On K-Banhatti, Revan Indices and Entropy Measures of MgO(111) Nanosheets via Linear Regression

Abstract

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 $p=1,2,3,\dots ,j$. Additionally, we conducted a linear regression analysis to establish the correlation between indices and entropies.

1. Introduction

Chemical graph theory facilitates establishing a connection between the graphical and chemical structure of molecules through quantitative structure-property relationships and quantitative structure-activity relationships [1,2,3]. Due to the extensive production growth in the chemical industry, the importance of chemical graph theory has increased. Consequently, it is important to thoroughly analyze the chemical properties of these novel medications and chemicals to utilize them effectively. A lot of research has been carried out to determine how chemical properties like molecular structure, toxicity, melting point, and freezing point are related [4,5].

The main goal of “quantitative structure-property relationships” and “quantitative structure-activity relationships” is to examine the connections between molecular structures and the properties or activities they have in different areas like medicine, pharmaceuticals, medical research, rational drug design, and experimental science [6]. The researchers analyzed different behaviors of chemical compounds in quantitative structure-property relationships [7,8] through topological indices. Kirmani et al. [9] examined the different topological variants and the physicochemical attributes of drugs employed in treating coronavirus. Hosseini and colleagues classify the degree-based indices according to their predictive capacity [10]. The study conducted by Hosseini and Shafiei [10] investigated the correlation between several chemical indices and thermodynamic properties.

The K-Banhatti indices and entropy measures are also very well-known indices to study molecular graphs of compounds. These indices have been applied to analyze the topological characteristics of natural polymers, such as cellulose networks [11]. Ghani et al. looked into the entropies and K-Banhatti indices of $C_6{H}_6$ in various chemical networks [12]. The main goal of their study was to find out how the K-Banhatti indices work with other molecules. V. R. Kulli developed the modified K-Banhatti indices [13,14]. Chaluvaraju came up with the Zagreb version of the K-Banhatti index of a graph, which has been used to study the topological properties of graphs [15]. Kiran Naz et al. studied the polycyclic random chains and computed their multiplicative and hyper K multiplicative K Banhatti indices [16]. The types of polycyclic chains that they studied include polyphenyl and spiro chains. Hussain et al. [17] studied k Banhatti indices and entropy measures of rhodium (III) chloride. Using the line-fit method, they conducted their research using linear regression analysis and established the relationship between indices and entropy. Ref. [18] calculated the M-polynomial of $C_3$ and $H_6$ nanosheets and computed some topological indices by using M-polynomials. Ref. [12] calculated the precise values of K-Banhatti Indices of $C_6{H}_6$ by using atom-bond partitioning method based on valencies K-Banhatti indices provide valuable information about the connectivity and complexity of graph structures. In the context of $MgO\left(111\right)$, understanding the topological properties of its molecular graph is essential for predicting its behavior.

Along with studying the topological indices, researchers studied the entropy of these indices to thoroughly study the behavior of molecules. The computation of indices and entropy measures provides valuable insights into the structural and topological characteristics of polymers. Entropy is a fundamental concept in various fields such as information theory, thermodynamics, and statistical mechanics, and is important in understanding the behavior and characteristics of systems. Mansoor et al. [11] looked into determining molecular descriptors along with entropy measures for isomeric natural polymers. Shannon introduced the concept of entropy to quantify a system’s randomness and information content. The mathematical formula of Shannon entropy is given as under:

$$E_{\varphi }\left(M\right)=\sum _{xy\in W\left(M\right)}\frac{\varphi \left(xy\right)}{{\sum }_{xy\in W\left(M\right)\varphi \left(xy\right)}}log[\frac{\varphi \left(xy\right)}{{\sum }_{xy\in W\left(M\right)}\varphi \left(xy\right)}]$$

(1)

Metal oxides are a significant category of materials with an extensive range of features, including insulating, semiconducting, and conducting characteristics. These materials have been utilized in several fields, including technological devices, personal care products, and catalysts. One case is the typical rock salt configuration of bulk $MgO$ (magnesium oxide), which serves various purposes. $MgO$ is a diamagnetic oxide with ionic properties, exhibiting insulating behavior. It possesses a significant band gap of 7.8 eV; this material is inert and has an extremely high melting point.

