Studies of Metal Organic Networks via M-Polynomial-Based Topological Indices

Hussain, Muhammad Tanveer; Javaid, Muhammad; Parveen, Sajida; Awais, Hafiz Muhammad; Alam, Md Nur

doi:https://doi.org/10.1155/2022/7407924

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Volume 2022 | Article ID 7407924 | https://doi.org/10.1155/2022/7407924

Studies of Metal Organic Networks via M-Polynomial-Based Topological Indices

Muhammad Tanveer Hussain,¹Muhammad Javaid,¹Sajida Parveen,¹Hafiz Muhammad Awais,¹and Md Nur Alam²

Academic Editor: Gohar Ali

Received09 Feb 2022

Accepted09 Mar 2022

Published12 May 2022

Abstract

Topological index (TI) is a graph-theoretic tool that is used to study different physical and structural properties of the networks in various disciplines of science such as computer science, chemistry, and information technology. In this article, we study transition metal tetra-cyano-benzene organic networks by computing their M-polynomials and various topological indices (TIs). At the end, a comparison is also included between all the computed degree-based topological indices to show their betterness.

1. Introduction

Metal-organic networks (MONs) is a modern class of porous materials constructed with organic linkers based on secondary building units and metal clusters, with significant attention in the last years. Consequently, their structural diversity, high-grade porosity, and compositional tenability represent the exciting of the various physicochemical characteristics such as the grafting active group [1], postsynthetic ligand, changing organic ligand [2], preparing composites with various substances [3, 4], ions exchange [5], and impregnating suitable active materials [6], methane, and hydrogen storage materials that involve magnesium-decorated fullerenes within metal-organic frameworks [7, 8]. Seetharaj et al. [9] studied the relationship among different solvents, architecture, temperature, molar ratio, and pH of the MONs. Therefore, MONs are also studied for the purification and separation of various liquids and gases, energy storage tools in batteries [10], and precursor for the formation of nanostructures [11–13] and later on, the idea of composition of linkers with the assistance of MONs for topological insight [14–16].

Graph theory presents the modern tools which are used to study the chemical compounds and to discuss their properties. The calculated TIs of the molecular graphs are the numeric values that describe chemical reactivity, biological activity, and physical property of chemical compounds such as heat of evaporation and formation; melting, boiling and flash points, surface tension, retention time in chromatography, pressure, temperature, density, and partition coefficients [17]. Bruckler et al. [18] considered -indices that consist of contributions of vertex pairs that exponentially decrease with distance; Gonzalez-Diaz et al. [19] reviewed and commented on the “quo vadis” and challenges in the definition of TIs as we enter the new century; Klavzar and Gutman [20] used the bound that implies that and MTI are linearly correlated not only in the case of acyclic molecules but also in the case of molecules with arbitrarily many cycles; Matamala and Estrada [21] provided the role of topological structure in the study of molecular physicochemical properties; Randic [22] reported that the sensitivity of TIs for structural changes within comprehensive groups of cyclic saturated hydrocarbons is evaluated. For more details, see [23–25].

Moreover, Wiener defined a distance-based topological index in graph theory for the boiling point of paraffin [26]. Gutman and Trinajstic [27] defined TIs named as 1st and 2nd Zagreb indices to compute the total -electrons energy of different organic molecules [27]. TIs represent a valuable role in the quantitative structure-activity/property relationships to characterize a graph (network) with a biological and chemical property or activity. This mathematical relationship is represented as , where is an activity or property and is any network [28, 29]. Therefore, TIs of many networks such as silicate, honeycomb, and hexagonal networks [30], rhombous silicate and rhombus oxide networks [31], titania nanotube [32], superconducting materials [33], MOFs [34], hexagonal parallelogram nanotube [35], fullerene networks and carbon nanotube [36], regular hexagonal lattice [37], graphs with given number of cut vertices [38], and explore trees in terms of given number of vertices of maximum degree [39] are studied in the literature.

In this article, we solve the new M-polynomials, and by using these M-polynomials, the different TIs are calculated for the metal-organic networks. In the end, some comparisons between the computed TIs with the help of graphs are also included. The remaining portion is arranged as follows: Section 2 contains various definitions as well as techniques that are used in calculated results. Sections 3 and 4 present the main results of M-polynomials which are utilized in topological indices. Section 5 involves the numerical and graphical demonstration and final remarks.

