A computational study of entropy measures and topological indices for Magnesium Nitride using curve fitting techniques.

Aims: In this article, the drugs used in the treatment of Diabetic Nephropathy are analyzed using the topological indices, and a QSPR analysis of the chemical properties is done. Study Design: The degree based topological indices are used for this structural analysis of chemical compounds of these drugs. The graph structure of a chemical structure is derived from a chemical component by considering the atoms as vertices and the bonds connecting two atoms as edges, the edge-partitions are computed and used to obtain the topological indices. Results: The topological indices of these drugs are calculated and compared with the theoretical properties of the drugs. The diagrammatic visualization of the data is plotted in Box graph, Line graph and in scatter diagram. The theoretical values are compared with these topological indices and the regression equations are found. The relation between the properties of chemicals in terms of topological indices is given as regression equations. These equations contain vital information about the structural relationship and the numerical invariants and the properties in terms of numerical values and may be used for drug design. Conclusion: The computed topological indices are visualized in various aspects and the regression analysis among the chemical properties of the drugs and the topological indices is done. This degree-based QSPR analysis helps to design medicines for the diabetic nephropathy.

The physicochemical properties of molecules are correlated with their chemical structure using degree-based topological indices. The graph entropy zone has captured the imagination of scientists because of its possible application in chemistry. Boron nanotubular structures are high-interest materials with new electronic characteristics due to the presence of multi center bonds. Mechanical and thermal stability are two challenges that these materials have in nanodevice applications. Therefore, they necessitate theoretical research into the other qualities. In this article, study the graph entropy measure by using topological invariants of different versions of atom-bond connectivity index Zagreb indices (including redefined forms and and the forgotten topological index F(G)) which gives a better prediction of the chemical properties of different chemical compounds. We computed these indices for Boron Nanotube because of its significance in nanotechnology. To ensure the correctness of the output, a computational analysis and graphical comparison of generated indices is provided.

In [13] Gutman introduced a novel graph invariant called Sombor index SO, defined via $sqrt{deg(u)^{2}+deg(v)^{2}}.$ In this paper we provide relations between Sombor index and some degree-based topological indices: Zagreb indices, Forgotten index and Randi' {c} index. Similar relations are established in the class of triangle-free graphs.

Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. There are certain types of topological indices such as degree-based topological indices, distance-based topological indices, and counting-related topological indices. These topological indices correlate certain physicochemical properties such as boiling point, stability, and strain energy of chemical compounds. In this paper, we compute an exact expression of Wiener index, vertex-Szeged index, edge-Szeged index, and total-Szeged index of single-walled titania nanotubes TiO2(m, n) by using the cut method for all values of m and n.

Graph theory plays substantial role in mathematics, chemistry, QSAR and physical sciences. The basic layout of the graph theoretic model is a molecular structure in which vertices of the graph correspond to atoms, and edges correspond to chemical bonds. The study of this graph model provides information about the chemical structure. A line graph has many useful applications in physical chemistry. M-polynomial is rich in producing closed forms of many degree-based topological indices which correlates chemical properties of the material under investigation. This polynomial is used in computing closed formulas of many degree-based topological invariants of the molecular structures. The molecular graph of carbon nanocones has a conical structure with a cycle of length k at its core and n layers of hexagons placed at the conical surface around its centre. In this study, we transformed the molecular structure of carbon nanocones into graph theoretic model and produced its line graph. Thereafter, we determined closed formulas for M-polynomials of line graphs of nanocones. We also recovered important topological degree-based indices of the line graph of nanocones. Moreover, we provide different graphs of topological indices and their relations with the parameters of the line graph of nanocones. These graphs depict the actual dependencies of the topological indices on the parameters of the carbon nanocones.

Theory of networks serves as a mathematical foundation for the construction and modeling of chemical structures and complicated networks. In particular, chemical networking theory has a wide range of utilizations in the study of chemical structures, where examination and manipulation of chemical structural information are made feasible by utilizing the numerical graph invariants. A network invariant or a topological index (TI) is a numerical measure of a chemical compound which is capable to describe the chemical structural properties such as melting point, freezing point, density, pressure, tension, and temperature of chemical compounds. Wiener initiated the first distance‐based TI which is considered to be the most important TI to preserve the chemical and physical properties of chemical structures. Later on, degree‐based TI was introduced to find the π ‐electron energy of molecules. Recently, connection number‐based TIs are studied which are more efficient than degree and distance‐based TIs. In this paper, we compute the connection number‐based TIs of the structure of crystal cubic carbons which are one of the most significant and interesting composites in modern resources of science due to the involvement of carbon atoms.

