On topological indices and entropy measures of beryllonitrene network via logarithmic regression model

Chemical graph theory, a subfield of graph theory, is used to investigate chemical substances and their characteristics. Chemical graph analysis sheds light on the connection, symmetry, and reactivity of molecules. It supports chemical property prediction, research of molecular reactions, drug development, and understanding of molecular networks. A crucial part of computational chemistry is chemical graph theory, which helps researchers analyze and manipulate chemical structures using graph algorithms and mathematical models. Beryllonitrene , a compound of interest due to its potential applications in various fields, is examined through the lens of graph theory and mathematical modeling. The study involves the calculation and interpretation of topological indices and graph entropy measures, which provide valuable insights into the structural and energetic properties of Beryllonitrene’s molecular graph. Logarithmic regression models are employed to establish correlations between these indices, entropy, and other relevant molecular attributes. The results contribute to a deeper understanding of Beryllonitrene’s complex characteristics, facilitating its potential applications in diverse scientific and technological domains. In this study, degree-based topological indices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{TI}$$\end{document}TI are determined, as well as the entropy of graphs based on these \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{TI}$$\end{document}TI.

www.nature.com/scientificreports/been discovered to correlate with logP values, implying a link between molecular structure and hydrophobic characteristics 13 .
Topological indices can reveal information about a molecule's chemical reactivity.The Szeged index, for example, or the edge-connectivity index, can be used to predict a compound's stability or reactivity.Liu et al. 14,15 analyses of some structural properties of networks.Higher values of these indices may imply stronger chemical stability or resistance 16 .While topological indices can provide useful information about molecular structure and potential correlations with physicochemical features, they can not capture the full complexity of intermolecular interactions 17,18 .Nadeem et al. 19 discussed the topological aspects of metal-organic structures.Ahmad et al. 20,21 analysis the theoretical study of energy of phenylene and anthracene.Koam et.al 22 computed the valency-based topological descriptor for Hexagon Star Networks.Liu et al. 23,24 compute Hosoya index of some graphs based on connection number.They cannot predict all aspects of compound behavior.Other elements that influence physicochemical qualities include electronic structure, stereochemistry, and intermolecular forces 25 .As a result, a thorough understanding of compound properties frequently necessitates the consideration of many elements in addition to topological indices.Some Topological index are given in Table 1.

Topological indices for beryllonitrene BeN 4
The structural organization of the chemical beryllonitrene is distinctive and fascinating.It is made up of a covalently linked network of beryllium (Be) and nitrogen (N) atoms.In a typical beryllonitrene molecule, four nitrogen atoms are connected to each beryllium atom, which forms the core of a tetrahedral coordination 35 .A crystal lattice or molecular network that resembles a three-dimensional honeycomb pattern is produced as a result of this arrangement shown in Fig. 1.Beryllonitrene has unique electrical and mechanical properties due to the alternation of beryllium and nitrogen atoms.Because of its extraordinary stability and electrical conductivity capabilities, beryllonitrene is of interest in a variety of sectors, including materials science and electronics.This is because beryllium, which is lightweight, forms strong covalent bonds with nitrogen 36 .
Table 1.Topological indies TIs along with their general formulas.

Graph entropy
Entropy is the measurement of disorders of a system while the measurement of unpredictability of information content or the measurement of uncertainty of a system also called the entropy of a system, the concept was introduce in 1948 37 .The concept of graph entropy was applied in chemistry, biology, and other sciences 38 .There are different types of graphs for measuring entropy, for exploring the network the degree power is most significant. where is the weight of the edge ρ̺ see 37 .By using Tables 1 and 3 and Eq.(1), we have following formulas and their calculation.
In chemistry and related sciences, topological indices are mathematical descriptors that describe the topology of molecular structures.In relation to these indices, entropy may be defined as the degree of randomness or disorder in the distribution of specific structural characteristics.The calculation of entropy using topological indices in the context of molecular structures can offer several benefits. ) = 12.8571mn − 2.0571m − 6.0571n + 0.4905 www.nature.com/scientificreports/ • The structural diversity of molecular compounds can be quantitatively evaluated using entropy measures that are derived from topological indices.Greater structural diversity may be indicated by higher entropy values, which would add to a more varied chemical space.• Entropy measurements are correlated with a number of molecular properties, both chemical and physical.
Properties like solubility, boiling points, and reaction rates can be predicted by using topological indices in entropy calculations.• Entropy makes it possible to compare various molecular sets or chemical databases according to the structural diversity of each set.Entropy values can be used by researchers to rank or screen compounds for additional testing.

