On computation of neighbourhood degree sum - based topological indices for zinc - based metal – organic frameworks

: The permeable materials known as metal – organic frameworks ( MOFs ) have a large porosity volume, excellent chemical stability, and a unique structure that results from the potent interactions between metal ions and organic ligands. Work on the synthesis, architectures, and properties of various MOFs reveals their utility in a variety of applications, including energy storage devices with suitable electrode materials, gas storage, heteroge - neous catalysis, and chemical assessment. A topological index, which is a numerical invariant, predicts the physi - cochemical properties of chemical entities based on the underlying molecular graph or framework. In this article, we consider two di ﬀ erent zinc - based MOFs, namely zinc oxide and zinc silicate MOFs. We compute 14 neighbour - hood degree sum - based topological indices for these fra - meworks, and the numerical and graphical representations of all the aforementioned 14 indices are made.


Introduction
Metal-organic frameworks (MOFs) are chemically coordinated, consistently organized, porous structures consisting of metal clusters and organic linkers. The purification and separation of various gases, biocompatibility, heterogeneous catalysis, and biomedical applications are only a handful of the multiple uses for MOFs. These are viable candidates for drug delivery, nitric oxide storage, imaging and sensing, and disease diagnostics due to their precise structure, wide range of pore creation, huge surface area, high porosity, customizable frameworks, and simple chemical functionalization. Recently, the field of biomedical applications has gained attention as an attractive and promising avenue for the creation and use of multifunctional MOFs. An effective method of drug distribution is a crucial aspect of this research. In biomedical activities, notably in drug delivery systems, nontoxic MOFs are viewed as more effective than toxic MOFs.
A recent discussion focused on the antibacterial activity of MOFs based on nontoxic antimicrobial cations (Zn 2+ ) (Taghipour et al., 2018;Wang et al., 2011). Due to the low toxicity of zinc ions, zinc-based MOFs are being studied for use in biomedical applications. In dermatology, zinc ions are frequently used to moisturize skin and treat wounds. In addition, the synthesis processes of zinc-based MOFs were discussed, along with their biological applications and biocompatibility (Prasad, 2008). In addition to detecting and sensing, MOFs can also predict a wide range of chemical and physical properties, including biosensors that increase response time, selectivity, and sensitivity (Ding et al., 2017), the impregnation of suitable active materials (Thornton et al., 2009), the change of photosynthetic and organic legends (Yin et al., 2015), grafting (Hwang et al., 2008), and exchange of ions (Kim et al., 2012).
The silicate mineral class is an amazing class among all the mineral classes. Sand and either metal carbonate or metal oxide are combined to create silicate. Silica usually appears as tetrahedra. The vertices in the corners represent oxygen ions in chemistry, whereas the vertices in the middle represent silicon ions. A wide diversity of silicate structures can be obtained by shifting about the tetrahedral building blocks. Zinc silicate coating can give long-term protection for steel in marine settings and has been utilized in quick coating work for over 50 years.
These metals also benefit the human body; they are found in red blood cells and produce a number of events connected to carbon dioxide metabolism. Zinc silicate networks are cost-effective because of their thin coating.
Cheminformatics is an active area of research where the relationship between quantitative structural behaviour and structural properties helps predict the biological activity and properties Aslam et al., 2017;Doley et al., 2020).
Chemical graph theory should be considered complementary to other theoretical chemistry fields as well as being on par with them in order to fully comprehend the nature of the molecular structure. The language of chemistry is not the same as the language of graph theory. As a result, graph theoretical terminology is suggested for common usage in chemistry as well as for the associated chemical concepts. Chemical graphs can represent chemical systems by using a straightforward conversion method that treats atoms as vertices and the bonds between them as edges. Chemical structure topological characterization enables the classification of molecules and the modelling of unknown structures with desired properties.
Molecular descriptors are numerical representations of the molecule's structural, physicochemical, biological attributes, and so on. In order to understand how a molecule functions and interacts with the body's physiological system, molecular descriptors attempt to measure these properties. Because the exact mechanism of pharmacological activity is often a mystery, it is best to begin with descriptors covering as many molecular properties as feasible and then assess their predictive potential.
Topological indices, also called connectivity indices, are a type of molecular descriptor that is computed from the chemical compound's molecular graph (Timmerman et al., 2002). Predicting the physicochemical, biological, environmental, and toxicological properties of chemicals directly from their molecular structure relies heavily on graph invariants and, by extension, topological indices. Numerous topological indices have been proposed in the literature, each of which describes structural complexity based on different features such as degree, distance, and independency, suggesting that indices cannot be precisely defined. It is not surprising, then, that new descriptors for comparing and characterizing systems have relied heavily on a wide range of ideas and approaches.
The basic topological indices are defined for connected, undirected molecular graphs. They exclude hydrogen atoms ("hydrogen suppressed") and do not take into account double bonds. Topological matrices (e.g. distance or adjacency matrices) are used to define the relationships between the atoms, and they can be mathematically manipulated to yield a single real number. Therefore, the topological index can be understood as a set of two-dimensional descriptors (González-Díaz et al., 2007, 2008) that can be readily computed from molecular graphs and that are insensitive to the specifics of how the graph is represented or labelled, as well as requiring no energy minimization on the part of the chemical structure. A topological index is a numerical graph invariant that characterizes the structure of a graph. The degree of a vertex is given by d u or d(u) (West, 2001) and denotes the number of edges at that vertex. The sum of the degrees of all adjacent nodes of node u, expressed as δ(u)/δ u , is the neighbourhood degree of node u ϵ V.
The topological index is a useful technique for describing the physical, chemical, and biological properties of chemical substances (Asif et al., 2020;De, 2016;Patil and Yattinahalli, 2020). In quantitative structureactivity relationship/quantitative structure-property relationship (Hayat et al., 2019;Randic, 1996), topological indices play a crucial role in associating the diverse chemical compound structures with bioactivities and chemical properties. Mondal et al. (2019) defined the neighbourhood versions of Zagreb indices M 1N , M 2N , HM N , and F N , and the indices are as follows: Furthermore, Mondal et al. (2021) defined some more neighbourhood indices, and the indices are as follows: Ravi and Desikan (2021) proposed some open neighbourhood degree sum-based topological indices. They computed those indices for graphene structures. The open neighbourhood indices introduced by them are as follows: Various researchers (Abbas et al. 2023;Ali 2021;Awais et al., 2020;Chu et al., 2020;Haoer 2021;Hong et al., 2020;Hussain et al., 2021;Manzoor et al., 2021;Ravi and Desikan, 2022;Siddiqui et al., 2021a,b;Xu et al., 2020;Yu et al., 2022;Zaman et al., 2023;Zhao et al., 2021a,b) have computed different topological indices for various MOFs. M-polynomial of MOFs was studied by Kashif et al. (2021). Ahmad et al. (2022) recently computed the degree-based polynomial for some MOFs.
Recently, researchers have computed various topological indices for the zinc oxide and zinc silicate networks, namely Zagreb connection-based indices (Javaid and Sattar, 2022;Sattar and Javaid, 2023) and degree-based indices (Zaman et al., 2023).
Let ZnO(n) be a molecular graph of zinc oxide of growth n. The total number of the vertices and edges of the zinc oxide network is 70n + 46 and 85n + 55, respectively. Let ZnSi(n) be a molecular graph of zinc silicate of growth n. The total number of vertices and edges of the zinc silicate network is 82n + 50 and 103n + 61, respectively. One can see the molecular structures of zinc oxide and zinc silicate network (ZnO(n) and ZnSi(n)), for growth n ≥ 1, from the article by Javaid and Sattar (2022). Inspired by their work, we compute the neighbourhood degree sum-based topological descriptors for these two MOFs. Figures 1 and 2 depict the molecular structures of zinc oxide and zinc silicate networks ZnO(n) and ZnSi(n), for growth n = 1.

