On K-Banhatti indices and entropy measure for rhodium (III) chloride via linear regression models

Rhodium (III) chloride is a metallic compound characterized by its shiny and silvery-white appearance. It possesses high reflectivity and exhibits excellent resistance to corrosion. This makes it a popular choice for applications such as plating materials in jewelry and other decorative items, imparting a lustrous and reflective surface to the coated objects. Topological indices are numerical parameters employed to characterize the topology of a molecular structure. These indices are derived from the connectivity of atoms within the molecule and serve as predictors for various molecular properties, including reactivity, stability, and solubility. On the other hand, the Shannon entropy of a graph finds extensive applications in network science. It is utilized in the analysis of diverse networks, such as social networks, biological networks, and transportation networks. The Shannon entropy allows for the characterization of a network's topology and structure, aiding in the identification of crucial nodes or structures that play significant roles in network functionality and stability. In this paper, our primary objective is to compute different K-Banhatti indices and employ them to evaluate the entropy measure of Rhodium (III) chloride RhCl3. Additionally, we conducted an examination through linear regression analysis involving various indices and entropies associated with Rhodium chloride. Moreover, we established a correlation between degree-based Banhatti indices and entropies via the line fit method.


Introduction
A graph consists of a collection of vertices (also referred to as nodes or points) and a collection of edges (also known as links or arcs) that establish connections between pairs of vertices.This branch (Graph Theory) of mathematics provides a framework for analyzing and comprehending intricate networks and relationships across various disciplines, such as computer science, engineering, physics, and social sciences.It enables the exploration of concepts like connectivity, paths, cycles, optimization, and many other properties associated with graphs and their applications.
In chemistry, graph theory finds extensive utility in representing the molecular structure of chemical compounds as graphs.In a chemical graph, the vertices correspond to the atoms within the molecule, while the edges signify the bonds linking these atoms.By scrutinizing the topology and properties of chemical graphs, chemists can acquire valuable insights into the physical and chemical characteristics of molecules, including reactivity, stability, and biological activity.Chemical graph theory boasts numerous applications in chemistry and related disciplines, such as drug design, chemical synthesis, materials science, and catalysis.For instance, it can aid in the prediction of properties for newly synthesized compounds, the development of more efficient catalysts for chemical reactions, and the comprehension of structure-activity relationships pertaining to drugs and bioactive molecules.
Throughout the paper, the graph is denoted as .If  =  represents an edge connecting vertices  and , the degree of the edge  in , denoted as   , is defined as   = () + () − 2. Kulli [18] examined the K-Banhatti index of graphs.Kulli [19] also investigated the multiplicative and hyper K-Banhatti indices of nanotubes.Wang et al. [38] discussed the K-Banhatti indices of acid curcumin.Rehman [31] examined the K-Banhatti indices of nanotubes.Anjum et al. [2] discussed the K-Banhatti and Hyper K-Banhatti indices of nanotubes.Table 1 presents various Banhatti Indices.
Where () obtain by adding degree of  and degree  and then subtract degree , similarly we obtain () by adding degree of  and degree  and then subtract degree , () = Δ() + () − (), where Δ() be the maximum degree, () be the minimum degree and () be the degree of vertex .
The Banhatti Index keywords in different areas, such as graph entropy, complex networks, transfer entropy, inclusion degree, entropy weight, and agent-based modeling and risk evaluation, respectively, were used.We presented the bibliometric analysis of Banhatti Index keywords in Fig. 1 as shown below.This Fig. 1 illustrates that the concept of the Banhatti Index is related to many research areas in recent years and interlinks with many other topological indices.
We have presented a bibliometric analysis of different authors who work on the topic of the Banhatti Index.We use the Scopus database for this analysis with 845 research articles.This figure shows that different coauthors are working on this topic and writing articles together.Moreover, the main idea of the Banhatti Index is introduced by Kulli in 2016.Fig. 2 shows that most of the work is done by Kulli and its co-authors.
Shannon entropy, named after the mathematician Claude Shannon, measures a system's uncertainty or randomness.It is commonly used in information theory, communication theory, and other fields, such as physics, chemistry, and biology.Shannon first introduced entropy in his renowned article [34], which said that the entropy of a probability distribution is viewed as a measure of the ambiguity of a system.It was created for analyzing graph structure data and the related chemical networks.The aforementioned graph entropy measures come in a wide variety of specific forms [29].Zhang et al. [39,40] discuss the topological indices of generalized bridge molecular graphs, Carbon Nanotubes, and products of chemical graphs.Shazia et al. [26] computed the Entropy Measures for Isomeric Natural Polymers.
Zhang et al. [41,42] provided the physical analysis of heat for the formation and entropy of Ceria Oxide.Dehmer [11] introduced information-based function graph entropies that examine properties and capture structural information.Zhao et al. [37] reported the research of graphitic carbon nitride based on molecular topological indices.Arockiaraj, M., et al. [4,5] discussed the entropies for different structures.Zhang et al. [43,44] gives an analysis of different Molecular Structures using Topological indices.Kavitha, S. R. J., et al. [17] discussed the topological characterization and graph entropies of tessellations of kekulene structures.
Shannon groundbreaking work in the late 1940 marked the inception of modern information theory [34].Information theory has been widely used in biology and chemistry after seeing early linguistics and electrical engineering applications.Entropy measurements play a crucial role in these applications [28].The structural complexity of graphs has been measured using Rashevsky's notion of graph entropy [12].
In their 2014 publication, Chen et al. [9] introduced the notion of entropy for edge-weighted graphs.They proposed a simple graph denoted as ( (); (); ()), where () represents the edge weight of the edge () in .Eq. ( 1) defines the entropy of the edge-weighted graph.

