Abstract
Oxide and silicate frameworks, known for their structural adaptability, play a pivotal role in gas storage, drug delivery, electronics, and catalysis. In this study, we explore the structural complexities of silicate and oxide networks through the lens of chemical graph theory, focusing on their molecular topology and its implications for real-world applications. By representing these materials as molecular graphs—where atoms are represented by vertices, and edges depict bonds—we employ various Revan topological indices namely, first Revan index, second Revan index, third Revan index, first modified Revan index, second modified Revan index, the first hyper Revan index, second hyper Revan index, sum connectivity Revan index, product connectivity Revan index, harmonic Revan index, geometric arithmetic Revan index, arithmetic geometric Revan index, F-Revan index, and Sombor Revan index for chain silicates, chain oxide frameworks, sheet oxide frameworks, and sheet silicate frameworks to quantitatively assess their structural and physicochemical properties. Through graphical and numerical analyses, this study offers new insights into the structure–property relationships of these networks. Our work opens the door for more efficient application of these materials across industries, particularly in nanotechnology, environmental remediation, and material science, where understanding topological features is critical to enhancing performance. The results also contribute to a deeper understanding of chemical networks, advancing both theoretical knowledge and practical applications in chemistry and material science.
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Introduction
The study of graphs that focused on the connection between networks of points and lines is known as graph theory. Graph theory finds numerous real-world applications in fields such as computer graphics, networking, biology, and various other domains. Molecular graph theory, a specialized branch, utilizes discrete mathematics to analyze and model the physical and biological properties of chemical compounds. Leonhard Euler1 was Swiss mathematician who introduced graph theory in eighteenth century.
Networks link nodes that are connected to each other in some way. A network is made up of several individual PCs that are linked together. A network can also be considered to be formed by cell phone users. Examining the optimal approach to network implementation is a step in the networking process. A new field of study called “cheminformatics” combines information science, mathematics, and chemistry. It attracted the interest of researchers worldwide.
Silicates make up the majority of minerals found in the crust of the Earth. In the majority of commonly found silicates, which include practically all silicate minerals found in Earth’s crust. Metal carbonates or oxides from sand can be combined to create silicates. By combining distinct tetrahedron silicates, we can obtain a variety of silicate structures. In a similar vein, distinct silicate structures construct silicate networks. The oxide networks are produced by taking silicon atoms out of the tetrahedra’s center. The copper II oxide (cupric oxide) network is also taken into consideration here. In medical science, copper is incredibly useful. It is necessary for the synthesis and stability of skin proteins in addition to containing powerful biocidal properties2. CuO is an inorganic chemical compound formed from copper II oxide, also known as cupric oxide. This mineral is essential to both plants and animals. Copper II oxide is a safe copper source that is used in vitamin and mineral supplements.
The properties of materials strongly depend on the molecular structure of materials. Therefore, it is very important to model and characterize the structure to predict and enhance the properties. A topological index is a numerical value that can be used to describe a certain feature of the molecular graph. Topological indices are used in theoretical chemistry to predict and evaluate the physical and biological characteristics of chemical compounds, including their boiling point, stability, enthalpy of vaporization, and other characteristics3,4,5,6,7,8,9,10,11,12,13,14. Many topological descriptors have been examined in theoretical chemistry and have found applications, particularly in QSPR/QSAR research15,16,17,18,19,20,21,22,23,24. Topological indices are categorized into three types which are as under: Degree based topological descriptors, distance based topological descriptors, and counting related polynomials indices25,26,27,28,29. Degree based topological descriptors play an important role in molecular graph theory particularly in chemistry7,30,31,32,33,34,35,36,37. The references listed in18,30,33,38,39,40,41,42,43,44,45,46,47,48 have utilized several novel topological descriptors for quantification of molecular structures. This research examines various silicate and oxide networks through Revan topological indices.
