*Article* **Entropies via Various Molecular Descriptors of Layer Structure of** *H***3***BO***<sup>3</sup>**

**Muhammad Usman Ghani 1,†, Muhammad Kashif Maqbool 2,†, Reny George 3,\*,†, Austine Efut Ofem 4,† and Murat Cancan 5,†**


**Abstract:** Entropy is essential. Entropy is a measure of a system's molecular disorder or unpredictability, since work is produced by organized molecular motion. Entropy theory offers a profound understanding of the direction of spontaneous change for many commonplace events. A formal definition of a random graph exists. It deals with relational data's probabilistic and structural properties. The lower-order distribution of an ensemble of attributed graphs may be used to describe the ensemble by considering it to be the results of a random graph. Shannon's entropy metric is applied to represent a random graph's variability. A structural or physicochemical characteristic of a molecule or component of a molecule is known as a molecular descriptor. A mathematical correlation between a chemical's quantitative molecular descriptors and its toxicological endpoint is known as a QSAR model for predictive toxicology. Numerous physicochemical, toxicological, and pharmacological characteristics of chemical substances help to foretell their type and mode of action. Topological indices were developed some 150 years ago as an alternative to the Herculean, and arduous testing is needed to examine these features. This article uses various computational and mathematical techniques to calculate atom–bond connectivity entropy, atom–bond sum connectivity entropy, the newly defined Albertson entropy using the Albertson index, and the IRM entropy using the IRM index. We use the subdivision and line graph of the *H*3*BO*<sup>3</sup> layer structure, which contains one boron atom and three oxygen atoms to form the chemical boric acid.

**Keywords:** entropies via various molecular descriptors; *H*3*BO*<sup>3</sup> layer structure; subdivision of *H*3*BO*3; line graph of *H*3*BO*<sup>3</sup>

**MSC:** 05C07; 05C09; 05C31; 05C76; 05C99

### **1. Introduction**

Theoretical chemistry and graph theory are combined in chemical graph theory (CGT). It makes a contribution to the modeling of actual and fictitious chemical substances, examines the mathematical structure and connectedness, and then unifies the mathematical and chemical notions [1]. A chemical compound is modeled by displaying its structural formula as a chemical graph, in which atoms are represented by vertices and chemical bonds by edges [2].

We determine a structure's distance-based entropy by using some well-known topological indices, which are the numbers that help characterize its topological features after

**Citation:** Ghani, M.U.; Kashif Maqbool, M.; George, R.; Ofem, A.E.; Cancan, M. Entropies via Various Molecular Descriptors of Layer Structure of *H*3*BO*3. *Mathematics* **2022**, *10*, 4831. https://doi.org/ 10.3390/math10244831

Academic Editor: Vasily Novozhilov

Received: 9 November 2022 Accepted: 13 December 2022 Published: 19 December 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

it has been reproduced. The many pharmacological, physicochemical (such as melting point, boiling temperature, volume, molecular weight, density, etc.), and toxicological properties of a chemical molecule have a link with these invariants [3–5]. Topological indices have the amazing feature of remaining constant over graph isomorphisms, making them typically graph-invariant [6–14]. Numerous topological indices based on chemical graphs that rely on the number of vertices have been discovered and studied [15–19]. The atom–bond connectivity index and its modified form, the atom–bond sum connectivity index, the Albertson index, and the IRM index, as well as their mathematical equations, are introduced and defined in this section. For more explanation, see [20–27].