Additionally, it accelerates chemical reactions and offers a suitable base for various chemicals, including group III-V elements, metals, and high-pressure superconductors. Because of its non-toxic and ecologically favorable properties, it is extensively employed as a sorbent for eliminating dyes and metallic substances. It can also be utilized as an optical material and constituent of optical composites [19,20]. In the past decade, there has been a significant focus on investigating a range of nanostructures, such as nanosheets, nanowires, nano-belts, and nanoparticles, in both experimental and theoretical domains [19,21,22]. As of now, the effective synthesis of $MgO$ thin films with two distinct facets, namely (111) and (100), has been achieved. As a very effective catalyst and inert substrate, the $MgO$ nanostructure received a lot of interest [23]. Magnesium oxide nanosheets, which have a band gap between 2.75 and 4.38 eV, could be very useful in UV-electronic devices on the nanoscale level.

Understanding $MgO\left(111\right)$’s structural and topological properties is crucial for predicting its behavior and optimizing its performance in different applications. By representing the crystal structure of Magnesium oxide as a graph, it is possible to analyze its connectivity, complexity, and other topological characteristics. Topological indices of $MgO\left(111\right)$ can be used to quantify its structural features and predict its properties [24]. This study aims to investigate the characteristics of magnesium oxide(111) nanosheets across a range of p values, encompassing $p=1,2,3,\dots ,j$.

2. Preliminaries and Mathematical Framework

Chemical graph theory involves the representation of a molecule as a graph, where the atoms are denoted as vertices, and the bond between atoms is represented by as an edge. The graph M is an improvised representation of a magnesium oxide (111) nanosheet molecule. The symbol X denotes the vertex set, whereas W is the edge set representing the atoms and the bond between them, respectively. The cardinality of X in a graph is commonly referred to as the order, whereas the total edges represent the size of the graph. The chemical networks consist of nodes, represented by x and y, and are connected by edges labeled as $w=xy$. The vertex x degree is represented by $d\left(x\right)$ and indicates the total edges connected to that vertex. Recently, an innovative concept of the edge degree has been introduced, denoted as $d\left(e\right)=d\left(x\right)+d\left(y\right)-2$. The maximum degree of the graph can be represented by $\Delta \left(M\right)$ and $\delta \left(M\right)$, $\kappa \left(x\right)=\Delta \left(M\right)+\delta \left(M\right)-d\left(x\right)$ and $r\left(\overline{x}\right)=\Delta \left(M\right)+\delta \left(M\right)-d\left(x\right)$. Table 1 contains the formulas of all the indices under consideration.

Table 1. Topological Indices.

Indices	Notations-	Formula
The first K-Banhatti Index [14]	${\mathcal{B}}_1\left(M\right)$-	${\sum }_{xw}[d\left(x\right)+d\left(e\right)]$
The second K-Banhatti Index [14]	${\mathcal{B}}_2\left(M\right)$-	${\sum }_{xw}[d\left(x\right)·d\left(e\right)]$
The first K hyper-Banhatti Index [25]	${\mathcal{HB}}_1\left(M\right)$-	${\sum }_{xw}{[d\left(x\right)+d\left(e\right)]}^2$
The second K hyper-Banhatti Index [25]	${\mathcal{HB}}_2\left(M\right)$-	${\sum }_{xw}{[d\left(x\right)·d\left(e\right)]}^2$
The K-Banhatti harmonic Index [26]	${\mathcal{H}}_b\left(M\right)$-	${\sum }_{xw}\left[\frac{2}{d\left(x\right)+d\left(e\right)}\right]$
The first hyper Revan Index [25]	$\eta R_1\left(M\right)$-	${\sum }_{w=xy}{[\kappa \left(x\right)+\kappa \left(y\right)]}^2$
The second hyper Revan Index [25]	$\eta R_2\left(M\right)$-	${\sum }_{w=xy}{[\kappa \left(x\right)·\kappa \left(y\right)]}^2$
The third Revan Index [26]	$\eta R_3\left(M\right)$-	${\sum }_{w=xy}[\kappa \left(x\right)-\kappa \left(y\right)]$
The first Revan vertex Index [26]	${\mathcal{R}}_1\left(M\right)$-	${\sum }_{w=xy}{\left[\kappa \left(\overline{x}\right)\right]}^2$

Various methodologies are employed to calculate the results, including vertex and edge division and combinatorial techniques. The indices associated with degrees are computed manually utilizing a simple calculator, and the results’ reliability is validated using Python. The chemical structures of magnesium oxide are constructed using Chem-Draw. The molecular graph of magnesium oxide (111) is given in Figure 1.

