Characterizations of Chemical Networks Entropies by K -Banhatii Topological Indices

: Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of conﬁgurations that molecules might take. Distance-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. We explain the basic applications of distance-based entropy to chemical phenomena. These applications include signal processing, structural studies on crystals, molecular ensembles, and quantifying the chemical and electrical structures of molecules. In this study, we examine the characterisation of polyphenylenes and boron ( B 12 ) using a line of symmetry. Our ability to quickly ascertain the valences of each atom, and the total number of atom bonds is made possible by the symmetrical chemical structures of polyphenylenes and boron B 12 . By constructing these structures with degree-based indices, namely the K Banhatti indices, ReZG 1 -index, ReZG 2 -index, and ReZG 3 -index, we are able to determine their respective entropies.


Introduction
In mathematical chemistry, topological indices are numerical values that describe the topology of molecular structures.The chemical process conceptual framework is a significant area of applied mathematics.This theory can be used to model issues in the real world.Since their inception, chemical networks have drawn the attention of researchers because of their widespread applications.The correlation coefficient (r) between the physicochemical characteristics and topological indices is determined in order to assess the utility of a topological index to forecast the physicochemical behavior of a chemical compound [1,2].Additionally, it falls within a class of challenging chemical graph theory applications for precise molecular issue resolutions.In the fields of chemical sciences and chemical graph theory, this theory is crucial.The QSAR/QSPR analysis included physiochemical properties and topological indices such as the 1 st multiple Zagreb index, 2 nd multiple Zagreb index, and hyper Zagreb index [3,4].Recently, Ali et al. defined the atom-bond sum-connectivity index in [5].
Let G(V G , E G ) be a graph, with the vertices and edges denoted by V G and E G , respectively.In chemical graph theory, a molecular graph is a simple connected graph that contains chemical atoms and bonds, which are commonly referred to as atoms and atom-bonds, respectively [6,7].
The valency of atom bonds (line segments) and a few Banhatti indices, each of which had the following description [8], was used by Kulli to begin constructing valency-based topological indices in 2016 [9][10][11].
The 1 st and 2 nd K-Banhatti indices are as follows, respectively: The redefined Zagreb indices ReZG1, ReZG2, and ReZG3 were 1 st proposed by Ranjini [12] in 2013: The 3 rd redefined Zagreb index is defined as Entropy is the measurement of the amount of thermal energy per unit of temperature in a system that cannot be used for productive labour.Entropy is a measure of a system's molecular disorder or unpredictability since work is produced by organised molecular motion, see new works on graph theory in [13,14].Entropy was initially discussed by Shannon in his well-known [15] from 1948.The entropy of a probability distribution measures the unpredictable nature of information content or the uncertainty of a system.Entropy was then used to analyse chemical networks and graphs in order to comprehend the structural information contained within these networks.Recently, graph entropies have become more well-liked in a variety of academic disciplines, including biology, chemistry, ecology, and sociology.Graph theory and network theory have both undertaken substantial study on invariants, which are utilised as information functionals in science and have been around for a long time.The degree of every atom is vitally crucial [16][17][18].The following sections will discuss graph entropy measures that have been applied to analyse biological and chemical networks in chronological order for further information [19].

Entropy Related to Valency-Based Indices
The entropy of an edge-weighted graph G is defined in [34], which was published in 2009.Ghani et al. defined the modified definition of entropy in [35].A network with lines that are weighted has the equation G = (V G , E G ), µ(a i a j )), where V G , E G , and the vertex set, edge set, and edge-weight of edge (a i a j ) are each represented by µ(a i a j ): • Entropy related to the 1 st K-Banhatti index Assume µ(a i a j ) = d a i + d a j .Then, the 1 st K-Banhatti B 1 index (1) is thus provided by By putting these parameters into Equation ( 6), the 1 st K-Banhatti entropy is • Entropy related to the 2 nd K-Banhatti index Assume µ(a i a j ) = d a i × d a j .Then, the 2 nd K-Banhatti B 2 index (1) is thus provided by µ(a i a j ).
By putting these parameters into Equation (6), the 2 nd K-Banhatti entropy is • Entropy related to the 1 st K hyper Banhatti index Assume µ(a i a j ) = (d a i + d a j ) 2 .Then, the 1 st K hyper Banhatti HB 1 index (2) is thus provided by By putting these parameters into Equation ( 6), the 1 st K hyper Banhatti entropy is • Entropy related to the 2 nd K hyper Banhatti index Assume µ(a i a j ) = (d a i × d a j ) 2 .Then, the 2 nd K hyper Banhatti index (2) is thus provided by By putting these parameters into Equation ( 6), the 2 nd K hyper Banhatti entropy is •

