Abstract
Metal–Organic Networks (MONs) are made by chemical molecules that contain metal ions and organic ligands. A crystalline porous solid called Metal–Organic Networks (MONs) is made up of a \(3D\) metal network of ions held in place by a multidentate ligand. (MONs) can be used for gas storage, purification drug delivery, gas separation, catalysis, and sensing applications. There is enormous potential for effective integration and research of MONs in diverse applications. Molecular descriptors are arithmetic measures that reveal a chemical substance's physical and chemical characteristics in its foundational network in a natural relationship. They demonstrate an important role in theoretical and ecological chemistry, and in the field of medicine. In this research, we calculated various recently discovered molecular descriptors viz. the modified version of second zagreb index, harmonic index, reciprocal randic index, modified version of forgotten topological index, redefined first zagreb topological index, redefined second zagreb topological index and redefined third zagreb topological index for two separate metal–organic networks. The numerical and graphical comparative analysis of these considered molecular descriptors are also performed.
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Introduction
Every heavenly body is a combination of several constituents that significantly contributes to the composition of the earth. The three-element hydrogen, oxygen and nitrogen are the most significant ones on the planet1,2. Massively used compounds made by chemically like metal–organic network (MON) are thus made up of alloy ions/metallic ions and organic linkers. With the aid of the hydrothermal method, new MONs composed of zinc considered metal ions and benzene \(\mathrm{1,3}\)-dicarboxylic acid as the organic linker3. Biography of MONs is their superficial alteration and as well as their particle control six division4. Devices with luminous properties could be created using \(Zn\)-related MONs, are chemical sensors1. In reality,\({Zn}^{+2}\) an astringent, anti-dandruff, antibacterial, and anti-inflammatory autogenous simple powerless noxious conversion metal cation, is frequently worked in homoeopaths as a scarring catalyst and face ointment5. Additionally, the production processes of MONs related to zinc have lately been documented, and as well their toxicity, biological uses, and biocompatibility6. In modern chemistry, graph theory offers essential tools that depict chemical compounds' heats of formation, evaporation, flash points, temperatures, pressures, densities, and partition coefficients7. Zinc oxide is a white powder that is insoluble in water. Nanostructure of Zinc oxide \((Zno)\) can be synthesized into a range of different morphologies. A variety of skin conditions can be treated with zinc oxide. In marine environments, zinc silicate coating can provide long-term protection of steel and is used in rapid coating work for over fifty years. Zinc oxide is also used in toothpastes to prevent plaque. These metals also help the human body, it is present in the red blood cells and causes several reactions related to carbon dioxide metabolism. Zinc silicate networks are economical because they are relatively thin coating.
The zinc oxide and zinc silicate exhibit physicochemical characteristics including grafting active groups8, incorporating appropriate active material9, ion exchange10, creating composites using various materials11, modifying organic ligands, photosynthetic ligands, and improving the selectivity, sensitivity, and response times of biosensors.Yap et al.12 and Lin et al.13 presented the most current developments in precursors for a variety of nanostructures and MON-related applications, including lithium ion batteries, super capacitors, photocatalysis, electrocatalysis, and catalysts for the manufacture of fine chemicals. The field of modern chemistry can benefit from using graph theory to represent the physical and chemical characteristics of chemical compounds, such as their heats of formation and evaporation, flash points, melting points, boiling points, temperatures, pressures, densities, retention times in chromatography, tensions, and partition coefficients14,15. In order to investigate many characteristics of chemical compounds (such as the boiling point of paraffin), Wiener first developed the distance-based topological index (TI) in 1947. Gutman and Trinajsic's highly regarded first-degree-based TI was developed to test the chemical plausibility of the total π-electron energy of the chemical compounds (alternant hydrocar-bons).
