On Computation of Entropy of Hex-Derived Network

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 HDN1(n) and HDN2(n), respectively of order n. We also compute some degree-based entropies such as Randić, ABC, and GA entropy of HDN1(n) and HDN2(n).


Introduction and Preliminary Results
A graph G is a tuple G(V, E), where V is the set of vertices and E is the set of edges. A graph can be represented by a numerical quantity which is known as topological index. ese 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.
e topological indices have vast number of applications in the chemical graph theory which is the special branch of mathematical chemistry.
e 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. e theory of topological indices was started by Wiener [1], when he was working on the boiling point of paraffins. e Wiener index is stated as (u, v). (1) 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. e idea of entropy was given by Shannon in 1948 [2]. e entropy of a graph G is defined as follows.
Let G be a graph and V(G) � 1, 2, . . . , n { } be the vertex set of G. Let P � (p 1 , p 2 , . . . , p n ) be the probability density of V(G) and VP(G) is the vertex packing polytope of G.
en, entropy of G with respect to P 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 HX (2) (Figure 1) is obtained from six triangles and HX(3) (Figure 1) is obtained from HX(2) by adding the layer of triangles around the boundary. Similarly, by adding the layer of triangles, we got HX(n).
e number of arc-wise connected open sets obtained by the partition of plane by a graph are known as faces of G [6]. In Figure 1, (3) shows the faces of HX (2). By combining the faces of HX(n), with the vertices, we obtain HDN 1(n) hexderived network of type 1. Figure 2 shows the HDN 1(4) hex-derived network of type 1 with 4 vertices [7,8].
e vertex and edge partition of HDN 1(n) is shown in Tables 1 and 2, respectively.
As discussed before in the formation HDN 1(n), if we join the vertices of HDN 1(n) with each other, then the new figure formed by this is hex-derived network of type 2 HDN 2(n). It is clear that HDN 1(n) is a subgraph of HDN 2(n). e hex-derived network of type 2 with 4 vertices is shown in Figure 3. e vertex and edge partition of HDN 2(n) is shown in Tables 3 and 4, respectively.
where α � 1, − 1, (1/2), − (1/2). ABC 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 GA index and written as follows: 1.2. Degree-Based Entropy of Graph. e entropy of a graph G is defined as where d u i is the degree or vertex u i .
By using the hand shaking lemma, we have

Edge Partition-Based Entropy of Graph.
e edge partition entropy of a graph G was introduced in 2014 by Chen et al. [14].
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 ic � entropy, ABC entropy, and GA entropy for hex-derived network HDN 1(n).

e Atom Bond Entropy of HDN 1(n).
If H 1 � HDN 1(n), then by using equation (4) and Table 2, the ABC index is Using equation (11) and Table 2, the ABC entropy is 6 Complexity Complexity 7 where ABC index of HDN 1(n) is written in equation (24). HDN 1(n), then by using equation (5) and Table 2, GA index is

e Geometric Arithmetic Entropy of HDN 1(n). If
Using equation (10) and where GA index of HDN 1(n) is written in equation (26).

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 ic � entropy, ABC entropy, and GA entropy for hex-derived network HDN 2(n).

Complexity 15
Using equation (12) and where GA index of HDN 2(n) 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 HDN 1(n) and HDN 2(n). Furthermore, we construct the Tables 5 and  6 to estimate the degree-based entropies for various values of n. From Tables 5 and 6, we can see that as the value of n 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.

Conclusion
In this article, taking into account the definition of Shannon and Chen's entropy, we studied two classifications of hexderived network and also compute the entropies of them. We discuss the degree-based topological indices such as Randic, ABC, and GA 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
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.