Abstract
A graph’s entropy is a functional one, based on both the graph itself and the distribution of probability on its vertex set. In the theory of information, graph entropy has its origins. Hex-derived networks have a variety of important applications in medication store, hardware, and system administration. In this article, we discuss hex-derived network of type 1 and 2, written as and , respectively of order . We also compute some degree-based entropies such as Randić, , and entropy of and .
1. Introduction and Preliminary Results
A graph is a tuple , where is the set of vertices and is the set of edges. A graph can be represented by a numerical quantity which is known as topological index. These indices have a vast number of applications in various fields, biology, computer science, information technology, and chemistry. Topological indices are used in QSAR/QSPR studies.
In order to understand the properties and information contained in the connectivity pattern of graphs, there are a number of numerical quantities, known as structure invariants, topological indices, or topological descriptors, which have been derived and studied over the past few decades. The topological indices have vast number of applications in the chemical graph theory which is the special branch of mathematical chemistry.
The combination of mathematics, information technology, and chemistry is a new division known as cheminformatics. It deals with QSAR and QSPR studies which predict the bio and physical chemical activities of compounds. The theory of topological indices was started by Wiener [1], when he was working on the boiling point of paraffins. The Wiener index is stated as
A number of problems that occur in discrete mathematics, statistics, biology, computer science, chemistry, information theory, etc., investigate the entropy of structures to deal with them. The idea of entropy was given by Shannon in 1948 [2]. The entropy of a graph is defined as follows.
Let be a graph and be the vertex set of . Let be the probability density of and is the vertex packing polytope of . Then, entropy of with respect to is
Graph entropy has been utilized broadly to portray the structure of graph-based frameworks in numerical science [3]. Rashevsky gave the idea of graph entropy in 1955 [4]. He said that the graph entropy is dependent on order of vertices.
Hexagonal mesh was firstly proposed by Chen [5]. A hexagonal mesh consists of six triangles with dimension more than one. A 2-dimensional mesh (Figure 1) is obtained from six triangles and (Figure 1) is obtained from by adding the layer of triangles around the boundary. Similarly, by adding the layer of triangles, we got .
(a)
(b)
(c)
The number of arc-wise connected open sets obtained by the partition of plane by a graph are known as faces of [6]. In Figure 1, (3) shows the faces of . By combining the faces of , with the vertices, we obtain hex-derived network of type 1. Figure 2 shows the hex-derived network of type 1 with 4 vertices [7, 8].
The vertex and edge partition of is shown in Tables 1 and 2, respectively.
As discussed before in the formation , if we join the vertices of with each other, then the new figure formed by this is hex-derived network of type 2 . It is clear that is a subgraph of . The hex-derived network of type 2 with 4 vertices is shown in Figure 3.
The vertex and edge partition of is shown in Tables 3 and 4, respectively.
1.1. Degree-Based Topological Indices
The first degree-based topological index was presented by Milan Randić [9] and generalised by Bollobás and Erdos [10], and Amić et al. [11], in 1998.where .
index was introduced in 1998 by Estrada et al. [12]. It has the formulae:
Vukičević was the person who studied this index for the first time [13]. It is written as index and written as follows:
1.2. Degree-Based Entropy of Graph
The entropy of a graph is defined aswhere is the degree or vertex .
By using the hand shaking lemma, we have . So,
1.3. Edge Partition-Based Entropy of Graph
The edge partition entropy of a graph was introduced in 2014 by Chen et al. [14].
1.4. Randic Entropy
Using equation (3), equation (9) is reduced as
1.5. Atom Bond Connectivity Entropy
Using equation (4), equation (9) is reduced as
1.6. The Geometric Arithmetic Entropy
Using equation (5), equation (9) is reduced as
2. Main Results
Dacheng Xu et al. [15], Zehui et al. [16], and Imran et al. [8] computed the metric dimension and topological indices for hex-derived networks, respectively. Here, we discuss the first two types of hex-derived networks in this work and calculate the exact results for entropies based on degree and edges. These entropies and their variants are currently subjected to extensive research activity, see [7, 17–27]. For basic notations and definitions, see [28–31].
2.1. Results on Hex-Derived Network of Type 1
In this section, we calculate certain degree-based entropies of hex-derived network of type 1. We compute Rand i entropy, entropy, and entropy for hex-derived network .
2.2. Degree-Based Entropy of
If , then by using equation (8) and Table 1, we get
2.3. Randic Entropy
If , then by using Table 2 and equation (3), we haveand for , Randic index is
For ,
For ,
For ,
Using equation (10) and Table 2, the Randic entropy is
For ,
For ,
For ,
For ,where for is written in equations (15)–(18), respectively.
2.4. The Atom Bond Entropy of
If , then by using equation (4) and Table 2, the index is
Using equation (11) and Table 2, the entropy iswhere index of is written in equation (24).
2.5. The Geometric Arithmetic Entropy of
If , then by using equation (5) and Table 2, index is
Using equation (10) and Table 2, we havewhere index of is written in equation (26).
2.6. Results on Hex-Derived Network of Type 2
In this section, we calculate certain degree-based entropies of hex-derived network of type 2. We compute Rand i entropy, entropy, and entropy for hex-derived network .
2.7. Degree-Based Entropy of
If , then by using equation (8) and Table 3, we have
2.8. Randic Entropy of
If , then by using equation (3) and Table 4, we have
For ,
For ,
For ,
For ,
Using equation (10) and Table 4, we have
For ,
For ,
For ,
For ,where for is written in equations (30)–(33), respectively.
2.9. The Atom Bond Entropy of
If , then by using equation (4) and Table 4, we have
Using equation (11) and Table 4, we havewhere index of is written in equation (39).
2.10. The Geometric Arithmetic Entropy of
If , by using equation (5) and Table 4, we have
Using equation (12) and Table 4, we havewhere index of is written in equation (41) section Discussion. Since entropy plays a vital role in various fields of science such as software engineering, medication, and pharmaceutical, that is why its numerical values and graphical representation are very much important for researchers. In this section, we calculate some exact values of degree-based entropies of hex-derived networks and . Furthermore, we construct the Tables 5 and 6 to estimate the degree-based entropies for various values of . From Tables 5 and 6, we can see that as the value of increases, the degree-based entropies of hex-derived network also increase. At the end, we construct some graphical representations of these entropies in Figures 4–15.
3. Conclusion
In this article, taking into account the definition of Shannon and Chen’s entropy, we studied two classifications of hex-derived network and also compute the entropies of them. We discuss the degree-based topological indices such as Randic, , and index and find their closed formulae of entropy for hex-derived network. We in like manner enlisted the mathematical assessments of these entropies in Tables 5 and 6. We gave the relation of these entropies which help us to know the physiochemical activity of these networks.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed to this article equally.
Acknowledgments
This study was supported by the 2018 Scientific Research Major Project of Higher Educational Institutes of Anhui Province (KJ2018ZD059).