The Zagreb type indices contributed tremendously in the various fields which can be seen in16,17,18,19,20,21. The details on other degree based molecular descriptors and structures are given in22,23,24,25,26,27. In 2021, the authors of28 computed the connection-based Zagreb indices such as first Zagreb connection index (ZCI), second ZCI, modified first ZCI, modified second ZCI, modified third ZCI, and modified fourth ZCI. We extended those results for other degree based topological indices as Modified version of second Zagreb index \({M}_{2}\left(G\right)\), Harmonic index \(H\left(G\right)\)29, Reciprocal Randic index \(RR\left(G\right)\), the Modified version of Forgotten topological index \({F}_{N}^{*}\), the Redefined First Zagreb topological index \({R}_{e}Z{G}_{1}(G)\)30, the Redefined Second Zagreb topological index \({R}_{e}Z{G}_{2}\left(G\right)\)31 as well as the Redefined third Zagreb topological index \({R}_{e}Z{G}_{3}(G)\)32 for MONs, namely Zinc oxide \(\left(ZNOX\left(n\right)\right)\) and zinc silicate \(\left(ZNSL\left(n\right)\right)\) as regards to the expanding layers,\(n\ge 3\). Our outcomes fascinate not only mathematician but also of theoretical chemists. The results of this study can be used to examine numerical quantities and guide future research into the physical properties of molecules. As a consequence, it is a beneficial procedure to eliminate costly and time-consuming laboratory studies. The findings of this research depict that \({R}_{e}Z{G}_{3}\left(G\right)\) achieved higher values than other classical Zagreb indices, which may have better correlation with the thirteen physicochemical characteristics of octane isomers.
Main results
Here, we initially present some significant definitions of the degree based molecular descriptors which will be useful to obtain the main results. In the whole study, we denote the adjacent vertices by \(p\) and \(q\), i.e. \(pq\in {E}_{G}.\)
Definition 1
The molecular descriptor \({M}_{2}\left(G\right)\) denotes modified version of second zagreb index that is described as33,
Definition 2
The molecular descriptor \(H\left(G\right)\) denotes harmonic index which is defined in34 as,
Definition 3
The molecular descriptor \(RR\left(G\right)\) denotes reciprocal randic index which is explained as35,
Definition 4
The molecular descriptor \({F}_{N}^{*}\) denotes modified version of forgotten topological index, that is described as36,
Definition 5
The molecular descriptor \({R}_{e}Z{G}_{1}\) denotes redefined first zagreb topological index, which is defined by36,
Definition 6
The molecular descriptor \({R}_{e}Z{G}_{2}(G)\) denotes the redefined second zagreb topological index, that is described as31,
Definition 7
The molecular descriptor \({R}_{e}Z{G}_{3}(G)\) denotes redefined third zagreb topological index, which is explained by31,
Theorem 1
Let \(H\cong ZNOX\left(n\right)\) is a zinc oxide network as depicted in Fig. 1, then
Proof
Based on the Definition 1 and Table 1, we have
Theorem 2
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as shown in Fig. 1, then
Proof
Based on the Definition 2 and Table 1, one has
Theorem 3
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as shown in Fig. 1, then
Proof
Based on the Definition 3 and Table 1, one has
Theorem 4
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as shown in Fig. 1, then
Proof
Based on the Definition 5 and Table 1, we have
The following corollary is a direct consequent of Theorem 4.
Corollary 5
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as depicted in Fig. 1, then
Theorem 6
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as shown in Fig. 1, then
Proof
Based on the Definition 7 and Table 1, we have
Theorem 7
Let \(H\cong ZNOX\left(n\right)\) be a zinc oxide network as depicted in Fig. 1, then
Proof
Based on the Definition 4 and Table 1, one has
Theorem 8
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as depicted by Fig. 2, then
Proof
Based on the Definition 1 and Table 2, we have
Theorem 9.
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as shown by Fig. 2, then
Proof
Based on the Definition 2 and Table 2, we have
Theorem 10
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as shown by Fig. 2, then
Proof
Based on the Definition 3 and Table 2, one has
Theorem 11
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as depicted by Fig. 2,
Proof
Based on the Definition 5 and Table 2, one has
The following corollary is a direct consequent of Theorem 11.