Let \({\mathbb{R}}^{ \otimes }\) be a connected graph with the vertex set \(V\left( {{\mathbb{R}}^{ \otimes } } \right)\) and edge set \(E\left( {{\mathbb{R}}^{ \otimes } } \right)\). The degree of a vertex \(\sigma^{ \oplus }\) given by \(\partial \left( {\sigma^{ \oplus } } \right)\) is the number of vertices adjacent at \(\sigma^{ \oplus }\). The maximum and minimum degree of a graph \({\mathbb{R}}^{ \otimes }\) is given as \(\psi \left( {{\mathbb{R}}^{ \otimes } } \right)\) and \(\chi \left( {{\mathbb{R}}^{ \otimes } } \right)\), respectively. The Revan vertex degree of a vertex \(u^{ \oplus } \in {\mathbb{R}}^{ \otimes }\) is defined as \({ \mathchar'26\mkern-9mu\lambda }\left( {\sigma^{ \oplus } } \right) = \psi \left( {{\mathbb{R}}^{ \otimes } } \right)\) + \(\chi \left( {{\mathbb{R}}^{ \otimes } } \right) - \partial \left( {\sigma^{ \oplus } } \right)\). The edge \(\sigma^{ \oplus } \varepsilon^{ \oplus }\) denotes the Revan edge connecting the Revan vertices \(\sigma^{ \oplus }\) and \(\varepsilon^{ \oplus }\). One can consult refs.49,50,51 for more details about these topological indices.
The 1st, 2nd and 3rd Revan indices52,53,54 of \({\mathbb{R}}^{ \otimes }\) are expressed as follows:
The 1st and 2nd modified Revan indices55,56 of \({\mathbb{R}}^{ \otimes }\) are expressed as follows:
The 1st and 2nd hyper-Revan indices57,58 of \({\mathbb{R}}^{ \otimes }\) are described as follows:
Sum connectivity Revan index59 of \({\mathbb{R}}^{ \otimes }\) are expressed as follows:
Product connectivity Revan index60 of \({\mathbb{R}}^{ \otimes }\) is expressed as follows:
Harmonic Revan index61 of \({\mathbb{R}}^{ \otimes }\) are defined as follows:
Geometric arithmetic Revan index62 of \({\mathbb{R}}^{ \otimes }\) is defined as follows:
Arithmetic geometric Revan index63 of \({\mathbb{R}}^{ \otimes }\) is described as follows:
F-Revan index64 of \({\mathbb{R}}^{ \otimes }\) is expressed as follows:
Sombor Revan index65 of \({\mathbb{R}}^{ \otimes }\) is defined as follows:
In this study, first Revan index \(\Re_{1} \left( {{\mathbb{R}}^{ \otimes } } \right)\) second Revan index \(\Re_{2} \left( {{\mathbb{R}}^{ \otimes } } \right)\), third Revan index \(\Re_{3} \left( {{\mathbb{R}}^{ \otimes } } \right)\), first modified Revan index \({}^{m}\Re_{1} \left( {{\mathbb{R}}^{ \otimes } } \right)\), second modified Revan index \({}^{m}\Re_{2} \left( {{\mathbb{R}}^{ \otimes } } \right)\), The first hyper-Revan index \(H\Re_{1} \left( {{\mathbb{R}}^{ \otimes } } \right)\), second hyper-Revan index \(H\Re_{2} \left( {{\mathbb{R}}^{ \otimes } } \right)\), sum connectivity Revan index \(S^{c} \Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), product connectivity Revan index \(P^{c} \Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), harmonic Revan index \(H^{r} \Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), geometric arithmetic Revan index \(GA\Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), arithmetic geometric Revan index \(GA\Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), F-Revan index \(F\Re \left( {{\mathbb{R}}^{ \otimes } } \right)\), and Sombor Revan index \(S^{m} \Re \left( {{\mathbb{R}}^{ \otimes } } \right)\) are computed for chain silicate and chain oxide networks. For more details regarding network modeling, the reader is referred to the works18,66,67,68,69,70,71.
Main results
Metal carbonates or oxides from sand can be combined to create silicates. By combining distinct tetrahedron silicates, we can obtain a variety of silicate structures. In a similar vein, distinct silicate structures construct silicate networks. The oxide networks are produced by taking silicon atoms out of the tetrahedra’s center. The copper II oxide (cupric oxide) network is also taken into consideration here72.
This section presents the main results for different types of Revan topological indices for the graphs of oxide and silicate chain networks \(n\).