Distributed fiber-optic sensors employ the conventional fiber as the sensing medium and allow all locations along the optical fiber to be measured simultaneously. Thus the sensing scheme provides an efficient way of simultaneously achieving multi-points sensing with a convenient and low-cost configuration. One of the most popular technologies that enables distributed sensing is based on Brillouin scattering. Characteristics of Brillouin scattering are determined mostly by the material properties of the optical fiber and the Brillouin Frequency Shift (BFS) has linear dependence on both the temperature and strain of the fiber under test (FUT). Based on this principle, intensive research efforts have been made in distributed Brillouin fiber sensors during the past two decades. Among them, Brillouin Optical Time-Domain Analysis (BOTDA) has attracted intensive research interest due to its promising properties for temperature and strain monitoring. In BOTDA the local Brillouin Gain Spectrum (BGS) is reconstructed by scanning the frequency offset of a continuous wave and an optical pulse around BFS when they are counter-propagating inside FUT, and the local BFS and hence the temperature and/ or strain information are retrieved accordingly. Usually curve fitting techniques are employed in determining the BFSs of the measured BGSs along the FUT, in which the measured BGS is fitted using an ideal curve and the BFS is found to be the frequency with the peak gain on the fitted curve. Fitting techniques using different lineshape for ideal curves, e.g. Lorentzian curve fitting (LCF), Gaussian curve fitting (GCF), pseudo-Voigt curve fitting (pVCF), and quadratic curve fitting (QCF) have been used to find the BFS distribution along FUT. However, the curve fitting techniques require careful initialization of model parameters and as many data points as possible collected on the measured BGS to ensure the fitting accuracy, otherwise the accuracy in BFS determination will degrade significantly. In addition, long processing time is needed for the iteration of fitting algorithms to find the BFS, especially when the sensing distance is long with a large number of BGSs collected and processed. Recently we have utilized the nonlinear mapping capability of Artificial Neural Network (ANN) in a BOTDA system and has successfully extracted temperature distribution from the measured BGSs along the FUT. Before temperature extraction ANN is trained to acquire the knowledge about the BGS patterns under different temperatures, thus it allows better accuracy even if the data points on the collected BGS become fewer when large frequency scanning step is adopted during the acquisition of BGSs. Moreover, the processing speed of ANN for temperature extraction is much faster than that of curving fitting techniques. In this presentation, we will review our work on using ANN for temperature extraction in a BOTDA system. Different idea spectrum profiles are used for training ANN, and the corresponding performances are compared with those using conventional curve fitting techniques. Some potential applications using ANN in BOTDA are also discussed. We believe that ANN can be an attractive tool for direct temperature or strain extraction in BOTDA system at high accuracy and fast speed.

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction of the characteristics provided by the chemical structure of the molecule is a key feature of these topological indices. The concepts from graph theory are presented in a brief discussion of one of its many applications to chemistry, namely, the use of topological indices in quantitative structure-activity relationship (QSAR) studies and quantitative structure-property relationship (QSPR) studies. This study uses the M-polynomial approach, a newly discovered technique, to determine the topological indices of the medication fenofibrate. With the use of degree-based topological indices, we additionally construct a few novel degree based topological descriptors of fenofibrate structure using M-polynomial. When using M-polynomials in place of degree-based indices, the computation of the topological indices can be completed relatively quickly. The topological indices are also plotted. Using M-polynomial, we compute novel formulas for the modified first Zagreb index, modified second Zagreb index, first and second hyper Zagreb indices, SK index, S K 1 index, S K 2 index, modified Albertson index, redefined first Zagreb index, and degree-based topological indices.

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randic index, inverse Randic index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.