2. Preliminaries

A network , where and are the organic ligands (metal node) and bonds between the nodes of , respectively. and are defined as the order and size of a graph . In connected and simple graphs, a path is occurring between two atoms (vertices) and the distance between two atoms and is the length of the shortest path that is shown as , in [22, 40]. 1st and 2nd Zagreb indices: let be a simple graph; then, its 1st and 2nd Zagreb indices are General Randic index: for a graph , set of real number , and , the general Randi index is Symmetric division deg index: let be a simple and connected graph; then, the symmetric division deg index is Harmonic index: let be a simple graph; then, its harmonic index is Inverse sum index: let be a simple graph; the inverse sum index is Augmented Zagreb index: let be a simple graph; then, the augmented Zagreb index is M-polynomial: for a graph and be the number of bonds (edges) of in such a way . Then, M-polynomial of is

In Tables 1 and 2, the relationship among the aforesaid TIs and the M-polynomial is given.

Moreover, , , , , , and , where . For further detailed discussion, refer to [41]. Metal-organic networks are considered such types of networks that are investigated for constant porosity. Therefore, these porous materials are filled with gas or liquid. Such types of pores are made with the help of metal nodes and organic ligands. The organic molecules are organic ligands that might be held together to form coordination. In organic chemistry, ligands are recognized as molecules or ions that are related to different types of functional groups attached to a central atom to form complicated coordination, i.e., coordinate covalent bonds between the central atom and ligands. These networks with porosity form in such a way that ligands attached in different places and metals place only in the middle of the network. Moreover, MONs are considered to be in two or three-dimensional networks. In the current discussion, we are dealing with MON, TM-TCNB (TM, transition metals of the 3rd series: titanium, vanadium, chromium, and zinc, where transition metals (group B elements) occupy the middle section (d-block) of the periodic table; TCNB, tetra-cyano-benzene). The two-dimensional MON of TM-TCNB is shown in Figure 1 which represents four colors consisting of blue, cyano, red, and yellow, representing N, TM, H, and N, respectively. Therefore, N stands for nitrogen, TM for transition metals, H for hydrogen, and C for carbon atom. The degree-based TI is a modern tool that opens up exciting discussion for many networks in the field of degree-based TIs. The modern contribution of these TIs based on degree is to compute the exact form of more than ten degrees and connection number-based descriptors.

Now, we take TM-TCNB as the primary network, and it does not recognize double or triple bonds (that is the primary reason to eliminate the hydrogen atoms). Furthermore, we eliminated eight hydrogen atoms from the network of TM-TCNB that were available at paranodes (vertices) of every benzene ring (present at the middle of every metal). After eliminating hydrogen’s atoms from the middle of the benzene ring, the rest of the network is presented as (TM-TCNB)¹, and this is our first metal-organic network. After that, we are going through the importance of the gained network of (TM-TCNB)¹. So, we are left at the point to add four new benzene rings (adjacent to middle benzene into first network (TM-TCNB)¹). Now, the resultant structure is TM-TCNB², and we know it as our second metal-organic network, as shown in Figure 2.

3. Main Results

In this section, we find the M-polynomial and different TIs with the help of this M-polynomial for the of the transition metal tetra-cyano-benzene organic network type-I (TM-TCNB)¹ and type-II (TM-TCNB)².

Theorem 1. Let be a transition metal tetra-cyano-benzene organic network type-I. Then, the M-polynomial of is

Proof. From Figure 2, we note that 3 distinct types of vertices exist in (TM-TCNB)¹ as follows:such thatMoreover,We have 4 distinct types of edges that is based on the degrees of end vertices in that are , such thatNow, the partition of the vertex set and edge set of the type-1 MON are given in Tables 3 and 4.
Now, by using definitions of M-polynomial Tables 3 and 4, we have

Theorem 2. Let be the transition metal tetra-cyano-benzene organic network type-I. Then,

Proof. Let be the M-polynomial of the type- (Theorem 1) MON; then,First, we find the required partial derivatives and integrals asNow, by substituting ,Using these values in formulae of Table 1,

Theorem 3. Let be the transition metal tetra-cyano-benzene organic network type-I. Then,

Proof. Let be the M-polynomial of MON; then,therefore,

Theorem 4. Let be the transition metal tetra-cyano-benzene organic network type-II. Then, the M-polynomial of is

Proof. From Figure 3, we show the partition of the vertex set and the edge set of according to degrees of the vertices. In this way, we get 3 distinct types of vertices in as follows:such thatAlso,Now, we get distinct types of edges based on the degree of end vertices of that areTherefore, partition of the vertex set and the edge set of the metal-organic network are given in Tables 5 and 6.
By using definition of M-polynomial, Tables 5 and 6, we have