The atom–bond connectivity index was established by Estrada et al. [28] and is a modified version of the connectivity index. It is described as

$$ABC(G, \mathbf{x}) = \sum\_{a\_i \sim d\_2} \mathbf{x} \sqrt{\frac{\frac{(V\_{d\_1} + V\_{d\_2} - 2)}{(V\_{d\_1} \times V\_{d\_2})}}{(V\_{d\_1} \times V\_{d\_2})}} \qquad \& \qquad ABC = \sum\_{b\_1 \sim d\_2} \sqrt{\frac{(V\_{d\_1} + V\_{d\_2} - 2)}{(V\_{d\_1} \times V\_{d\_2})}} \tag{1}$$

Zhou and Trinajstic [29] proposed the sum-connectivity index, <sup>∑</sup>*u*,*v*∈*ξ<sup>g</sup>* <sup>1</sup> *Vai* +*Vaj* , an alternative to the connectivity index. The atom–bond sum-connectivity (ABS) index is a recently proposed modification of the atom–bond connectivity index that makes use of the fundamental concept of the sum-connectivity index [30]. A definition of the ABS index is

$$ABS(G, \mathbf{x}) = \sum\_{a\_i \sim a\_j} \mathbf{x} \sqrt{\frac{(V\_{a\_i} + V\_{a\_j} - 2)}{(V\_{a\_i} + V\_{a\_j})}} \qquad \& \qquadABS = \sum\_{a\_i \sim a\_j} \sqrt{\frac{(V\_{a\_i} + V\_{a\_j} - 2)}{(V\_{a\_i} + V\_{a\_j})}} \tag{2}$$

To determine a graph's irregularity, the authors in [31] established the Albertson index A(G).

$$A(G, \mathbf{x}) = \sum\_{a\_i \sim a\_j} \mathbf{x}^{|V\_{a\_i} - V\_{a\_j}|} \qquad \text{ \textit{\&}} \qquad A(G) = \sum\_{a\_i \sim a\_j} |V\_{a\_i} - V\_{a\_j}| \tag{3}$$

The irregularities of the graph are gauged using the Albertson, Bell, and IRM indices [32]. The definition of IRM(G) is

$$IRM(G, \mathbf{x}) = \sum\_{\mathbf{a}\_i \sim a\_j} \mathbf{x}^{[V\_{\mathbf{a}\_i} - V\_{\mathbf{a}\_j}]^2} \qquad \text{ \&} \qquad IRM(G) = \sum\_{\mathbf{a}\_i \sim a\_j} [V\_{\mathbf{a}\_i} - V\_{\mathbf{a}\_j}]^2 \tag{4}$$

In this paper, we work with Boric acid *H*3*BO*3. It is an acid made up of four oxygen atoms, one phosphorus atom, and three hydrogen atoms. Boric acid is sometimes referred to as orthoboric acid, boracic acid, hydrogen borate, or acidum boricum. It possesses antiviral, antifungal, and antiseptic qualities and is a weak acid. Figure 1 depicts the boric acid complex, which consists of one boron atom, three oxygen atoms, and three hydrogen atoms. The floral pattern structure (base unit) depicted in Figure 1 is created by polymerizing the *H*3*BO*<sup>3</sup> unit structure, which consists of six repeating units of *H*3*BO*3.

The degree of unpredictability (or disorder) in a system is measured by entropy. It may also be considered a measurement of how evenly the molecules in the system distribute their energy. The number of alternative configurations of molecule position and the amount of kinetic energy at a specific thermodynamic state is known as a microstate.

**Figure 1.** Boric acid *H*3*BO*3.

#### *Entropies via Various Molecular Descriptors*

Ghani et al. in [33] and Manzoor et al. in [34] recently offered another strategy that is a little bit novel in the literature: applying the idea of Shannon's entropy [35] in terms of topological indices. The following formula represents the graph entropy:

$$ENT\_{\mu(G)} = -\sum\_{a\_i \sim a\_j} \frac{\mu(V\_{a\_i}V\_{a\_j})}{\sum\_{a\_i \sim a\_j} \mu(V\_{a\_i}V\_{a\_j})} \log \left\{ \frac{\mu(V\_{a\_i}V\_{a\_j})}{\sum\_{a\_i \sim a\_j} \mu(V\_{a\_i}V\_{a\_j})} \right\}.\tag{5}$$

where *a*1, *a*<sup>2</sup> represents atoms, *ξ<sup>G</sup>* represents the edge set, and *μ*(*Vai Vaj* ) represents the edge weight of edge (*Vai Vaj* ).
