Entropy Related to K-Banhatti Indices via Valency Based on the Presence of C6H6 in Various Molecules

Entropy is a measure of a system’s molecular disorder or unpredictability since work is produced by organized molecular motion. Shannon’s entropy metric is applied to represent a random graph’s variability. Entropy is a thermodynamic function in physics that, based on the variety of possible configurations for molecules to take, describes the randomness and disorder of molecules in a given system or process. Numerous issues in the fields of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines are resolved using distance-based entropy. These applications cover quantifying molecules’ chemical and electrical structures, signal processing, structural investigations on crystals, and molecular ensembles. In this paper, we look at K-Banhatti entropies using K-Banhatti indices for C6H6 embedded in different chemical networks. Our goal is to investigate the valency-based molecular invariants and K-Banhatti entropies for three chemical networks: the circumnaphthalene (CNBn), the honeycomb (HBn), and the pyrene (PYn). In order to reach conclusions, we apply the method of atom-bond partitioning based on valences, which is an application of spectral graph theory. We obtain the precise values of the first K-Banhatti entropy, the second K-Banhatti entropy, the first hyper K-Banhatti entropy, and the second hyper K-Banhatti entropy for the three chemical networks in the main results and conclusion.


Introduction
In the late 1990s, researchers began investigating the information content of complex networks, [1] and graphs based on Shannon's entropy work [2]. Numerous quantitative measures for analyzing complex networks have been proposed [3,4] spanning a wide range of issues in discrete mathematics, computer science, information theory, statistics, chemistry, biology, and other fields [5,6]. Graph entropy measures, for example, have been widely used to characterize the structure of graph-based systems [7,8] in mathematical chemistry, biology, and computer science-related areas. The concept of graph entropy [9], developed by Rashevsky [10] and Trucco [11] has been used to quantify the structural complexity of graphs [12,13].
Chemical indices are important tools for studying different physico-chemical properties of molecules without having to conduct several tests. In the investigation of medicines, quantitative structure-activity relationships (QSAR) use mathematical computations to decipher the chemical properties [14,15]. Some researchers have analyzed the topological and K-Banhatti indices in [16,17]. Mowshowitz [18] introduced the entropy of a graph as an information-theoretic quantity. The complexity is evident here. The well-known Shannon's entropy [2] is used to calculate the entropy of a graph. Importantly, Mowshowitz interpreted his graph entropy measure as a graph's structural information content and demonstrated that this quantity satisfies important properties when used with product graphs [18]. Inspired by Dehmer and Kraus [19], it was discovered that determining the minimal values of graph entropies is difficult due to a lack of analytical methods to address this specific problem.
The first-order valence-based K-Banhatti indices [17,20,21] are, respectively, as follows: and where a i and a j denote the atoms of a molecule, V a i and V a j represent the valency of each atom, and, if a i and a j are connected or have atom bonds, then we denote this by a i ∼ a j . Accordingly, the second valence-based K-Banhatti index [22] and polynomial are as follows: and The hyper K-Banhatti index and first and second polynomial types [21] are as follows: The Banhatti indices were proposed by the Indian mathematician Kulli as a result of Milan Randic's 1972 Zagreb indices. With various techniques, such as modified and hyper-Banhatti indices, Kulli offered a number of studies on Banhatti indices. These indices have excellent associations with several chemical and nonchemical graph properties. The amount of thermal energy per unit temperature in a system that cannot be used for useful work is known as entropy [23,24]. In this article, we investigate the graphs of different molecules, namely the pyrene network, the circumnaphthalene series of benzenoid, and the honeycomb benzenoid network, to determine the K-Banhatti entropies' use of K-Banhatti indices [21,25].

Definitions of Entropies via K-Banhatti Indices
Manzoor et al. in [26] and Ghani et al. in [27] recently offered another strategy that is a little bit novel in the literature: applying the idea of Shannon's entropy [28] in terms of topological indices. The following formula represents the valency-based entropy: where a 1 , a 2 represents the atoms, E G represents the edge set, and µ(V a i V a j ) represents the edge weight of edge (V a i V a j ). •

The first K-Banhatti entropy
Let µ(V a i V a j ) = V a i + V a j . The first-order K-Banhatti index (1) is thus provided as The use of (5), the first K-Banhatti entropy, is shown below: • The Second K-Banhatti entropy Then, the second K-Banhatti index (2) is given by The use of (5), the second K-Banhatti entropy, is shown below: • Entropy related to the first K-hyper Banhatti index Then, the first K-hyper Banhatti index (3) is given by The use of (5), the first K-hyper Banhatti entropy, is shown below: •

