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
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
and
nanosheets and computed some topological indices by using
M-polynomials. Ref. [
12] calculated the precise values of
K-Banhatti Indices of
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
, 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:
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 (magnesium oxide), which serves various purposes. 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
thin films with two distinct facets, namely (111) and (100), has been achieved. As a very effective catalyst and inert substrate, the
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
’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
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
.
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
. The vertex
x degree is represented by
and indicates the total edges connected to that vertex. Recently, an innovative concept of the edge degree has been introduced, denoted as
. The maximum degree of the graph can be represented by
and
,
and
.
Table 1 contains the formulas of all the indices under consideration.
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.
The Magnesium Oxide
nanosheet has order
and size
. The vertex set is divided into three categories (or subsets)
depending upon their degrees given in
Table 2.
The edge set of
has 4 categories
depending upon the degrees of their end vertices that are given in
Table 3.
has two edges where
and
.
has two edges where
.
with
and
contains
edges.
contains
edges where
.
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
The second
K-Banhatti entropy
The first hyper
K-Banhatti entropy
The second hyper
K-Banhatti entropy
The
K-Banhatti harmonic entropy
The first hyper Revan entropy
The second hyper Revan entropy
The first Revan vertex entropy
Upon assessing
Table 5 and
Figure 3, it becomes clear that the expansion rate of the
index and the
index shows almost same growth as
p increases. Similarly, it is noticeable that when the value of
p increases, the growth rate of the
index surpasses that of
.
Moving on, it can be observed that as p increases, the index grows far faster than . Similarly, the index grows faster than the and indices as p increases. 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.