Figure 1. Molecular Graph of $MgO\left(111\right)$.

The Magnesium Oxide $MgO\left(111\right)$ nanosheet has order $2(4p^2+4p+1)$ and size $12p^2+8p+1$. The vertex set is divided into three categories (or subsets) $X_1,X_2,X_3$ depending upon their degrees given in Table 2.

Table 2. Vertex Division of $MgO\left(111\right)$.

$\mathit{d}\left(\mathit{x}\right)$	Cardinality-	$\mathit{r}\left(\overline{\mathit{x}}\right)$
$X_1$	2-	3
$X_2$	$8p$-	2
$X_3$	$8p^2$-	1

The edge set of $MgO\left(111\right)$ has 4 categories $W_1,$$W_2,W_3,W_4$ depending upon the degrees of their end vertices that are given in Table 3. $W_1$ has two edges where $d\left(x\right)=1$ and $d\left(y\right)=3$. $W_2$ has two edges where $d\left(x\right)=2=d\left(y\right)$. $W_3$ with $d\left(x\right)=2$ and $d\left(y\right)=3$ contains $4(4p-1)$ edges. $W_4$ contains $4p(3p-2)+1$ edges where $d\left(x\right)=d\left(y\right)=3$.

Table 3. Edge Division of $MgO\left(111\right)$.

Edges	$\left(\mathit{d}\right(\mathit{x}),\mathit{d}(\mathit{y}\left)\right)$	Frequency	$\mathit{d}\left(\mathit{e}\right)$	$\mathit{\kappa }\left(\mathit{x}\right)$	$\mathit{\kappa }\left(\mathit{y}\right)$
$W_1$	(1,3)	2	2	3	1
$W_2$	(2,2)	2	2	2	2
$W_3$	(2,3)	$4(4p-1)$	3	2	1
$W_4$	(3,3)	$4p(3p-2)+1$	4	1	1

3. Results and Discussion

We computed the K-Banhatti and Revan indices using multiple approaches, such as vertex degree analysis and edge partitioning. Following that, we calculated entropy utilizing these indices. Python has been used for result validation and correlation analysis. By employing the data shown in Table 3 and Table 4, we can compute K-Banhatti indices, as given below.

Table 4. K-Banhatti and Revan indices of MgO(111).

p	${\mathcal{B}}_1\left(\mathit{M}\right)$	${\mathcal{B}}_2\left(\mathit{M}\right)$	${\mathcal{HB}}_1\left(\mathit{M}\right)$	${\mathcal{HB}}_2\left(\mathit{M}\right)$	${\mathcal{H}}_{\mathit{b}}\left(\mathit{M}\right)$	$\mathit{\eta }{\mathit{R}}_1\left(\mathit{M}\right)$	$\mathit{\eta }{\mathit{R}}_2\left(\mathit{M}\right)$	$\mathit{\eta }{\mathit{R}}_3\left(\mathit{M}\right)$	${\mathcal{R}}_1\left(\mathit{M}\right)$
1	234	404	1354	2988	15.790	192	103	16	58
2	802	1316	5074	12,924	43.523	448	195	32	114
3	1706	2804	11,146	29,772	84.971	800	311	48	186
4	2946	4868	19,570	53,532	140.133	1248	451	64	274
5	4522	7508	30,346	84,204	209.00	1792	615	80	378
6	6434	10,724	43,474	121,788	291.6	2432	803	96	498
7	8682	14,516	58,954	166,284	387.90	3168	1015	112	634
8	11,266	18,884	76,786	217,692	497.92	4000	1251	128	786
9	14,186	23,828	96,970	276,012	621.65	4928	1511	144	954
10	17,442	29,348	119,506	341,244	759.10	5952	1795	160	1138