Rhodium (III) chloride structure
Due to the distinctive feature of a variable oxidation state, transition metal forms a variety of salt with other periodic table elements, such as halogens.Rhodium chloride is one of them, where transition metal rhodium (Rh) forms a salt with chlorine from the halogen group.In this inorganic salt oxidation state of rhodium is (III).This inorganic salt is diamagnetic, and it has octahedral geometry with three chloride ions bonded with Rh (III) metal centers, as shown in Fig. 3.It is present in both hydrated and anhydrous forms, and its color depends upon the number of water molecules present with it.In its anhydrate form, it's in brown solid and octahedral molecular geometry [10].
Rhodium chloride with three molecules of water is soluble in commonly used solvents, and it's widely used in the laboratory for preparative purposes.Anhydrous Rhodium chloride is not soluble, so it is not used in laboratories [14].The sodium salt of rhodium   chloride is used to synthesize rhodium chloride through ion exchange chromatography.The trisodium salt of rhodium is obtained from common ores of platinum and iridium group metals.Hydrated rhodium is collected after crystallization of product [8].To prepare anhydrous rhodium chloride, both are reacted in metallic form at the higher temperature of 200 − 300 • [32].Rhodium chloride trihydrate has been used to synthesize various complexes, despite its complexity in the formation of its solutions.It reacts with various chemical species to produce a variety of complexes such as ammonia, pyridine, phosphines, thioethers, arsines, acetylacetone, 1, 5-cyclooctadiene, and alkenes [13,15,24,27,30,33,35,36]. Being a transition metal salt and having a variable oxidation state leads to the application of rhodium chloride in various catalytic applications such as asymmetric ring opening reactions [25], activation of C-H bond in organic compounds [16], decomposition of nitrous oxides [7], conversion of chlorobenzene [1].The considered structure to be the subgraph of the structure [3].
Let  be the number of the unit cells, which are arranged linearly.In Fig. 3, 4, and 5, the number of vertices of degree 1 is represented by blue color denoted by  (1), the number of vertices of degree 2 is represented by red color denoted by  (2), and the number of vertices of degree 6 is represented by purple color denoted by  (6).The cardinality of vertices of degree 1, 2 and 6 is 8 + 20, 14 + 24 and 6 + 10.The cardinality of edges between vertices of degrees (1, 6) is 8 + 20 and is denoted by  (1,6).The cardinality of edges between vertices of degrees (2, 2) is 4 and is denoted by  (2,2).The cardinality of edges between vertices of degrees (2, 6) is 14 + 22 and is denoted by  (2,2).The order and size of the Rhodium (III) Chloride molecular graph are 28 + 54 and 22 + 46, respectively.The vertex and edge partition are dedicated in Table 2 and Table 3 respectively.

Methodology
We computed K-Banhatti indices using various techniques, including edge partitioning, vertex degree approaches, neighborhood degree counts, and combinatorial computation.After that, we compute the entropy using the K-Banhatti indices.We make use of Matlab to generalize and validate calculations.To perform correlation analysis, specialized software called   was used.