The first redefined Zagreb entropy
Assume µ(a i a j ) = d a i +d a j d a i d a j . Then, the 1 st redefined Zagreb index (4) is thus provided by By putting these parameters into Equation ( 6), the 1 st redefined Zagreb entropy is •

The second redefined Zagreb entropy
Assume µ(a i a j ) = . Then, the 2 nd redefined index ( 4) is thus provided by By putting these parameters into Equation ( 6), the 2 nd redefined Zagreb entropy is •

The third redefined Zagreb entropy
Assume µ(a i a j ) = (d a i d a j )(d a i + d a j ) .Then, the 3 rd redefined Zagreb index ( 5) is thus provided by By putting these parameters into Equation ( 6), the 3 rd redefined Zagreb entropy is

The Boron Network
We discuss the topological characteristics of boron B 12 in this article.The icosahedral network of boron B 12 has two dimensions.The existence of icosahedral structures containing B 12 was confirmed by a recent investigation of high-pressure solid boron [36].However, prior theoretical and practical investigations of multiple boron clusters have demonstrated that the B 12 structure is unstable in the gas phase [37][38][39].Figure 1 displays the molecular graph of boron B 12 .The dotted line in Figure 1 represents the line of symmetry; using this line, we can easily obtain the edge-partition of B 12 .

Results and Discussion
Now, (s, t) are the units of B 12 , where s and t show the number of B 12 in horizontal rows and vertical columns.In the boron network, the edge set E(G) is divided into seven groups based on the degree of each edge's end vertices.The set that is disjoint is shown by the symbols ξ (d(u i ),d(v j )) .The 1 st set that is disjoint is ξ (2,4) , the 2 nd set that is disjoint is ξ (2,5) , the 3 rd set that is disjoint is ξ (3,4) , the 4 th set that is disjoint is ξ (3,5) , the 5 th set that is disjoint is ξ (4,4) , the 6 th set that is disjoint is ξ (4,5) , and the 7 th set that is disjoint is ξ (5,5) .
From the symmetrical chemical structure of boron, B 12 , we find the edge-partition of B 12 easily: This partition provides Assume that B 12 is an icosahedral network of boron.Then, by using Equation ( 1) and the edge-partition of B 12 , the 1 st K-Banhatti index is By using the edge-partition of B 12 and Equation ( 7) as described below, 10 .
After simplifying the preceding expression, the following equation yields the precise value of the 1 st K-Banhatti entropy: × (9st + 7s + 7t + 3)(10) 10 .( • The second K-Banhatti entropy of B 12 Assume that B 12 is an icosahedral network of boron.Then, by using Equation ( 1) and the edge-partition of B 12 , the 2 nd K-Banhatti entropy index is By using the edge-partition of B 12 and Equation (8) as described below, × 2(s + t)10 10 × (3st + s + 5)12 12 × (3st + 2s + 3t + 4)15 15 × (s + 2t + 1)16 16   × (9st + 7s + 6t + 5)20 20 × (9st • The 1 st hyper K-Banhatti entropy of B 12 Assume that B 12 is an icosahedral network of boron.Then, by using the Equation ( 2) and the edge-partition of B 12 , the 1 st K hyper Banhatti index is By using the edge-partition of B 12 and Equation (9) as described below; Assume that B 12 is an icosahedral network of boron.Then, by using Equation ( 2) and the edge-partition of B 12 , the 2 nd K hyper Banhatti index is HB 2 (B 12 ) = 8233s + 8290t + 9972st + 5151 By using the edge-partition of B 12 and Equation (10) as described below, Assume that B 12 is an icosahedral network of boron.Then, by using Equation ( 4) and the edge-partition of B 12 , the 1 st redefined Zagreb index is ReZG 1 (B 12 ) = 11st + 11s + 11t + 9 By using the edge-partition of B 12 and Equation ( 11) as described below, ) ) ×(s ) ) • The 2 nd redefined Zagreb entropy of B 12 Assume that B 12 is an icosahedral network of boron.Then, by using Equation ( 4) and the edge-partition of B 12 , the 2 nd redefined Zagreb index is By using the edge-partition of B 12 and Equation ( 12) as described below, ) ) ) ×(3st ) ) • The 3 rd redefined Zagreb entropy of B 12 Assume that B 12 is a hexagonal grid of benzenoid.Then, by using Equation ( 5) and the edge-partition of B 12 , the 3 rd redefined Zagreb index is Now, we compute the 3 rd redefined Zagreb entropy by using the edge-partition of B 12 and Equation ( 13) as described below,