Theorem 5. Let be a transition metal tetra-cyano-benzene organic network type-II. Then,

Proof. Let be the M-polynomial of second MON:The needed derivatives and integrals are calculated asNow, by substituting ,

Theorem 6. Let be a transition metal tetra-cyano-benzene organic network type-II. Then,

Proof. Let be the M-polynomial of second MON; then,We getNow, using Table 2,

4. Comparison and Conclusion

In this section, we calculate various degree-based topological indices and present the results in the form of tables and graphs as given Tables 7–10.

4.1. Comparison between , , and of (TM-TCNB)¹

The comparison of first Zagreb , second Zagreb , second modified Zagreb , and symmetry division deg of (TM-TCNB)¹ is computationally computed with the help of M-polynomials, as given in Table 3. We also computed these indices for different values of . When increases, then all TIs are increased in the same order. We show the graphical presentation of all the aforesaid topological indices in Figure 4 for the particular values of z.

4.2. Comparison between , , and of (TM-TCNB)²

The comparison of first Zagreb , second Zagreb , second modified Zagreb , and symmetry division deg of (TM-TCNB)² is computationally computed with the assistance of aforesaid M-polynomials given in Table 4. We calculated these indices for different values of . When increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 5 for the particular values of z.

4.3. Comparison between , , and of (TM-TCNB)¹

The comparison of the harmonic , inverse sum (IS), and augmented Zagreb index (AZI) of (TM-TCNB)¹ is computationally computed with the assistance of M-polynomial given in Table 5. We calculated these indices for different values of . When increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 6 for the particular values of z.

4.4. Comparison between , , and of (TM-TCNB)²

The comparison of the harmonic , inverse sum , and augmented Zagreb index of (TM-TCNB)² is computationally computed with the assistance of M-polynomial given in Table 6. We calculated these indices for different values of . When increases, then all the TIs are increased in the same order. We show the graphical representation of all the degree-based TIs in Figure 7 for the particular values of z.

In this work, we discussed the M-polynomials of (TM-TCNB)¹ and (TM-TCNB)². We also work out on the modern degree-based TIs, including 1st Zagreb , 2nd Zagreb , 2nd modified Zagreb , general Randic , symmetry division deg (SDD), harmonic , inverse sum (IS), and the augmented Zagreb index (AZI) and showed their behavior for (TM-TCNB)¹ and (TM-TCNB)². In addition, all the obtained results are also compared with the help of the numerical values provided in the tables, (see Tables 7–10). The problem is still open for the computational results related to different topological indices of the various metal-organic networks.

Data Availability

The data used to support this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