Results for the graph of chain oxide network \({\mathbb{C}}O_{n}\)
The graph of chain oxide is shown in Fig. 1. We partition the edges of the graph with respect to the degree of the end vertices. All vertices containing degrees according to edges connected with the respective vertices are computed as 2 and 4. Here we have three different kinds of edges whose end vertices have degree (2, 2), (2, 4) and (4, 4). Symbolically represented by \(E_{1} \left( {2,\,2} \right)\), \(E_{2} \left( {2,\,4} \right)\) and \(E_{3} \left( {4,\,4} \right)\). Total number of edges calculated of the type \(E_{1} \left( {2,\,2} \right)\), \(E_{2} \left( {2,\,4} \right)\) and \(E_{3} \left( {4,\,4} \right)\) are 2, 2n and n − 2 respectively. Table 1 provides a summary of all these findings.
Chain oxide network.
Theorem 1
Let \({\mathbb{C}}O_{n}\) is the graph of chain oxide network, then we have
Proof
By using the edge partitions mentioned in Table 1 and Eqs. (1–14), we can obtain the results as follows.
\(\square\)
Results for the graph of chain silicate network \({\mathbb{C}}S_{n}\)
The graph of chain silicate is shown in Fig. 2. The edge partition of the graph is performed according to the degree of the end vertices. All vertices with degrees according to edges connected with the respective vertices are computed as 3 and 6. Here we have three different types of edges whose end vertices have degree (3, 3), (3, 6) and (6, 6). Symbolically represented by \(E_{1} \left( {3,\,3} \right)\), \(E_{2} \left( {3,\,6} \right)\) and \(E_{3} \left( {6,\,6} \right)\). Total edge number calculated of the type \(E_{1} \left( {3,\,3} \right)\), \(E_{2} \left( {3,\,6} \right)\) and \(E_{3} \left( {6,\,6} \right)\) are \(n + 4\), \(4n - 2\) and \(n - 2\) respectively. All these results are summarized in Table 2.
Chain silicate network.
Theorem 2
Let \({\mathbb{C}}S_{n}\) is the chain silicate network, then
Proof
By using the edge partitions mentioned in Table 2 and Eqs. (1–14), we can obtain the results as follows.
\(\square\)
Results for the graph of sheet oxide network \(SO^{*}\)
The graph of sheet oxide network is shown in Fig. 3. The edge partition of the graph is performed according to the degree of the end vertices. All vertices with degrees according to edges connected with the respective vertices are computed as 2 and 4. Here we have two different types of edges whose end vertices have degree (2, 4) and (4, 4). Symbolically represented by \(E_{1} \left( {2,\,4} \right)\) and \(E_{2} \left( {4,\,4} \right)\). Total edge number calculated of the type \(E_{1} \left( {2,\,4} \right)\) and \(E_{2} \left( {4,\,4} \right)\) are \(12n\) and \(18n^{2} - 12n\) respectively. All these results are summarized in Table 3.
Sheet oxide network.
Theorem 3
Let \(SO^{*}\) is the graph of sheet oxide network then
Proof
By using the edge partitions mentioned in Table 3 and Eqs. (1–14), we can obtain the results as follows
\(\square\)
Results for the graph of sheet Silicate network \(SS^{*}\)
The graph of sheet silicate is shown in Fig. 4. The edge partition of the graph is performed according to the degree of the end vertices. All vertices having degrees according to edges connected with the respective vertices are computed as 3 and 6. Here we have three different types of edges whose end vertices have degree (3, 3), (3, 6) and (6, 6). Symbolically represented by \(E_{1} \left( {3,\,3} \right)\), \(E_{2} \left( {3,\,6} \right)\) and \(E_{3} \left( {6,\,6} \right)\). Total number of edges computed of the type \(E_{1} \left( {3,\,3} \right)\), \(E_{2} \left( {3,\,6} \right)\) and \(E_{3} \left( {6,\,6} \right)\) are \(6n\), \(18n^{2} + 6n\) and \(18n^{2} - 12n\) respectively. All these results are summarized in Table 4.
Sheet silicate network.
