Statistics and Calculation of Entropy of Dominating David Derived Networks

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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 application in various fields biology, computer science, information technology, and chemistry. Topological indices are used in QSAR/QSPR studies.
To comprehend the properties and data contained in the network example of graphs, there are a number of mathematical values, known as structure invariants, topological indices, or topological descriptors, which have been determined and concentrated in the course of recent many years. e topological indices have tremendous number of uses in the chemical graph which is the uncommon part of numerical science.
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 [17], when he was working on the boiling point of paraffins. e 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. Shannon, in 1948, gave the concept of entropy [12]. 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) be 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 [6]. Trucco gave the idea of graph entropy in 1955 [15]. He said, the graph entropy is dependent on order of vertices. e construction of dominating David derived networks is the same as the David derived networks. e David derived network is obtained from Star of David network SD(n) by splitting each edge of it into two by installing another vertex. In the same way, dominating David derived network, DDD(n), is obtained from honeycomb network HC(n). For the detailed construction, the reader can refer to [8,9].

Degree-Based Topological Indices.
e first degree-based topological index was presented by Milan Randic [11] and generalised by Bollabas and Erdos [2] and Amic et al. [1], in 1998: where α � 1, − 1, (1/2), − (1/2). ABC index was introduced in 1998, by Estrada et al. [7]. It has the formula Vukicevic and Furtula studied this index for the first time [16]. It is written as the GA index: 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, and we have p j�1 d(u j ) � 2q. So,

Main Results
Simonraj and George [13] computed the metric dimension of David network and Imran et al. [8] computed the degreebased topological indices of dominating David derived networks; also, Song et al. [14] computed the entropy-based indices of Hex derived networks. Here, we discuss the dominating David derived networks in this work and calculate the exact results for entropies based on edges.

Results on Dominating David Derived Network of Type 1.
Here, we calculate certain degree-based entropies of dominating David derived network of type 1. D 1 (n) is shown in Figure 1. e edge partition of D 1 (n) is shown in Table 1. We compute Randic entropy, ABC entropy, and GA entropy for D 1 (n).

e Atom Bond Entropy of D 1 (n).
If H � D 1 (n), then, by using equation (4) and Table 1, the ABC index is Using equation (11) and Table 1, the ABC entropy is where ABC index of D 1 (n) is written in equation (23).

e Geometric Arithmetic Entropy of D 1 (n).
If H � D 1 (n), then, by using equation (5) and Table 1, the GA index is Using equation (12) and Table 1, we have 6 Complexity where GA index of D 1 (n) is written in equation (25).

Results on Dominating David Derived Network of Type 2.
Here, we calculate certain degree-based entropies of Dominating David Derived network of type 2. e D 2 (n) shown in Figure 2, and edge partition is shown in Table 2.
We compute Randic entropy, ABC entropy and GA entropy for D 2 (n).

Results on Dominating David Derived Network of Type 3.
Here, we calculate certain degree-based entropies of dominating David derived network of type 3. D 3 (n) is shown in Figure 3, and edge partition is shown in Table 3. We compute Randic entropy, ABC entropy, and GA entropy for D 3 (n).

e Atom Bond Entropy of D 3 (n).
If H � D 2 (n), then, by using equation (4) and Using equation (11) and Table 3, we have

Complexity
where the ABC index of D 3 (n) is written in equation (51).