szeged-type indices of subdivision vertex-edge join (SVE-join)
  26. Fractional metric dimension of metal-organic frameworks
  27. On topological descriptors of ceria oxide and their applications
  28. On analysis of thermodynamic properties of cuboctahedral bi-metallic structure
  29. M-polynomial-based topological indices of metal-organic networks
  30. Two modified Zagreb indices for random structures
  31. Some degree-based topological indices of caboxy-terminated dendritic macromolecule
  32. Hosoya properties of the commuting graph associated with the group of symmetries
  33. On topological properties of hierarchical hypercube network based on Ve and Ev degree
  34. On vertex PI index of certain triangular tessellation networks
  35. Some topological properties of uniform subdivision of Sierpiński graphs
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