A Paradigmatic Approach to Find the Valency-Based K-Banhatti and Redefined Zagreb Entropy for Niobium Oxide and a Metal–Organic Framework

Entropy is a thermodynamic function in chemistry that reflects the randomness and disorder of molecules in a particular system or process based on the number of alternative configurations accessible to them. Distance-based entropy is used to solve a variety of difficulties in biology, chemical graph theory, organic and inorganic chemistry, and other fields. In this article, the characterization of the crystal structure of niobium oxide and a metal–organic framework is investigated. We also use the information function to compute entropies by building these structures with degree-based indices including the K-Banhatti indices, the first redefined Zagreb index, the second redefined Zagreb index, the third redefined Zagreb index, and the atom-bond sum connectivity index.


Introduction
The optical properties of metallic nanoparticles have drawn the attention of scientists and researchers. The heat created by the nanoparticles overwhelms cancer tissue while causing no harm to healthy cells. Niobium nanoparticles have the capacity to easily attach to ligands, making them ideal for optothermal cancer treatment. Chemical graph theory is a contemporary branch of applied chemistry, which has remained an attractive area of research for scientists during the past two decades, and significant contributions have been made by scientists in this area of research including [1][2][3][4][5][6][7]. We investigate the relationship between atoms and bonds using combinatorial approaches such as vertex and edge partitions. Topological indices are essential in providing directions for treating malignancies or tumors. These indices can be obtained experimentally or numerically. Although experimental data are valuable, they are also costly; therefore, computational analysis gives a cost-effective and time-efficient solution.
The transformation of a chemical structure into a number is used to generate a topological index. The topological index is a graph invariant that characterizes the graph's topology while remaining invariant throughout graph automorphism. A topological index is a numerical number defined only by the graph. The eccentricity-based topological indices are crucial in chemical graph theory [8]. Wiener, a chemist, first used a topological index in 1947 while researching the relationship between the molecular structure and the physical and chemical properties of certain hydrocarbon compounds [9]. In 2010, Damir et al. defined the redefined second Zagreb index as the same as the inverse sum indeg index [10].
We used the concept of valency-based entropies in this article, where vȧ 1 and vȧ 2 denote the valency of atoms,ȧ 1 andȧ 2 , within the molecule. Kulli started computing valency-based topological indices in 2016 using the valency of atom bonds and some Banhatti indices [11][12][13], each of which has the following definition: The first valence-based K-Banhatti polynomial and the first K-Banhatti index are as follows: The second valence based K-Banhatti polynomial and the second K-Banhatti index are as follows, respectively: The first valence based hyper K-Banhatti polynomial and the firstst hyper K-Banhatti index are as follows, respectively: In 2013, Ranjini [14] introduced a redefined version of the Zagreb indices ReZG 1 , and in 2021, Shanmukha [15] defined them as The third redefined Zagreb index was defined as Recently, Ali et al. amalgamated the atom-bond connectivity index and sum connectivity index and initiated the new molecular descriptor named the atom-bond sumconnectivity index [16], defined as: Shannon first popularized the concept of entropy in his 1948 work [17]. Entropy is the quantity of thermal energy per unit temperature in a system that is not accessible for meaningful work [18,19]. Because the work is derived from organized molecular motion, entropy is also a measure of a system's molecular disorder or unpredictability [20,21]. In this article, we build the Niobium dioxide NbO 2 and the metal-organic framework (MOF) to compute the K-Banhatti and redefined Zagreb entropies using K-Banhatti indices [22][23][24], and redefined Zagreb indices, respectively. The idea of entropy is extracted from Shazia Manzoor's paper [25].

Now, by inserting these values into Equation
•

Niobium Dioxide NbO 2
Niobium Nb, a refractory metal, is a good choice for the initial shell of nuclear fusion reactors. It does, however, have a strong attraction for O 2 and C, both of which are available in pyrotechnics and refrigerant-like liquids. As part of the first barrier, Nb is well known for its ability to interact very effectively with O 2 [27]. As a result, reliable thermodynamic data on NbO, NbO 2 , Nb 2 O 5 , and other intermediate phases, such as Nb 12 O 29 , are very effective. In transistors, niobium monoxide is used as a gate electrode, and a (NbO/NbO 2 ) junction may be used in robust switching devices. In this article, we will attempt to explain NbO 2 , which has a total atom count of 2 + 5s + 5t + 9st; see Figure 1. There are three types of atoms in NbO 2 based on their valency: eight atoms with valency 2, 8s + 8t + 4st − 8 atoms with valency 3, and 2 − 3s − 3t + 5st atoms with valency 4. Table 1 shows the atom-bond partitions of NbO 2 derived from these results. Table 1. Atom-bond partition of NbO 2 .

