Abstract
Chemical graph theory is the combination of mathematical graph theory and chemistry. To analyze the biocompatibility of the compounds, topological indices are used in the research of QSAR/QSPR studies. The degree-based entropy is inspired by Shannon’s entropy. The connectivity pattern such as planar octahedron network is used to predict physiochemical activity. In this article, we present some degree-based entropies of planar octahedron network.
1. Introduction
All the graphs in this article are finite and undirected. A graph is set of points, where each pair of points (also known as vertex) are connected by an edge (also known as link or line). In network, vertices are called nodes, and in chemical graph, vertices are called atoms. In network, edges are called links or lines, while in chemical graph, they are called covalent bonds. The subbranch of chemical graph theory is topological indices. Many articles have been written on the topic of topological index. The representation of molecular graph by a drawing, a polynomial, a sequence of numbers, a matrix, or a derived number is called a topological index. As such, under graph isomorphism, these numeric numbers are unique. Most of the time, molecules and molecular compounds are nicely presented by molecular graph for better understanding.
Topological descriptors assume fundamental job in QSAR/QSPR studies in light of the fact that they convert a compound graph into a numerical number. We compare other physicochemical properties of carbon-based compounds (such as nanotubes, hydrocarbons, nanocones, and fullerenes). Due to these properties, topological descriptors have many applications in organic chemistry, biotechnology, and nanotechnology.
Cheminformatics is a branch of science that participates in mathematics, chemistry, and IT. In chemical graph theory, we consider molecular graph’s solution using the graph theory techniques which is the subdivision of mathematical chemistry. Molecules or atoms are represented by vertices in chemical graph theory, also the bonds between them by edges [1].
The pioneer of topological indices is Wiener [2]. It is defined as
Randić presented first the vertex-degree-based topological index in 1975 [3], which is written by
Bollobás and Erdos [4] and Amić et al. [5] compute the “general Randić index” independently in . where .
index was introduced in , by Estrada et al. [6]. It has the formulae
Vukiević and Furtula were the persons who studied this index for the first time [7]. It is written as index and written as
Entropy is the uncertainty in a random variable or quantity. In other words, it is the information obtained by learning the values of some unknown variables. Entropy has many applications in information theory as information entropy, in chemistry as thermodynamic entropy, and in graph theory as graph entropy [8–16]. In general, entropy is defined as the following: Let be a discrete random variable and and be the probability distribution of set . Then, entropy of is
The definition of entropy was given by Shannon in 1948 [17]. In graph theory, the idea of graph entropy was given by Rashevsky in 1955 [18]. It has been used comprehensively to depict the design of graph-based systems in mathematical science [19]. The graph entropy is defined as the following:
For a graph , is finite vertex set. Let be the density of probability of vertex set and be the vertex packing polytope of . Then, entropy of with respect to is
Octahedron networks have its roots in physical world as natural crystals of diamond are octahedron; also, many metal ions have octahedron configuration. In physics, these networks can be used as circuits. The construction of planar octahedron network is based on silicate structure derived by Manuel and Rajasingh [20] and was derived by Simonraj and George [21] (for the complete construction of , see Figure 1, for triangular prism network , see Figure 2, and for hex planar octahedron network, see Figure 3; we refer the reader to read the article [22]).
Degree-based entropy is defined as From Equation (8), edge-based entropy can be deducted as
From Equation (3) and Equation (9), Randić entropy will be
From Equation (4) and Equation (9), entropy will be
From Equation (5) and Equation (9), entropy will be
2. Main Results
Planar octahedron network and its derived forms are inorganic structures used in chemistry. Here, we research some degree-based entropies for these networks. These days, there is a broad examination movement on entropies (for further studies, see [23, 24]; for basic definitions and notations, we refer the reader to [25, 26]).
2.1. Results on Planar Octahedron Network
In this section, we will compute Randić, , and entropies for planar octahedron network. The edge partition of is written in Table 1.
2.1.1. Randić Entropy
If , then from Table 1 and Equation (3), we have
For ,
For ,
For ,
For ,
Using Equation (10) and Table 1, we have
For ,
For ,
For ,
For ,
where for is written in (14), (15), (16), and (17), respectively.
2.1.2. Entropy
If , then from Table 1 and Equation (4), we have
Using Equation (11) and Table 1, we have
where index is written in (24).
2.1.3. Entropy
If , then from Table 1 and Equation (5), we have
Using Equation (12) and Table 1, we have
where index is written in (26).
2.2. Results on Triangular Prism Network
In this section, we will compute Randić, , and entropies for triangular prism network. The edge partition of is written in Table 2.
2.2.1. Randić Entropy
If , then from Table 2 and Equation (3), we have
For ,
For ,
For ,
For ,
Using Equation (10) and Table 2, we have
For ,
For ,
For ,
For , where for is written in (28), (29), (30) and (31).
2.2.2. Entropy
If , then from Table 2 and Equation (4), we have
Using Equation (11) and Table 2, we have
where index is written in (37).
2.2.3. Entropy
If , then from Table 2 and Equation (5), we have
Using Equation (12) and Table 2, we have
where index is written in (39).
2.3. Results on Hex Planar Octahedron Network
In this section, we will compute Randić, , and entropies for hex planar octahedron network. The edge partition of hex is written in Table 3.
2.3.1. Randić Entropy
If hex , then from Table 3 and Equation (3), we have
For ,
For ,
For ,
For ,
Using Equation (10) and Table 3, we have
For ,
For ,
For ,
For ,
where for is written in (42), (43), (44), and (45), respectively.
2.3.2. Entropy
If hex , then from Table 3 and Equation (4), we have
Using Equation (11) and Table 3, we have
where index is written in (51).
2.3.3. Entropy
If hex , then from Table 3 and Equation (5), we have
Using Equation (12) and Table 3, we have
where index is written in (53).
3. Discussion and Conclusion
In this article, we computed some degree-based topological indices of planar octahedron networks. After that, we used the definition of Shannon’s graph entropy to find some exact results of entropies for planar octahedron networks. For the variational change in the values of entropies for the degree-based indices, we construct some tables to enlist the numerical values for these networks. It is clear from Tables 4, 5, and 6 that the increase in the value of causes a proportional increase or decrease in the values of entropies for Randić, ABC and GA indices. These formulae and their numerical values will help the researchers to predict physio- and biochemical activities of these networks. These numerical values of entropies can also predict the amount of energy that is unavailable for the work done in a chemical system. Furthermore, our future work will based on entropies of some other complex networks.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare no conflict of interest.
Acknowledgments
This work was supported in part by the Natural Science Fund of Education Department of Anhui Province under Grant KJ2020A0478.