• The first K-Banhatti Index

  $$\begin{array}{cc}\hfill {\mathcal{B}}_1\left(M\right)& =\sum _{xw}[d\left(x\right)+d\left(e\right)]\hfill \\ \hfill & =2\left[\right(1+2)+(3+2\left)\right]+2\left[\right(2+2)+(2+2\left)\right]+4(4p-1)\hfill \\ \hfill & \left[\right(2+3)+(3+3\left)\right]+\left(4p\right(3p-2)+1)\left[\right(3+4)+(3+4\left)\right]\hfill \\ \hfill & =32+44(4p-1)+56p(3p-2)+14\hfill \\ \hfill & =168p^2+64p+2\hfill \end{array}$$
• The second K-Banhatti Index

  $$\begin{array}{cc}\hfill {\mathcal{B}}_2\left(M\right)& =\sum _{xw}[d\left(x\right)\times d\left(e\right)]\hfill \\ \hfill & =2\left[\right(1\times 2)+(3\times 2\left)\right]+2\left[\right(2\times 2)+(2\times 2\left)\right]+4(4p-1)\hfill \\ \hfill & \left[\right(2\times 3)+(3\times 3\left)\right]+\left(4p\right(3p-2)+1)\left[\right(3\times 4)+(3\times 4\left)\right]\hfill \\ \hfill & =32+32+15(16p-4)+46p(3p-2)+96\hfill \\ \hfill & =288p^2+48p+68\hfill \end{array}$$
• The first hyper K-Banhatti Index

  $$\begin{array}{cc}\hfill {\mathcal{HB}}_1\left(M\right)& =\sum _{xw}{[d\left(x\right)+d\left(e\right)]}^2\hfill \\ \hfill & =2[{(1+2)}^2+{(3+2)}^2]+2[{(2+2)}^2+{(2+2)}^2]+4(4p-1)\hfill \\ \hfill & [{(2+3)}^2+{(3+3)}^2]+(4p(3p-2)+1)[{(3+4)}^2+{(3+4)}^2]\hfill \\ \hfill & =132+244(4p-1)+98\left[4p\right(3p-2)+1]\hfill \\ \hfill & =1176p^2+192p-14\hfill \end{array}$$
• The second hyper K-Banhatti Index

  $$\begin{array}{cc}\hfill {\mathcal{HB}}_2\left(M\right)& =\sum _{xw}{[d\left(x\right)\times d\left(e\right)]}^2\hfill \\ \hfill & =2[{(1\times 2)}^2+{(3\times 2)}^2]+2[{(2\times 2)}^2+{(2\times 2)}^2]+4(4p-1)\hfill \\ \hfill & [{(2\times 3)}^2+{(3\times 3)}^2]+(4p(3p-2)+1)[{(3\times 4)}^2+{(3\times 4)}^2]\hfill \\ \hfill & =74+468(4p-1)+288\left[4p\right(3p-2)+1]\hfill \\ \hfill & =3456p^2-432p-36\hfill \end{array}$$
• The K-Banhatti harmonic Index

  $$\begin{array}{cc}\hfill {\mathcal{H}}_b\left(M\right)& =\sum _{xw}[\frac{2}{d\left(x\right)+d\left(e\right)}]\hfill \\ \hfill & =2[\frac{2}{(1+2)}+\frac{2}{(3+2)}]+2[\frac{2}{(2+2)}+\frac{2}{(2+2)}]+4(4p-1)\hfill \\ \hfill & [\frac{2}{(2+3)}+\frac{2}{(3+3)}]+(4p(3p-2)+1)[\frac{2}{(3+4)}+\frac{2}{(3+4)}]\hfill \\ \hfill & =\frac{62}{15}+\frac{44}{15}(4p-1)+\frac{p}{7}(3p-2)+\frac{4}{7}\hfill \\ \hfill & =\frac{48p^2}{7}+\frac{752p}{105}+\frac{62}{35}\hfill \end{array}$$
• The first hyper Revan Index

  $$\begin{array}{cc}\hfill \eta R_1\left(M\right)& =\sum _{w=xy}{[\kappa \left(x\right)+\kappa \left(y\right)]}^2\hfill \\ \hfill & =2{(3+1)}^2+2{(2+2)}^2+4(4p-1){(2+1)}^2\hfill \\ \hfill & +[4p(3p-2)+1]{(1+1)}^2\hfill \\ \hfill & =32+32+36(4p-1)+4\left[4p\right(3p-2)+1]\hfill \\ \hfill & =48p^2+112p+32\hfill \end{array}$$
• The second hyper Revan Index