Neighbourhood degree sum-based topological descriptors for MOFs
In this section, we compute the neighbourhood degree sum-based topological descriptors for both MOFs. We compute the topological indices using the neighbourhood  degree-sum of the end vertices. Tables 1 and 2 provide the edge partitions of the first and second MOFs, respectively, based on the neighbourhood degree sum of the end vertices. Let ZnO(n) be the zinc oxide metal-organic network of growth n, for, n ≥ 1. Then, by applying the neighbourhood degree-sum edge partitions given in Table 1 in Eqs 1-14, we obtain      Let ZnSi(n) be the zinc silicate metal-organic network of growth n, for, n ≥ 1. Then, by applying the neighbourhood degree-sum edge partitions given in Table 2        3 Numerical and graphical interpretations of computed results for the zinc-based metal-organic networks In this section, we present the numerical computations of the 14 neighbourhood-based indices for the 2 zinc-based metal-organic networks. From the expressions for the topological indices computed in the previous section for ZnO(n) and ZnSi(n) structures, we observe that the values of the indices vary linearly with n, the growth. Similar relationship exists between the topological indices and the growth for the indices discussed in the articles by Javaid and Sattar (2022), Sattar and Javaid (2023), and Zaman et al. (2023).
The computed index values for the 14 indices are shown in Tables 3 and 4 for growth n up to 5. Graphs in Figure 3 depict the computed results for ZnO(n) for n up to 10.
Furthermore, the results concerning zinc silicatebased MOF are shown in Tables 5 and 6 for growth n up to 5. Graphs in Figure 4 depict the computed results for ZnSi(n) for n up to 10.

Conclusions
Computing various topological indices of chemical graphs allows for the investigation of chemical molecules and research on how the indices connect to the physiochemical properties. In this article, we determined the cardinality of the neighbourhood degree sum-based edge partitions corresponding to two MOFs, ZnO(n) and ZnSi(n). These edge partitions were used to compute 14 neighbourhood degree sum-based topological indices for ZnO(n) and ZnSi(n), respectively. As a future work, we plan to apply these descriptors to various transformations of MOFs.
Acknowledgments: The authors would like to acknowledge the editor and reviewers for their valuable comments towards the improvement of this manuscript.
Funding information: The authors state no funding involved.