Entropy related to the second K-hyper Banhatti index
Let µ(V a i V a j ) = (V a i × V a j ) 2 . Then, the second K-hyper Banhatti index (4) is given by The use of (5), the second K-hyper Banhatti entropy, is shown below:

The Pyrene Network
The precise arrangement of rings in the benzenoid system offers a transformation within a sequence of benzenoid structures of the benzenoid graph, which changes the structure. The Pyrene network PY n is a collection of hexagons, and it is a simple, connected 2D planner graph, where n represents the number of hexagons in any row of PY n (see Figure 1). Accordingly, the Pyrene network is a series of benzenoid rings, and the total number of benzenoid rings is n 2 in PY(n). We sum up the Zagreb polynomial and topological indices of PY(n) in this section.

Results and discussion
The number of atoms and the total number of atomic bonds for PY n are now determined. Let us consider the line of symmetry that divides PY n into two symmetric parts, as shown in Figure 1. Let us denote the number of atoms in one symmetric portion of PY n by x and the number of layers by l. In one symmetric part of PY n , there are l layers of carbon atoms for 1 ≤ l ≤ n, as indicated in Figure 1. Then, an lth layer contains 2l + 1 carbon atoms. Accordingly, we have The number of atoms in PY n is 2x = 2n 2 + 4n because of the two symmetric parts in PY n . Furthermore, a PY n corner atom and an atom other than a corner atom have valencies two and three, respectively. Thus, out of 2n 2 + 4n atoms, 4n + 2 atoms have valency two, and 2(n 2 − 1) atoms have valency three. So, by using Formula (1), the number of atomic bonds in PY n is 3n 2 + 4n − 1. According to the valencies (two and three) of the atoms, there are three types of atomic bonds, which are (2,2), (2,3), and (3,3) in PY n . On the basis of valency, Table 1 provides the partition of the set of atomic bonds.
The edge partition of PY n is: This partition provides: Table 1. Atomic bond partition of PY n .

Atomic Bond Type
Number of atom bonds 6 8(n − 1)

Entropy related to the first K-Banhatti index of PY n
Let PY n be the Pyrene network of C 6 H 6 . The first K-Banhatti polynomial is calculated using Equation (1) and Table 1.
Following the simplification of Equation (10), we obtain the first K-Banhatti index, which is given at x = 1 via differentiation.
Here, we calculate the first K-Banhatti entropy of PY n using Table 1 and Equation (11) inside Equation (6) in the following manner: • The second K-Banhatti entropy of PY n Let PY n be the Pyrene network of C 6 H 6 . Then, using Equation (2) and Table 1, the second K-Banhatti polynomial is To differentiate (34) at x = 1, we obtain the second K-Banhatti index: Here, we calculate the second K-Banhatti entropy of PY n using Table 1 and Equation (13) in Equation (7) as described below: •

Entropy related to the first K-hyper Banhatti index of PY n
Let PY n be the Pyrene network of C 6 H 6 . Then, using Equation (3) and Table 1, the first K-hyper Banhatti polynomial is To differentiate (15) at x = 1, we obtain the first K-hyper Banhatti index Here, we calculate the first K-hyper Banhatti entropy of PY n using Table 1 and Equation (16) in Equation (9) as described below: • Entropy related to the second K-hyper Banhatti index PY n Let PY n be the Pyrene network of C 6 H 6 . Then, using Equation (4) and Table 1, the second K-hyper Banhatti polynomial is To differentiate (18) at x = 1, we obtain the second K-hyper Banhatti index Here, we calculate the second K-hyper Banhatti entropy of PY n using Table 1 and Equation (19) in Equation (9) as described below:

Characteristics of K-Banhatti Indices of PY n
Here, we contrast the K-Banhatti indices, namely B 1 , B 2 , HB 1 , and HB 2 for PY n quantitatively and visually in Table 2 and Figure 2, respectively.