Logarithmic regression model and its analysis
A dependent variable and one or more independent variables are modeled, and the connection between them is examined using the statistical approach known as regression analysis 39 .It is frequently used to comprehend the effects of independent factors on the dependent variable and create forecasts or estimates in various domains, including economics, finance, social sciences, and engineering 40 .Regression analysis' fundamental premise is to identify the line or curve that best captures the connection between the variables.The variable you seek to predict or explain is the dependent variable, called the response variable.The variables expected to impact the dependent variable are referred to as independent variables, often known as predictor variables or explanatory variables 41 .We used the SPSS software for these analysis (https:// www.ibm.com/ produ cts/ spss-stati stics).Regression analysis may have many different forms, but the most popular one is basic linear regression, which only requires one independent variable.The relationship between the variables is considered linear in basic linear regression 42 .The line's equation is displayed as: where, Y is the dependent variable, β 0 is the Y-intercept, β i is the Coefficients of independent variable for i = 1...z , X is the Independent variable and, ε is the Error.
To minimize the sum of squared differences between the observed values of Y and the anticipated values from the model, regression analysis aims to estimate the values of β 0 and β 1 .The least squares method is com- monly used for this estimating process.Regression analysis also offers several statistical measures to evaluate the model's quality, such as the coefficient of determination (R 2 ) , which shows the percentage of the dependent variable's variance that can be accounted for by the independent variables.Regression analysis is a potent tool for figuring out how variables relate to one another, formulating predictions, and investigating cause-and-effect relationships.It is widely used in many disciplines for data analysis, decision-making, and research 43 .
A statistical method for modeling the relationship between a dependent variable and one or more independent variables where a logarithmic scale may better represent the relationship is known as logarithmic regression analysis, logarithmic transformation, or log-linear regression 44 .
where, Y is the dependent variable, β 0 is the Y-intercept, β i is the Coefficients of independent variable for i = 1 . . .z , X is the Independent variable, log() is the log function, and ε is the Error.
The logarithmic transformation enables the modeling of relationships in which the independent variables' effects on the dependent variable are multiplicative rather than additive.It is frequently employed when the relationship between the variables is curvilinear, with declining returns or increasing rates of change 45 .Logarithmic regression can be applied to data analysis in various domains, including economics, finance, biology, and environmental sciences 46 .It enables researchers to record and evaluate non-linear correlations between variables, as well as make predictions or draw insights using the logarithmic scale.

Discussion on computed results
Using the SPSS software, basically two regression models (logarithmic and power) are applied to examine the relationship between TI and graph entropy.It is noticed that the curve of logarithmic model is more closer then the power model because curve of logarithmic model touches almost each point of the observed data set, so we conclude that logarithmic model is more significant then the power, that is why logarithmic regression is applied to check the relationship between graph topological indices and entropy.The basic purpose of applying regression is to check the best predictor, the variable having good relation are the best predictor.In this case variables are curvilinear, so the best model to show their relationship is logarithmic regression..As curve of logarithmic model passes through exactly each point of GA(BeN 4 ) , so we may say that the relationship between GA(BeN 4 ) and its corresponding entropy ENT GA (G) is much more better than the other TI .Here we use different symbols for indices and entropy in the Figures that are It can be seen that GA(BeN 4 ) and ENT GA(BeN 4 ) has best relationship having R = 1 , R 2 = 1 , S E = 0.011 and F = 186557 : 243 .A model with maximum value of R, R 2 and F, while minimum S E is best model.So we may conclude that GA(BeN 4 ) is the best predictor of complexity of BeO 4 .

Conclusion
This study delved into the intricate realm of Beryllonitrene's molecular structure through the lens of graph theory and mathematical modeling.The computation and analysis of topological indices and graph entropy have illuminated crucial insights into the compound's unique structural and energetic attributes.By employing logarithmic regression models, we established meaningful correlations between these indices, entropy, and other molecular characteristics, offering a comprehensive perspective on Beryllonitrene's complex properties.
The findings underscore the significance of computational methodologies in deciphering the properties of novel materials, such as Beryllonitrene, which holds promise for diverse applications.The successful application of logarithmic regression models showcases their utility in capturing nuanced relationships within complex systems.Furthermore, the insights gained from this study provide a valuable foundation for potential applications of Beryllonitrene in various scientific and technological domains.As we move forward, this research sets the stage      for further investigations into the molecular properties of Beryllonitrene and similar compounds.Additionally, the methodologies employed here could be extended to the analysis of other novel materials, contributing to the advancement of materials science and fostering innovation across disciplines.Ultimately, the integration of computational techniques and mathematical models in this study serves as a testament to their pivotal role in unraveling the mysteries of emerging materials and compounds.The degree-based topological indices TI are determined, as well as the entropy of graph based on these TI to the complexity of BeN 4 .It is noticed that by increasing the number of unit cell of BeN 4 the value of TI and its corresponding entropy is also increasing which shows that as number of unit cell increases complexity of the BeN 4 also increases.Using the SPSS software, logarithmic and power regression is applied to examine the rela- tionship between TI and graph entropy.It is noticed that the line of logarithmic model is more closer then the power model because curve of logarithmic model touches almost each point of the observed data set so we conclude that logarithmic model is more significant then the power.As curve of logarithmic model passes through exactly each point of GA(BeN 4 ) , so we may say that the relationship between GA(BeN 4 ) and its corre- sponding entropy ENT GA (G) is much more better than the other TI e.g ( R 1 (BeN 4 ) , R −1 (BeN 4 ) , R 1 2 (BeN 4 , R − 1 2 (BeN 4 ) , ABC(BeN 4 ) , AZI(BeN 4 ) , M 1 (BeN 4 ) , M 2 (BeN 4 ) , HM(BeN 4 ) , F(BeN 4 ) , ReZG 1 (BeN 4 ) , ReZG 2 (BeN 4 ) , and ReZG 3 (BeN 4 ) ) because it has highest value of R = 1 , R 2 = 1 and F = 186557 : 243 , while the vale of S E = 0.011 is minimum as compared to the other TI .So we may conclude that GA(BeN 4 ) is best predictor of complexity BeN 4 of among all these indices.

Table 2 .
The vertex division for the chemical graph of beryllonitrene BeN 4 .

Table 13 .
The statistical values for logarithmic model.

Table 14 .
The statistical values for logarithmic model.

Table 15 .
The statistical values for logarithmic model.

Table 16 .
The statistical values for logarithmic model.

Table 18 .
The statistical values for logarithmic model.