Corollary 12
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as shown by Fig. 2, then
Theorem 13
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as shown by Fig. 2, then
Proof
Based on the Definition 7 and Table 2, one has
Theorem 14
Suppose \(K\cong ZNSL\left(n\right)\) be a zinc silicate network as shown by Fig. 2, then
Proof
Considering Table 2 and Definition 4, we can write
Graphical interpretations
In this part, we worked out all indices numerically and presented the results in the table below. From Figs. 3, 4, 5, 6, 7, 8, 9 and Table 3, it is easy to see a positive relationship between n and the considered topological indices. As we increase n, the topological indices also increase. The comparative graphs of \({\mathrm{M}}_{2}\left(\mathrm{G}\right)\), \(\mathrm{H}\left(\mathrm{G}\right)\), \(\mathrm{RR}\left(\mathrm{G}\right)\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{1}(\mathrm{G})\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{2}(\mathrm{G})\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{3}(\mathrm{G})\) and \({\mathrm{F}}_{\mathrm{N}}^{*}\left(\mathrm{G}\right)\) indices of \(\mathrm{ZNOX}\) for various values are presented in Fig. 10. Thus, Fig. 10 describe that all indices for \(\mathrm{ZNOX}\) increase for increasing value of n. The increasing rate of \({R}_{e}Z{G}_{3}(G)\) is higher than that of other topological indices. This depict that, \({R}_{e}Z{G}_{3}\left(G\right)\) achieved higher values than other classical Zagreb indices, which may have better correlation with the thirteen physicochemical characteristics of octane isomers.
From Figs. 11, 12, 13, 14, 15, 16, 17 and Table 4, it is easy to see a positive relationship between n and the considered topological indices. As we increase n, the topological indices also increase. Meanwhile, the comparative relationship of \({\mathrm{M}}_{2}\left(\mathrm{G}\right)\), \(\mathrm{H}\left(\mathrm{G}\right)\), \(\mathrm{RR}\left(\mathrm{G}\right)\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{1}(\mathrm{G})\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{2}(\mathrm{G})\), \({\mathrm{R}}_{\mathrm{e}}{\mathrm{ZG}}_{3}(\mathrm{G})\) and \({\mathrm{F}}_{\mathrm{N}}^{*}\left(\mathrm{G}\right)\) indices of \(ZNSL\) for various values are presented in Fig. 18. Thus, Fig. 18 describe that all indices for \(\mathrm{ZNOX}\) increase for increasing value of n. The increasing rate of \({R}_{e}Z{G}_{3}(G)\) is higher than that of other topological indices. This depicts that, \({R}_{e}Z{G}_{3}\left(G\right)\) achieved higher values than other classical Zagreb indices, which may have better correlation with the thirteen physicochemical characteristics of octane isomers.
Conclusion
In this research, we calculated various recently discovered molecular descriptors for two separate metal–organic networks. The molecular descriptors which we considered are \({M}_{2}\left(G\right)\), \(H\left(G\right)\), \(RR\left(G\right)\), \({F}_{N}^{*}(G)\), \({R}_{e}Z{G}_{1}(G)\), \({R}_{e}Z{G}_{2}(G)\) as well as \({R}_{e}Z{G}_{3}(G)\). The two metal–organic networks we considered are, Zinc oxide \(\left(ZNOX\left(n\right)\right)\) and zinc silicate \(\left(ZNSL\left(n\right)\right)\). The numerical and graphical comparative analysis of the considered molecular descriptors are also performed. The obtained results depict that \({R}_{e}Z{G}_{3}\left(G\right)\) achieved higher values than other classical Zagreb indices, which may have better correlation with the thirteen physicochemical characteristics of octane isomers. It is quite motivating to study the distance based topological indices for the Metal–Organic Networks. In the near future we will carry it out.
Data availability
All data generated or analysed during this study are included in this article.
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Funding
The author Asad Ullah gratefully acknowledge the financial support for this study derived from the Higher Education Commission of Pakistan (Grant No. 20–11682/NRPU/R&D/HEC/2020).
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Zaman, S., Jalani, M., Ullah, A. et al. Mathematical analysis and molecular descriptors of two novel metal–organic models with chemical applications. Sci Rep 13, 5314 (2023). https://doi.org/10.1038/s41598-023-32347-4
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DOI: https://doi.org/10.1038/s41598-023-32347-4
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