Circumnaphthalene Series of Benzenoid
Circumnaphthalene is similar to the benzenoid polycyclic aromatic hydrocarbons with the formula C 32 H 14 and the ten peri-fused six-member rings in figure CNB 2 . Ovalene is a chemical that is reddish-orange in color. It is only slightly soluble in solvents, such as benzenoid, toluene, and dichloromethane. The circumnaphthalene series of benzenoids is designated by CNB n , where "n" is the number of benzenoid rings in the corner, as seen in

Results and Discussion
In Figure 3, we have the following three partitions of the carbon atoms in CNB n : These partitions provide us with the atomic bond partition of the CNB n network (see Table 3). Table 3. Atomic bond partition of CNB n network.

Entropy related to the 1st K-Banhatti index of CNB n
Let CNB n be the circumnaphthalene series of benzenoid of C 6 H 6 . Then, using Equation (1) and Table 3, the first K-Banhatti polynomial is Following the simplification of Equation (21), we obtain the first K-Banhatti index, which is given at x = 1 via differentiation.
Here, we calculate the first K-Banhatti entropy of CNB n using Table 1 and Equation (24) in Equation (6) in the following manner: •

The second K-Banhatti entropy of CNB n
Let CNB n be the circumnaphthalene series of benzenoid of C 6 H 6 . Then, using Equation (2) and Table 1, the second K-Banhatti polynomial is To differentiate (23) at x = 1, we obtain the second K-Banhatti index Here, we calculate the second K-Banhatti entropy of CNB n using Table 3 and Equation (24) in Equation (7) as described below: •

Entropy related to the first K-hyper Banhatti index of CN B n
Let CNB n be the circumnaphthalene series of benzenoid of C 6 H 6 . Then, using Equation (3) and Table 3, the first K-hyper Banhatti polynomial is To differentiate (26) at x = 1, we obtain the first K-hyper Banhatti index Here, we calculate the first K-hyper Banhatti entropy of CNB n using Table 1 and Equation (27) in Equation (9) as described below: = log 4(81n 2 − 168n + 70) − 1 4(81n 2 − 168n + 70) log 6(4 32 ) • Entropy related to the second K-hyper Banhatti index CNB n Let CNB n be the circumnaphthalene series of benzenoid of C 6 H 6 . Then, using Equation (4) and Table 3, the second K-hyper Banhatti polynomial is To differentiate (29) at x = 1, we obtain the second K-hyper Banhatti index Here, we calculate the second K-hyper Banhatti entropy of CNB n using Table 3 and Equation (30) in Equation (9) as described below: log 6(4) 32

Characteristics of K-Banhatti Indices of CNB n
Here, we contrast the K-Banhatti indices, namely B 1 , B 2 , HB 1 , and HB 2 for CNB n quantitatively and visually in Table 4 and Figure 4, respectively.

The Honeycomb Benzenoid Network
In this section, we introduce a chemical compound that has received more and more attention in recent years, partly due to its applications in chemistry. Honeycomb networks are formed when hexagonal tiling is used recursively in a specific pattern. HB n denotes an n-dimensional honeycomb network, where n is the number of Benzene rings from center to top, center to bottom, or center to each corner of HB n , as shown in Figure 5.

Results and Discussion
The honeycomb network HB n is created by adding a layer of hexagons around the boundary of HB (n−1) . In the honeycomb benzenoid network, a 6n amount of atoms has valency two, and 6n 2 − 6n atoms have valency three. According to the valency of each atom in HB n , the atomic bonds are classified into three types: 2 ∼ 2, 2 ∼ 3, and 3 ∼ 3 (see Figure 5).
Thus, according to the above partition of the atomic bonds, there is 3n(3n − 1) total number of atomic bonds used in the honeycomb benzenoid network. The atomic bond partition of HB n is shown in Table 5.

Types of Atomic Bonds
Cardinality of atomic bonds 6 12(n − 1) (9n 2 − 15n + 6) • Entropy related to the first K-Banhatti index of HB n Let HB n be the honeycomb benzenoid network of C 6 H 6 . Then, using Equation (1) and Table 5, the first K-Banhatti polynomial is Following the simplification of Equation (32), we obtain the first K-Banhatti index given at x = 1 via differentiation.

Characteristics of K-Banhatti Indices of HB n
Here, we contrast the K-Banhatti indices, namely B 1 , B 2 , HB 1 , and HB 2 for HB n quantitatively and visually in Table 6 and Figure 6, respectively. different fields and serving as the basis for future interdisciplinary research. We intend to extend this idea to different chemical structures in the future, opening up new directions for study in this area.