References

Y. K. Hwang, D.-Y. Hong, J.-S. Chang et al., “Amine grafting on coordinatively unsaturated metal centers of MOFs: consequences for catalysis and metal encapsulation,” Angewandte Chemie International Edition, vol. 47, no. 22, pp. 4144–4148, 2008.
View at: Publisher Site | Google Scholar
E. Mohamed, K. Jaheon, R. Nathaniel et al., “Systematic design of pore size and functionality in isoreticular MOFs and their application in methane storage,” Science, vol. 295, pp. 469–472, 2002.
View at: Google Scholar
J. Juan-Alcañiz, J. Gascon, and F. Kapteijn, “Metal-organic frameworks as scaffolds for the encapsulation of active species: state of the art and future perspectives,” Journal of Materials Chemistry, vol. 22, no. 20, pp. 10102–10118, 2012.
View at: Publisher Site | Google Scholar
D. Bradshaw, A. Garai, and J. Huo, “Metal-organic framework growth at functional interfaces: thin films and composites for diverse applications,” Chemical Society Reviews, vol. 41, no. 6, pp. 2344–2381, 2012.
View at: Publisher Site | Google Scholar
K. Min, F. C. John, F. Honghan, A. P. Kimberley, and M. C. Seth, “Postsynthetic ligand and cation exchange in robust metal-organic frameworks,” Journal of the American Chemical Society, vol. 134, pp. 18082–18088, 2012.
View at: Google Scholar
W. T. Aaron, M. N. Kate, M. H. James, J. H. Anita, and R. H. Matthew, “Metal−Organic frameworks impregnated with magnesium-decorated fullerenes for methane and hydrogen storage,” Journal of the American Chemical Society, vol. 131, pp. 10662–10669, 2009.
View at: Google Scholar
M. Azari and A. Iranmanesh, “Generalized Zagreb index of graphs,” Studia Universitatis Babeș-Bolyai, vol. 56, no. 3, pp. 59–70, 2011.
View at: Google Scholar
K. Won-Tea, Q. Shaopeng, F. O. Alan et al., “Hollow Pd-Ag composite nanowires for fast responding and transparent hydrogen sensors,” ACS Appl. Matter. Interfaces, vol. 9, no. 45, pp. 39464–39474, 2017.
View at: Google Scholar
R. Seetharaj, P. V. Vandana, P. Arya, and S. Mathew, “Dependence of solvents, pH, molar ratio and temperature in tuning metal organic framework architecture,” Arabian Journal of Chemistry, vol. 12, no. 3, pp. 295–315, 2019.
View at: Publisher Site | Google Scholar
L. Hao, W. Kecheng, S. Yujia, T. Christina, L. Jialuo, and Z. Hong-Cai, “Recent advances in gas storage and separation using metal–organic frameworks,” Materials Today, vol. 21, pp. 221–235, 2018.
View at: Google Scholar
H. Y. Min, L. F. Kam, and Z. C. George, “Study suggests choice between green energy or economic growth,” Green Energy and Environment, vol. 2, pp. 218–245, 2017.
View at: Google Scholar
L. R. Biao, X. Shengchang, X. Huabin, Z. Wei, and C. Banglin, “Exploration of porous metal–organic frameworks for gas separation and purification,” Coordination Chemistry Reviews, vol. 378, pp. 87–103, 2019.
View at: Google Scholar
V. Alexander, “Upper and lower bounds of symmetric division deg index,” Iran J Math Chem, vol. 52, pp. 91–108, 2014.
View at: Google Scholar
C. W. Megan, L. Jiafei, I. Timur, and K. F. Omar, “Linker competition within a metal-organic framework for topological insights,” American Chemical Society, vol. 58, no. 2, pp. 1513–1517, 2019.
View at: Google Scholar
C. Petit and T. J. Bandosz, “MOF-graphite oxide composites: combining the uniqueness of graphene layers and metal-organic frameworks,” Advanced Materials, vol. 21, no. 46, pp. 4753–4757, 2009.
View at: Publisher Site | Google Scholar
J. Y. Seung, Y. C. Jae, K. C. Hee, H. C. Jung, S. N. Kee, and R. P. Chong, “Preparation and enhanced hydrostability and hydrogen storage capacity of CNT@MOF-5 hybrid composite,” Chemistry of Materials, vol. 21, pp. 1893–1897, 2009.
View at: Google Scholar
G. Rücker and C. Rücker, “On topological indices, boiling points, and cycloalkanes,” Journal of Chemical Information and Computer Sciences, vol. 39, no. 5, pp. 788–802, 1999.
View at: Publisher Site | Google Scholar
F. M. Bruckler, T. Doslic, A. Graovac, and I. Gutman, “On a class of distance-based molecular structure descriptors,” Chemical Physics Letters, vol. 503, pp. 336–338, 2011.
View at: Google Scholar
H. Gonzalez-Diaz, S. Vilar, L. Santana, and E. Uriarte, “Medicinal chemistry and bioinformatics-current trends in drugs discovery with networks topological indices,” Current Topics in Medicinal Chemistry, vol. 7, no. 10, pp. 1015–1029, 2007.
View at: Google Scholar
S. Klavzar and I. Gutman, “A comparison of the Schultz molecular topological index with the wiener index,” Journal of Chemical Information and Computer Sciences, vol. 36, pp. 1001–1003, 1996.
View at: Google Scholar