Types of Atom Bonds
Cardinality of Atom bonds 16 The first K-Banhatti entropy of NbO 2 Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (1) and Table 1, the first K-Banhatti polynomial is After simplifying Equation (18), we obtain the first K-Banhatti index by taking the first derivative at x = 1.
Now, we compute the first K-Banhatti entropy of NbO 2 by using Table 1 and Equation (19) in Equation (10) in the following way: • The second K-Banhatti entropy of NbO 2 Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (2) and Table 1, the second K-Banhatti polynomial is Taking the first derivative of Equation (20) Now, we compute the second K-Banhatti entropy of NbO 2 by using Table 1 and Equation (21) in Equation (11) in the following way: 16 .
• The first K-hyper Banhatti entropy of NbO 2 Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (3) and Table 1, the first K-hyper Banhatti polynomial is Taking the first derivative of Equation (22) at x = 1, we obtain the first K-hyper Banhatti index Now, we compute the first K-hyper Banhatti entropy of NbO 2 by using Table 1 and Equation (23) in Equation (13) in the following way: • The second K-hyper Banhatti entropy of NbO 2 Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (4) and Table 1, the second K-hyper Banhatti polynomial is Taking the first derivative of Equation (24) Now, we compute the second K-hyper Banhatti entropy of NbO 2 by using Table 1 and Equation (25) in Equation (13) in the following way: • The first redefined Zagreb entropy of NbO 2 Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (5) and Table 1, the first redefined Zagreb polynomial is Taking the first derivative of Equation (26) at x = 1, we obtain the first redefined Zagreb index ReZG 1 (NbO 2 ) = 9st + 5s + 5t + 2.
Now, we compute the first redefined Zagreb entropy by using Table 1 and Equation (27) in Equation (14) in the following way: •

The second redefined Zagreb entropy of NbO 2
Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (6) and Table 1, the second redefined Zagreb polynomial is Taking the first derivative of Equation (28) at x = 1, we obtain the second redefined Zagreb index Now, we compute the second redefined Zagreb entropy by using Table 1 and Equation (29) in Equation (15) in the following way: . •

The third redefined Zagreb entropy of NbO 2
Let NbO 2 be a network of a niobium dioxide molecule. Then, by using Equation (7) and Table 1, the third redefined Zagreb polynomial is Taking the first derivative of Equation (30) at x = 1, we obtain the third redefined Zagreb index Now, we compute the third redefined Zagreb entropy by using Table 1 and Equation (31) in Equation (16) in the following way: •

Atom-bond sum connectivity entropy of NbO 2
Let NbO 2 be a network of a niobium dioxide molecule. Then, using Equation (8) and Table 1, the atom-bond sum connectivity polynomial is Taking the first derivative of Equation (32) at x = 1, we obtain the atom-bond sum connectivity index Now, we compute the atom-bond sum connectivity entropy by using Table 1 and Equation (33) in Equation (17) in the following way:

Metal-Organic Framework
Metal-organic frameworks are distinguished by their three-dimensional frameworks composed of metallic ions. This metal-organic framework has the molecular formula FeTPyP-Co, where Fe denotes iron, TPyP denotes tetrakis pyridyl porphyrin, and Co denotes cobalt [28]. All metal ions and organic molecules in the MOF (s,t) network can accommodate a wide range of guest molecules. Metal-organic frameworks have several uses, including as energy storage devices, gas storage, heterogeneous catalysis, and chemical evaluation. We will examine a 2D structure of a metal-organic framework called MOF (s,t) , where s and t are the unit cells in a row and column, respectively. The MOF (2,2) is shown in Figure 3. There are 74st atoms in the MOF (s,t) , and 2(44st − s − t) + 1 atom-bonds are used, as Figure 3 of MOF (2,2) demonstrates. The atom-bonds partition of the MOF (s,t) is shown in Table 3. Table 3. Atom-bonds partition of MOF (s,t) .
• The first K-hyper Banhatti entropy of MOF (s,t) Let MOF (s,t) be a metal-organic framework. Then, using Equation (3) and Table 3, the first K-hyper Banhatti polynomial is Taking the first derivative of Equation (39) at x = 1, we obtain the first K-hyper Banhatti index Now, we compute the first K-hyper Banhatti entropy of MOF (s,t) by using Table 3 and Equation (40) in Equation (12) in the following way: After simplification, we obtain • The second K-hyper Banhatti entropy of MOF (s,t) Let MOF (s,t) be a metal-organic framework. Then, by using Equation (4) and Table 3, the second K-Banhatti polynomial is x (3×4) 2 = (24st + 1)x 9 + 6(s + t − 1))x 36 + 2(28st − 2s − 2t + 1)x 81 Taking the first derivative of Equation (41) at x = 1, we obtain the second K-hyper Banhatti index Now, we compute the second K-hyper Banhatti entropy of MOF (s,t) by using Table 3 and Equation (42) in Equation (13) in the following way: After simplification, we obtain •

The first redefined Zagreb entropy of MOF (s,t)
Let MOF (s,t) be a metal-organic framework. Then, using Equation (5) and Table 3, the first redefined Zagreb polynomial is Taking the first derivative of Equation (44) at x = 1, we obtain the first redefined Zagreb index ReZG 1 (MOF (s,t) ) = 2(37st + 2).
Now, we compute the first redefined Zagreb entropy using Table 3 and Equation (45) in Equation (14) in the following way: .
After simplification, we obtain • The second redefined Zagreb entropy of MOF (s,t) Let MOF (s,t) be a metal-organic framework. Then, using Equation (6) and Table 3, the second redefined Zagreb polynomial is Taking the first derivative of Equation (46) at x = 1, we obtain the second redefined Zagreb index Now, we compute the second redefined Zagreb entropy by using Table 3 and Equation (47) in Equation (15) in the following way: . .