The association of M-polynomial to chemical compounds and chemical networks is a relatively new idea, and it gives good results about the topological indices. These results are then used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this paper, an effort is made to compute the general form of the M-polynomials for two classes of dendrimer nanostars and four types of nanotubes. These nanotubes have very nice symmetries in their structural representations, which have been used to determine the corresponding M-polynomials. Furthermore, by using the general form of M-polynomial of these nanostructures, some degree-based topological indices have been computed. In the end, the graphical representation of the M-polynomials is shown, and a detailed comparison between the obtained topological indices for aforementioned chemical structures is discussed.

For a graph $G$, the M-polynomial is defined as \(M(G;x,y)= \sum_{\delta \leq \alpha \leq \beta \leq \Delta}m_{\alpha \beta}(G)x^{\alpha}y^{\beta}\), where \(m_{\alpha \beta} (\alpha, \beta \geq 1)\), is the number of edges \(ab\) of \(G\) such that \(\deg_{G}(a)=\alpha\) and \(\deg_{G}(b)=\beta\), and \(\delta\) is the minimum degree and \(\Delta\) is the maximum degree of \(G\). The physiochemical properties of chemical graphs are found by topological indices, in particular, the degree-based topological indices, which can be determined from an algebraic formula called M-polynomial.\ We compute the closest forms of M-polynomial for Mycielskian of paths and cycles. Further, we plot the 3-D graphical representation of M-polynomial.\ Finally, we derive some degree-based topological indices with the help of M-polynomial.

The structure and topology of chemical compounds can be determined using chemical graph theory. Using topological indices, we may uncover much about connectivity, complexity, and other important aspects of molecules. Numerous research investigations have been conducted on the K-Banhatti indices and entropy measurements in various fields, including the study of natural polymers, nanotubes, and catalysts. At the same time, the Shannon entropy of a graph is widely used in network science. It is employed in evaluating several networks, including social networks, neural networks, and transportation systems. The Shannon entropy enables the analysis of a network’s topology and structure, facilitating the identification of significant nodes or structures that substantially impact network operation and stability. In the past decade, there has been a considerable focus on investigating a range of nanostructures, such as nanosheets and nanoparticles, in both experimental and theoretical domains. As a very effective catalyst and inert substrate, the MgO nanostructure has received a lot of interest. The primary objective of this research is to study different indices and employ them to look at entropy measures of magnesium oxide(111) nanosheets over a wide range of p values, including p=1,2,3,…,j. Additionally, we conducted a linear regression analysis to establish the correlation between indices and entropies.

Shannon entropy, also known as information entropy or entropy, measures the uncertainty or randomness of probability distribution. Entropy is measured in bits, quantifying the average amount of information required to identify an event from the distribution. Shannon’s entropy theory initiates graph entropies and develops information-theoretic magnitudes for structural computational evidence of organic graphs and complex networks. Graph entropy measurements are valuable in several scientific fields, such as computing, chemistry, biology, and discrete mathematics. In this study, we investigate the entropy of fractal-type networks by considering cycle, complete, and star networks as base graphs using degree-based topological indices.

The use of silicon carbide is increasing significantly in the fields of research and technology. Topological indices enable data gathering on algebraic graphs and provide a mathematical framework for analyzing the chemical structural characteristics. In this paper, well-known degree-based topological indices are used to analyze the chemical structures of silicon carbides. To evaluate the features of various chemical or non-chemical networks, a variety of topological indices are defined. In this paper, a new concept related to the degree of the graph called "bi-distance" is introduced, which is used to calculate all the additive as well as multiplicative degree-based indices for the isomer of silicon carbide, Si2C3-1[t, h]. The term "bi-distance" is derived from the concepts of degree and distance in such a way that second distance can be used to calculate degree-based topological indices.

In this study, we investigate entropy measurements for subdivided cage networks based on topological indices. We specifically calculate different entropy, redefining Zagreb entropy, HM ( G ) , M 1 ( G ) , M 2 ( G ) entropy, atom bond connection entropy, and Randic entropy. We examine the graphical behavior of various entropy measures using the line fit approach. The results highlight patterns in the distribution of entropy values and interactions between them, which shed light on the intricate connectivity and structural properties of segmented cage networks. This work improves our understanding of cage network dynamics and provides a visual framework for interpreting their behavior.