Main results
We performed calculations for several K-Banhatti Indices in this section and evaluated our numerical results.We also visually illustrated the response of each topological index to different parameter values.To achieve the desired outcomes, we utilized various strategies.Using values Table 3 in Table 1, we obtain different K-Banhatti indices as follows.
A numerical comparison between different K-Banhatti indices is shown in Table 4.
Analyzing Table 4 and Fig. 6(), it is evident that as the value of  increases, the  1 index demonstrates a significantly rapid growth compared to  2 .Similarly, in Fig. 6(), it can be observed that with an increasing value of , the  2 index exhibits a more pronounced growth rate in comparison to  1 .
Moving on to Table 4 and Fig. 7(), it can be seen that as  increases, the  1 index shows a substantially higher growth rate compared to   and  2 .Likewise, in Fig. 7(), it is apparent that as  increases, the  1 index displays a more rapid growth rate compared to  2 and  3 indices.

Computing K-Banhatti entropy
In this section, we proceed to compute the entropy of the K-Banhatti index and assess the numerical results obtained.Furthermore, we present visual representations showcasing the response of each topological index to various parameter values.To achieve the desired outcomes, we employ a combination of strategies.By utilizing the computed K-Banhatti indices and corresponding values from Table 3 in Equation ( 1), we derive the following entropy values.

Linear regression between K-Banhatti indices and Shannon entropy
This section delves into the relationship between K-Banhatti indices and entropy values, examining their correlation.Researchers rely on graphical and numerical representations of these results to save time and minimize laboratory work.Consequently, we computed the degree-based entropies for various values of n in the ℎ 3 structures.The line fitting of entropy measure is shown in Fig. 10 In this study, we applied this approach to examine the relationship between entropy formation and multiple indices.Through the manipulation of underlying factors, we utilized linear curve fitting to determine the degree of alignment between entropy and the indices.The accuracy of the analysis was assessed using metrics such as linear regression, standard error () estimation, correlation coefficient (), coefficient of determination ( 2 ), and coefficients  and .Our primary focus was placed on highlighting the significance of  2 .The simulations were conducted using Microsoft Excel.Table 6 presents the correlation coefficient values    between entropy and each of the indices, providing a quantitative measure of their significance.Additionally, Table 7 showcases the predicted entropy values obtained from the analysis.In Table 6 and Fig. 14, the index  3 exhibits the highest values for both  and  2 , indicating a strong correlation with entropy.Additionally, it has the lowest value for standard error () in comparison to the other indices.Therefore, based on these criteria,  1 can be considered the best predictor of entropy.Table 5 displays the entropy values obtained using Eq. ( 1), while Table 7 presents the predicted entropy values obtained through linear regression models.

Conclusion
In this article, we computed K-Banhatti indices.Using K-Banhatti topological descriptors of ℎ 3 , we generalized analytical expressions.Then use it in entropy formulas and find the entropies of K-Banhatti indices of ℎ 3 .Following a considerable increase in disorder, the transition state of ℎ 3 with rising  values exhibits the biggest modification in entropy.
By employing a line fit technique, we were able to explore the relationship between entropy formation and various indices, thereby enabling the investigation of connections among multiple types of variables.This analysis involved utilizing the line fit method to estimate the correlation between entropy and the K-Banhatti index while manipulating several underlying parameters.To assess the accuracy of our findings, we employed various evaluation metrics such as linear regression, standard error estimation, ,  2 , and coefficient analysis (specifically coefficients a and b).Emphasis was primarily placed on the significance indicated by the  2 value.According to the results of the investigation into these structures; in this regard, the First Revan K-Banhatti index consistently delivers the most optimal and progressively enhancing outcomes due to its superior performance of  2 and  .All the obtained results were presented both in numerical form and through graphical representations.

Declaration of competing interest
All authors declare that there are no conflicts of interest regarding the publication of this paper.

Fig. 2 .
Fig. 2. Bibliometric analysis of the citations of authors-wise research based on K-Banhatti Index.

Table 2
Division of vertices.

Table 3
Division of edges.

Table 5
Numerical comparison of entropy of K-Banhatti Indices.

Table 6
Statistical parameter of linear model.