The Polyphenylenes Network
Polyphenylenes include benzenoid aromatic nuclei that are linked together by a bond [40].Polyphenylenes have been the subject of research for many years.Up until 1979, there was a lot of interest in polyphenylenes because of its thermal and thermo-oxidative stability.The quest for a single group of polymers that can be converted to another, such as an electrical insulator that can be transformed into an electrical conductor by utilizing doping with an electron acceptor or donor, is a key ongoing subject in polyphenylene [41].
Figure 3 for polyphenylenes P [s,t] has been shown.In 2011, Zhen Zhou [42] indicated that two-dimensional polyphenylene is a conventional semiconductor with a large band gap and that the porous structure gives it a remarkable selectivity for H 2 permeability compared to CO 2 , CO, and CH 4 .This porous graphene, which has been tested, is likely to find use in a hydrogen-powered civilization.In Figure 3, the dot lines represent the line of symmetry, with 6s(t + 1) atoms on the left side and the same on the right.Here, in the P [s,t] , there are two sorts of atoms v i and a j such that d v i = 2 and d a j = 3, where d v i and d a j mean the valency of atoms ∀ v i , v j ∈ P [ Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.Equation (1) and Table 2 are then used to calculate the 1 st K Banhatti index:

Line of symmetery
By using Table 2 and Equation ( 7) as follows: After simplification, we obtain • Entropy related to 2 nd K-Banhatti index of P [s,t] Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.Then, by using Equation (1) and Table 2, the 2 nd K-Banhatti index is By using Table 2 and Equation ( 8) as described below: • Entropy related to the 1 st K hyper Banhatti index of P [s,t] Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.The 1 st hyper Banhatti index is then determined using Equation (2) and Table 2 HB 1 (P [s,t] ) = 8(118st + 17s + 25t) By using Table 2 and Equation (10) as described below, Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.The 2 nd K hyper Banhatti index is then calculated using the Equation (2) and Table 2: By using Table 2 and Equation (10) as described below, • Entropy related to the 1 st redefined Zagreb index of P [s,t] Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.The 1 st redefined Zagreb index is then obtained by using Equation ( 4) and Table 2 ReZG 1 (P [s,t] ) = 8(12st + s + 2t) By using Table 2 and Equation ( 11) as described below, Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.The 3 rd redefined Zagreb index is then obtained using Equation ( 5) and Comparison of K-Banhatti and Redefined Zagreb Indices of P (s,t) In this section, we present numerical and graphical comparison of B 1 , B 2 3, HB 1 , HB 1 , ReZG 1 , ReZG 2 and ReZG 3 , of polyphenylenes P (s,t) for s, t = 1, 2, 3, . .., 11, in Table 3 and Figure 4 respectively.

Table 1 .
Comparison of K-Banhatti and redefined Zagreb indices of B 12 .

Table 2 .
s,t] .The order and size of P [s,t] , is From the symmetrical chemical structure of polyphenylene, P [s,t] , we find the edgepartition of P [s,t] easily.The edge partition of polyphenylenes P [s,t] is shown in Table 2. Edge-partition of polyphenylenes P [s,t] .

related to the 2 nd redefined Zagreb in the index of P
[s,t]Assume that P [s,t] is a network of benzenoid aromatic nuclei in polyphenylenes.The 2 nd redefined Zagreb index is then obtained by utilizing Equation (4) and

the 3 rd redefined Zagreb index of
P[s,t]