  $$\begin{array}{cc}\hfill \eta R_2\left(M\right)& =\sum _{w=xy}{[\kappa \left(x\right)\times \kappa \left(y\right)]}^2\hfill \\ \hfill & =2{\left(3\right)}^2+2{(2\times 2)}^2+4(4p-1){(2\times 1)}^2\hfill \\ \hfill & +[4p(3p-2)+1]{(1\times 1)}^2\hfill \\ \hfill & =18+32+16(4p-1)+\left[4p\right(3p-2)+1]\hfill \\ \hfill & =12p^2+56p+35\hfill \end{array}$$
• The third Revan Index

  $$\begin{array}{cc}\hfill \eta R_3\left(M\right)& =\sum _{w=xy}[\kappa \left(x\right)-\kappa \left(y\right)]\hfill \\ \hfill & =2(3-1)+2(2-2)+4(4p-1)(2-1)\hfill \\ \hfill & +\left[4p\right(3p-2)+1](1-1)\hfill \\ \hfill & =16p\hfill \end{array}$$
• The first Revan Vertex Index

  $$\begin{array}{cc}\hfill {\mathcal{R}}_1\left(M\right)& =\sum _{w=xy}{\left[r\left(\overline{x}\right)\right]}^2\hfill \\ \hfill & =2{\left(3\right)}^2+8p{\left(2\right)}^2+8p^2{\left(1\right)}^2\hfill \\ \hfill & =8p^2+32p+18\hfill \end{array}$$

Upon assessing Table 4 and Figure 2, it becomes clear that the expansion rate of the ${\mathcal{B}}_1\left(M\right)$ index is significantly higher than that of the ${\mathcal{B}}_2\left(M\right)$ index as p increases. Moreover, it is noticeable that when the value of p increases, the growth rate of the ${\mathcal{HB}}_2\left(M\right)$ index surpasses that of ${\mathcal{HB}}_1\left(M\right)$. Moving on, it can be observed that as p increases, the ${\mathcal{R}}_1\left(M\right)$ index grows far faster than ${\mathcal{H}}_b\left(M\right)$. Similarly, the $\eta R_1\left(M\right)$ index grows faster than the $\eta R_2\left(M\right)$ and $\eta R_3\left(M\right)$ indices as p increases. $\eta R_3\left(M\right)$ shows a very steady increase.

Figure 2. Graphical Representation of K-Banhatti and Revan Indices.

4. K-Banhatti Entropy and Linear Regression Analysis

In this section, the results of the entropy of the indices are given, along with the discussion on corresponding numerical outcomes. In addition, we include visualizations of topological indices for different parameter values. The following entropy values are found using the above-estimated indices and the values from Table 3 in Equation (1).