Theorem 4
Let \(SS^{*}\) is the graph of sheet Silicate network then
Proof
By using the edge partitions mentioned in Table 4 and Eqs. (1–14), we can obtain the results as follows
\(\square\)
Numerical results and chemical applicability of the Revan indices
Numerical and graphical results of above computed topological indices for the graph of chain oxide network of dimension n, chain silicate network of dimension n, sheet oxide network and sheet silicate network are mentioned in Tables 5, 6, 7 and 8 and visualized in Figs. 5, 6, 7 and 8 respectively. Here, we observed that all indices for chain oxide network of dimension n, chain silicate network of dimension n, sheet oxide network and sheet silicate network increase for rising value of n. The increasing rate of \(H\Re_{2}\) and \(H\Re_{1}\) are higher than other topological indices. These computed topological indices have good correlations with many characteristics in network and chemistry. In order to test the chemical applicability and property prediction ability of the Revan topological indices, we have tested them against the experimental data of octane isomers.
Graphical representation of Revan topological indices for \({\mathbb{C}}O_{n}\).
Graphical representation of Revan topological indices for \({\mathbb{C}}S_{n}\).
Graphical representation of Revan topological indices for \(SO^{*}\).
Graphical representation of Revan topological indices for \(SS^{*}\).
Following the guidelines set by the International Academy of Mathematical Chemistry (IAMC), regression analysis is employed to evaluate the applicability of topological indices in modeling physicochemical properties73. Octane isomers are often used for such analyses due to their structural diversity, which encompasses variations in branching and non-polar characteristics. These organic compounds provide an ideal test case because their numerous structural isomers allow for robust statistical evaluation, and comprehensive experimental data is readily accessible. According to Randić and Trinajstić74, theoretical invariants should be correlated with experimental physicochemical properties of octane isomers to assess their predictive capabilities19,75,76,77.
Here, the correlation of the 1st Revan index and second Revan index with entropy and the acentric factor was analyzed. Experimental data for the physicochemical properties of octane isomers were sourced from the IAMC-recommended database73, and its associated datasets78. Calculations were conducted following the methodology outlined in “Main results” section above. As illustrated in Figs. 9 and 10, both 1st and 2nd Revan indices demonstrated good correlation with the acentric factor (AF) and entropy (S). Among these two indices, the 1st Revan index has the highest correlation with both Acentric Factor and Entropy. This shows the potential chemical applicability of the considered topological indices.
Correlation of 1st Revan index with acentric factor and entropy for octane isomers.
Correlation of 2nd Revan index with acentric factor and entropy for octane isomers.
Conclusion
Topological indices help us understand the physical properties, biological activity, and chemical activity of a molecular structure. In this research we have computed degree based topological indices, namely, first Revan index, second Revan index, third Revan index, first modified Revan index, second modified Revan index, The first hyper- Revan index, second hyper- Revan index, sum connectivity Revan index, product connectivity Revan index, harmonic Revan index, geometric arithmetic Revan index, arithmetic geometric Revan index, F-Revan index, and Sombor Revan index for the graph of chain oxide network of dimension n, chain silicate network of dimension n, sheet oxide network and sheet silicate network. It can be observed from the obtained results that the first and second hyper Revan indices acquire higher values than other computed topological indices. These Revan type indices have good correlations with many properties in networking and Chemistry. The chemical applicability testing results of these indices show that, they have strong potential to predict important physico-chemical properties like Entropy and Acentric factor. Thus, the computed results can provide a good basis to understand the topology and properties of these graphs and networks in a better way. These findings may also have significant contributions in the field of chemical and materials sciences. For future research some other molecular structures can be considered for studying these Revan topological indices in order to test their robustness.
Data availability
All data generated or analyzed during this study are included in this article.
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Acknowledgements
The authors extend their appreciation to Taif University, Saudi Arabia, project number (TU-DSPP-2024-47) for supporting this work.
Funding
This research was funded by Taif University, Saudi Arabia, Project No. (TU-DSPP-2024-47).
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All the authors Qun Zhang, Zubair Ahmad, Asad Ullah, Y. S. Hamed, Muzher Saleem and Melaku Berhe Belay have equally contributed to this manuscript in all stages, from conceptualization to the write-up of final draft.
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Zhang, Q., Ahmad, Z., Ullah, A. et al. Mathematical study of silicate and oxide networks through Revan topological descriptors for exploring molecular complexity and connectivity. Sci Rep 15, 8116 (2025). https://doi.org/10.1038/s41598-025-91960-7
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DOI: https://doi.org/10.1038/s41598-025-91960-7