A. R. Matamala and E. Estrada, “Generalised topological indices: optimisation methodology and physico-chemical interpretation,” Chemical Physics Letters, vol. 410, no. 4-6, pp. 343–347, 2005.
View at: Publisher Site | Google Scholar
M. Randic, “Characterization of molecular branching,” Journal of the American Chemical Society, vol. 97, no. 23, pp. 6609–6615, 1975.
View at: Publisher Site | Google Scholar
B. Furtula and I. Gutman, “A forgotten topological index,” Journal of Mathematical Chemistry, vol. 53, no. 4, pp. 1184–1190, 2015.
View at: Publisher Site | Google Scholar
X. Li and J. Zheng, “Extremal chemical trees with minimum or maximum general Randic index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 54, pp. 195–205, 2005.
View at: Google Scholar
A. M. Naji, N. D. Soner, and I. Gutman, “On leap Zagreb indices of graphs,” Communications in Combinatorics and Optimization, vol. 2, pp. 99–117, 2017.
View at: Google Scholar
H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947.
View at: Publisher Site | Google Scholar
I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972.
View at: Publisher Site | Google Scholar
I. Gutman and O. Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, New York, NY, USA, 1986.
J. Devillers, D. Domine, C. Guillon, S. Bintein, and W. Karcher, “Prediction of partition coefficients (LOGPoct) using autocorrelation descriptors,” SAR and QSAR in Environmental Research, vol. 7, no. 1-4, pp. 151–172, 1997.
View at: Publisher Site | Google Scholar
B. Rajan, A. William, C. Grigorious, and S. Stephen, “On certain topological indices of silicate, honeycomb and hexagonal networks,” Journal of Mathematical and Computational Science, vol. 3, no. 5, pp. 530–535, 2012.
View at: Google Scholar
M. Javaid, M. A. Ur Rehman, and J. Cao, “Topological indices of rhombus type silicate and oxide networks,” Canadian Journal of Chemistry, vol. 95, pp. 134–143, 2016.
View at: Google Scholar
M. Javaid, J.-B. Liu, M. A. Rehman, and S. Wang, “On the certain topological indices of titania nanotube TiO₂[m, n],” Zeitschrift für Naturforschung A, vol. 72, no. 7, pp. 647–654, 2017.
View at: Publisher Site | Google Scholar
X. Guiyin, N. Ping, D. Hui, D. Bing, L. Laiyang, and Z. Xiaogang, “Harvard enters superconducting spin cycle,” Materials Today, vol. 20, pp. 22–35, 2017.
View at: Google Scholar
W. Hailong, Z. Qi-Long, Z. Ruqiang, and X. Qiang, “Metal-organic frameworks for energy applications,” Chememistry, vol. 2, pp. 52–80, 2017.
View at: Google Scholar
S. Akhter and M. Imran, “On molecular topological properties of benzenoid structures,” Canadian Journal of Chemistry, vol. 94, no. 8, pp. 687–698, 2016.
View at: Publisher Site | Google Scholar
J. Baca, M. Horvathova, M. Mokrisova, and A. Suhanyiova, “On topological indices of fullerenes,” Applied Mathematics and Computation, vol. 251, pp. 154–161, 2015.
View at: Google Scholar
M. Baca, J. Horvathova, M. Mokrisova, A. Suhanyiova, and S.-F. Andrea, “On topological indices of carbon nanotube network,” Canadian Journal of Chemistry, vol. 93, pp. 1157–1160, 2015.
View at: Google Scholar
S. Wang and S. Ji, “On the sharp lower bounds of Zagreb indices of graphs with given number of cut vertices,” Journal of Mathematical Analysis and Applications, vol. 458, no. 1, pp. 21–29, 2018.
View at: Google Scholar
S. Wang, C. Wang, L. Chen, and J.-B. Liu, “On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree,” Discrete Applied Mathematics, vol. 227, pp. 166–173, 2017.
View at: Publisher Site | Google Scholar
P. S. Ranjini, V. Lokesha, and A. Usha, “On the symmetric division deg index of graph,” The International Journal of Graph Theory, vol. 1, pp. 116–121, 2013.
View at: Google Scholar
A. Kashif, S. Aftab, M. Javaid, and H. M. Awais, “M-polynomial-based topological indices of metal-organic networks,” Main Group Metal Chemistry, vol. 44, no. 1, pp. 129–140, 2021.
View at: Publisher Site | Google Scholar

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Copyright © 2022 Muhammad Tanveer Hussain et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Journal of Mathematics

Distances in Graph Theory

Studies of Metal Organic Networks via M-Polynomial-Based Topological Indices

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Comparison and Conclusion

4.1. Comparison between , , and of (TM-TCNB)1

4.2. Comparison between , , and of (TM-TCNB)2

4.3. Comparison between , , and of (TM-TCNB)1

4.4. Comparison between , , and of (TM-TCNB)2

Data Availability

Conflicts of Interest

References

Copyright

4.1. Comparison between , , and of (TM-TCNB)¹

4.2. Comparison between , , and of (TM-TCNB)²

4.3. Comparison between , , and of (TM-TCNB)¹

4.4. Comparison between , , and of (TM-TCNB)²