• The first K-Banhatti entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{B}}_1}& =log\left({\mathcal{B}}_1\right)-\frac{1}{{\mathcal{B}}_1}log[\prod _{xy\in W\left(M\right)}{(d\left(x\right)+d\left(e\right))}^{\left(d\right(x)+d(e\left)\right)}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{B}}_1}& =log(168p^2+64p+2)-\frac{log(2\times {\left(8\right)}^8)}{168p^2+64p+2}-\frac{log(2\times {\left(8\right)}^8)}{168p^2+64p+2}\hfill \\ \hfill & -\frac{log(4(4p-1)\times {\left(11\right)}^{11})}{168p^2+64p+2}-\frac{log((4p(3p-2)+1)\times {\left(14\right)}^{14})}{168p^2+64p+2}\hfill \end{array}$$
• The second K-Banhatti entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{B}}_2}& =log\left({\mathcal{B}}_2\right)-\frac{1}{{\mathcal{B}}_2}log[\prod _{xy\in W\left(M\right)}{(d\left(x\right)\times d\left(e\right))}^{\left(d\right(x)\times d(e\left)\right)}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{B}}_2}& =log(288p^2+48p+68)-\frac{log(2\times {\left(8\right)}^8)}{288p^2+48p+68}-\frac{log(2\times {\left(8\right)}^8)}{288p^2+48p+68}\hfill \\ \hfill & -\frac{log(4(4p-1)\times {\left(15\right)}^{15})}{288p^2+48p+68}-\frac{log((4p(3p-2)+1)\times {\left(24\right)}^{24})}{288p^2+48p+68}\hfill \end{array}$$
• The first hyper K-Banhatti entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{HB}}_1\left(M\right)}& =log\left({\mathcal{HB}}_1\left(M\right)\right)-\frac{1}{{\mathcal{HB}}_1\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left({(d\left(x\right)+d\left(e\right))}^2\right)}^{{\left(d\right(x)+d(e\left)\right)}^2}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{HB}}_1\left(M\right)}& =log(1176p^2+192p-14)-\frac{log(2\times {\left(34\right)}^{34})}{1176p^2+192p-14}\hfill \\ \hfill & -\frac{log(2\times {\left(32\right)}^{32})}{1176p^2+192p-14}-\frac{log(4(4p-1)\times {\left(61\right)}^{16})}{1176p^2+192p-14}\hfill \\ \hfill & -\frac{log\left((4p(3p-2)+1){\left(98\right)}^{98}\right)}{1176p^2+192p-14}\hfill \end{array}$$
• The second hyper K-Banhatti entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{HB}}_2\left(M\right)}& =log\left({\mathcal{HB}}_2\left(M\right)\right)-\frac{1}{{\mathcal{HB}}_2\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left({(d\left(x\right)\times d\left(e\right))}^2\right)}^{{\left(d\right(x)\times d(e\left)\right)}^2}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{HB}}_2\left(M\right)}& =log(3456p^2-432p-36)-\frac{log(2\times {\left(40\right)}^{40})}{3456p^2-432p-36}\hfill \\ \hfill & -\frac{log(2\times {\left(32\right)}^{32})}{3456p^2-432p-36}-\frac{log(4(4p-1)\times {\left(117\right)}^{117})}{3456p^2-432p-36}\hfill \\ \hfill & -\frac{log((4p(3p-2)+1)\times {\left(288\right)}^{288})}{3456p^2-432p-36}\hfill \end{array}$$
• The K-Banhatti harmonic entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{H}}_b\left(M\right)}& =log\left({\mathcal{H}}_b\left(M\right)\right)-\frac{1}{{\mathcal{H}}_b\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left(\frac{2}{d\left(x\right)+d\left(e\right)}\right)}^{\left(\frac{2}{d\left(x\right)+d\left(e\right)}\right)}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{H}}_b\left(M\right)}& =log(6.857p^2+7.162p+1.77)-\frac{log(2\times {\left(\frac{16}{15}\right)}^{\frac{16}{15}})}{6.857p^2+7.162p+1.77}\hfill \\ \hfill & -\frac{log(2\times {\left(1\right)}^1)}{6.857p^2+7.162p+1.77}-\frac{log(4(4p-1)\times {\left(\frac{11}{15}\right)}^{\frac{11}{15}})}{6.857p^2+7.162p+1.77}\hfill \\ \hfill & -\frac{log((4p(3p-2)+1)\times {\left(\frac{4}{7}\right)}^{\frac{4}{7}})}{6.857p^2+7.162p+1.77}\hfill \end{array}$$
• The first hyper Revan entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{\eta R_1\left(M\right)}& =log\left(\eta R_1\left(M\right)\right)-\frac{1}{\eta R_1\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left({(\kappa \left(x\right)+\kappa \left(y\right))}^2\right)}^{{(\kappa \left(x\right)+\kappa \left(y\right))}^2}\hfill \\ \hfill {\mathcal{E}}_{\eta R_1\left(M\right)}& =log(48p^2+112p+32)-\frac{log(2\times {\left(16\right)}^{16})}{48p^2+112p+32}\hfill \\ \hfill & -\frac{log(2\times {\left(16\right)}^{16})}{48p^2+112p+32}-\frac{log(4(4p-1)\times {\left(9\right)}^9)}{48p^2+112p+32}\hfill \\ \hfill & -\frac{log((4p(3p-2)+1)\times {\left(4\right)}^4)}{48p^2+112p+32}\hfill \end{array}$$
• The second hyper Revan entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{\eta R_2\left(M\right)}& =log\left(\eta R_2\left(M\right)\right)-\frac{1}{\eta R_2\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left({(\kappa \left(x\right)\times \kappa \left(y\right))}^2\right)}^{{(\kappa \left(x\right)\times \kappa \left(y\right))}^2}\hfill \\ \hfill {\mathcal{E}}_{\eta R_2\left(M\right)}& =log(12p^2+56p+35)-\frac{log(2\times {\left(9\right)}^9)}{12p^2+56p+35}\hfill \\ \hfill & -\frac{log(2\times {\left(16\right)}^{16})}{12p^2+56p+35}-\frac{log(4(4p-1)\times {\left(4\right)}^4)}{12p^2+56p+35}\hfill \\ \hfill & -\frac{log\left((4p(3p-2)+1){\left(1\right)}^1\right)}{12p^2+56p+35}\hfill \end{array}$$
• The third Revan entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{\eta R_3\left(M\right)}& =log\left(\eta R_3\left(M\right)\right)-\frac{1}{\eta R_3\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{(\kappa \left(x\right)-\kappa \left(y\right))}^{\left(\kappa \right(x)-\kappa (y\left)\right)}]\hfill \\ \hfill {\mathcal{E}}_{\eta R_3\left(M\right)}& =log\left(16p\right)-\frac{log(2\times {\left(2\right)}^2)}{16p}-\frac{log(2\times {\left(0\right)}^0)}{16p}\hfill \\ \hfill & -\frac{log(4(4p-1)\times {\left(1\right)}^1)}{16p}-\frac{log((4p(3p-2)+1)\times {\left(0\right)}^0)}{16p}\hfill \end{array}$$
• The first Revan vertex entropy

  $$\begin{array}{cc}\hfill {\mathcal{E}}_{{\mathcal{R}}_1\left(M\right)}& =log\left({\mathcal{R}}_1\left(M\right)\right)-\frac{1}{{\mathcal{R}}_1\left(M\right)}\hfill \\ \hfill & log[\prod _{xy\in W\left(M\right)}{\left({\left(r\left(\overline{x}\right)\right)}^2\right)}^{{\left(r\left(\overline{x}\right)\right)}^2}]\hfill \\ \hfill {\mathcal{E}}_{{\mathcal{R}}_1\left(M\right)}& =log(8p^2+32p+18)-\frac{log(2\times {\left(9\right)}^9)}{8p^2+32p+18}\hfill \\ \hfill & -\frac{log(8p\times {\left(4\right)}^4)}{8p^2+32p+18}-\frac{log(8p^2\times {\left(1\right)}^1)}{8p^2+32p+18}\hfill \end{array}$$

Upon assessing Table 5 and Figure 3, it becomes clear that the expansion rate of the ${\mathcal{E}}_{{\mathcal{B}}_1\left(M\right)}$ index and the ${\mathcal{E}}_{{\mathcal{B}}_2\left(M\right)}$ index shows almost same growth as p increases. Similarly, it is noticeable that when the value of p increases, the growth rate of the ${\mathcal{E}}_{{\mathcal{HB}}_2\left(M\right)}$ index surpasses that of ${\mathcal{E}}_{{\mathcal{HB}}_1\left(M\right)}$.

Table 5. Entropy of K-Banhatti and Revan indices of MgO(111).

p	${\mathcal{E}}_{{\mathcal{B}}_1\left(\mathit{M}\right)}$	${\mathcal{E}}_{{\mathcal{B}}_2\left(\mathit{M}\right)}$	${\mathcal{E}}_{{\mathcal{HB}}_1\left(\mathit{M}\right)}$	${\mathcal{E}}_{{\mathcal{HB}}_2\left(\mathit{M}\right)}$	${\mathcal{E}}_{{\mathcal{H}}_{\mathit{b}}\left(\mathit{M}\right)}$	${\mathcal{E}}_{\mathit{\eta }{\mathit{R}}_1\left(\mathit{M}\right)}$	${\mathcal{E}}_{\mathit{\eta }{\mathit{R}}_2\left(\mathit{M}\right)}$	${\mathcal{E}}_{\mathit{\eta }{\mathit{R}}_3\left(\mathit{M}\right)}$	${\mathcal{E}}_{{\mathcal{R}}_1\left(\mathit{M}\right)}$
1	5.019	5.616	6.655	7.181	2.442	4.634	3.904	2.343	3.540
2	6.556	7.062	8.383	9.276	3.595	5.831	4.873	3.165	4.453
3	7.379	7.881	9.251	10.218	4.334	6.530	5.484	3.642	5.045
4	7.951	8.457	9.843	10.842	4.870	7.02	5.933	3.972	5.487
5	8.392	8.902	10.295	11.311	5.290	7.42	6.289	4.223	5.842
6	8.752	9.265	10.662	11.689	5.636	7.744	6.586	4.426	6.139
7	9.056	9.571	10.971	12.006	5.930	8.020	6.841	4.595	6.395
8	9.319	9.837	11.238	12.279	6.185	8.262	7.065	4.741	6.620
9	9.552	10.071	11.474	12.519	6.412	8.476	7.265	4.868	6.822
10	9.760	10.281	11.684	12.733	6.615	8.670	7.446	4.982	7.004

Figure 3. Graphical Representation of Entropy of Banhatti and Revan Indices.

Moving on, it can be observed that as p increases, the ${\mathcal{E}}_{{\mathcal{H}}_b\left(M\right)}$ index grows far faster than ${\mathcal{E}}_{{\mathcal{R}}_1\left(M\right)}$. Similarly, the ${\mathcal{E}}_{{\eta R}_1\left(M\right)}$ index grows faster than the ${\mathcal{E}}_{\eta R_2\left(M\right)}$ and ${\mathcal{E}}_{{\eta R}_3\left(M\right)}$ indices as p increases. ${\mathcal{E}}_{{\eta R}_3\left(M\right)}$ shows a very steady increase.

This section explores the correlation between K-Banhatti and Revan indices and the corresponding entropy values. Academic researchers heavily depend on graphical and computational representations of their findings to optimize efficiency and reduce the need for costly laboratory procedures. The current investigation employed a particular research methodology to analyze the correlation between the advancement of entropy and many other variables. Linear regression is employed to assess the relationship between entropy and the indices. This methodology entailed the manipulation of fundamental factors. The analysis’s performance was evaluated using the RMSE. The simulations were performed utilizing Microsoft Excel software. The coefficients are displayed in Table 6.

Table 6. Linear Regression Analysis of Entropy of K-Banhatti and Revan indices of $MgO\left(111\right)$.

${\mathcal{E}}_{\mathcal{I}\left(\mathit{M}\right)}$	a	b	$\mathit{RMSE}$
${\mathcal{E}}_{{\mathcal{B}}_1\left(M\right)}$	1.43	12.40	1.11
${\mathcal{E}}_{{\mathcal{B}}_2\left(M\right)}$	1.43	12.94	1.11
${\mathcal{E}}_{{\mathcal{HB}}_1\left(M\right)}$	1.44	14.34	1.12
${\mathcal{E}}_{{\mathcal{HB}}_2\left(M\right)}$	1.45	15.40	1.15
${\mathcal{E}}_{{\mathcal{H}}_b\left(M\right)}$	1.41	9.24	1.07
${\mathcal{E}}_{{\eta R}_1\left(M\right)}$	1.38	11.23	1.03
${\mathcal{E}}_{{\eta R}_2\left(M\right)}$	1.34	9.92	0.96
${\mathcal{E}}_{{\eta R}_3\left(M\right)}$	0.86	6.36	0.443
${\mathcal{E}}_{{\mathcal{R}}_1\left(M\right)}$	1.34	9.50	0.97

5. Conclusions

We calculated the K-Banhatti indices in this article. We derived analytical formulas by generalizing the K-Banhatti topological descriptors of $MgO\left(111\right)$. Subsequently, we utilize our results in entropy equations to determine the entropies of K-Banhatti indices of $MgO\left(111\right)$. The transition state of $MgO\left(111\right)$ undergoes a significant rise as the p increases, resulting in the most substantial modification in entropy. This research employed linear regression to evaluate the relationship between entropy and the K-Banhatti index. To evaluate the precision of our findings, we utilized statistical metrics such as RMSE. Based on the findings of the analysis carried out, it has been seen that the third Revan index constantly yields the most favorable and progressively improving results. This may be attributed to its higher accuracy regarding the root mean square error (RMSE). The results were reported in both numerical and graphical formats. If the experimental values of the physical properties of $MgO\left(111\right)$ are known, it is easy to verify which one of these indices is better for predicting the corresponding physical properties of $MgO\left(111\right)$.