# Statistics and Calculation of Entropy of Dominating David Derived Networks

Authors: Xuemei Zhao; Haidar Ali; Bilal Ali; Muhammad Ahsan Binyamin; Jia-Bao Liu; Ali Raza
Journal: Complexity
 (2021)

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## 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. Dominating David derived networks have a variety of important applications in medication store, hardware, and system administration. In this study, we discuss dominating David derived network of type 1, 2, and 3 written asD1n, D2n, and D3n, respectively of order n. We also compute some degree-based entropies such as Randić, ABC, and GA entropy of D1n, D2n, and D3n.


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## Body

## 1. Introduction and Preliminary Results

A graphG 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. These 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. The topological indices have tremendous number of uses in the chemical graph which is the uncommon part of numerical science.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 [17], when he was working on the boiling point of paraffins. The Wiener index is stated as(1)WG=∑u,v⊆VGdu,v.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]. The entropy of a graph G is defined as follows.LetG be a graph and VG=1,2,…,n be the vertex set of G. Let P=p1,p2,…,pn be the probability density of VG and VPG be the vertex packing polytope of G. Then, entropy of G with respect to P is(2)HG,P=mina∈VPG∑i=1npilog1ai.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.The construction of dominating David derived networks is the same as the David derived networks. The David derived network is obtained from Star of David networkSDn by splitting each edge of it into two by installing another vertex. In the same way, dominating David derived network, DDDn, is obtained from honeycomb network HCn. For the detailed construction, the reader can refer to [8, 9].


### 1.1. Degree-Based Topological Indices

The 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:(3)RαG=∑e=uv∈EGdu×dvα,where α=1,−1,1/2,−1/2.ABC index was introduced in 1998, by Estrada et al. [7]. It has the formula(4)ABCG=∑e=uv∈EGdu+dv−2du×dv.Vukicevic and Furtula studied this index for the first time [16]. It is written as the GA index:(5)GAG=∑e=uv∈EG2du×dvdu+dv.


### 1.2. Degree-Based Entropy of Graph

The entropy of a graphG is defined as(6)ENTψG=−∑i=1pdui∑j=1pdujlogdui∑j=1pduj,where dui is the degree or vertex ui.(7)ENTdG=−∑i=1pdui∑j=1pdujlogdui−log∑j=1pduj,⟹ENTdG=log∑j=1pduj−1∑j=1pduj∑i=1pduilogdui,⟹ENTdG=log∑j=1pduj−1∑j=1pduj∑i=1plogduidui,by using the Hand Shaking Lemma, and we have ∑j=1pduj=2q. So,(8)ENTdG=log2q−12q∑i=1plogduidui.


### 1.3. Edge Weight-Based Entropy of Graph

The edge weight entropy of a graphG was introduced in 2014 by Chen et al. [5]:(9)ENTdG=−∑u′v′∈EGdu′v′∑uv∈EGduvlogdu′v′∑uv∈EGduv.


#### 1.3.1. Randic´ Entropy

Using equation (3), equation (9) reduced to(10)ENTRαG=logRα−1Rα∑i=1m∑uv∈EiGdu×dvαdu×dvα.


#### 1.3.2. Atom Bond Connectivity Entropy

Using equation (4), equation (9) reduced to(11)ENTABCG=logABC−1ABC∑i=1m∑uv∈EiGlogdu+dv−2du×dvdu+dv−2/du×dv.


#### 1.3.3. The Geometric Arithmetic Entropy

Using equation (5), equation (9) reduced to(12)ENTGAG=logGA−1GA∑i=1m∑uv∈EiGlog2du×dvdu+dv2du×dv/du+dv.


## 1.1. Degree-Based Topological Indices

The 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:(3)RαG=∑e=uv∈EGdu×dvα,where α=1,−1,1/2,−1/2.ABC index was introduced in 1998, by Estrada et al. [7]. It has the formula(4)ABCG=∑e=uv∈EGdu+dv−2du×dv.Vukicevic and Furtula studied this index for the first time [16]. It is written as the GA index:(5)GAG=∑e=uv∈EG2du×dvdu+dv.


## 1.2. Degree-Based Entropy of Graph

The entropy of a graphG is defined as(6)ENTψG=−∑i=1pdui∑j=1pdujlogdui∑j=1pduj,where dui is the degree or vertex ui.(7)ENTdG=−∑i=1pdui∑j=1pdujlogdui−log∑j=1pduj,⟹ENTdG=log∑j=1pduj−1∑j=1pduj∑i=1pduilogdui,⟹ENTdG=log∑j=1pduj−1∑j=1pduj∑i=1plogduidui,by using the Hand Shaking Lemma, and we have ∑j=1pduj=2q. So,(8)ENTdG=log2q−12q∑i=1plogduidui.


## 1.3. Edge Weight-Based Entropy of Graph

The edge weight entropy of a graphG was introduced in 2014 by Chen et al. [5]:(9)ENTdG=−∑u′v′∈EGdu′v′∑uv∈EGduvlogdu′v′∑uv∈EGduv.


### 1.3.1. Randic´ Entropy

Using equation (3), equation (9) reduced to(10)ENTRαG=logRα−1Rα∑i=1m∑uv∈EiGdu×dvαdu×dvα.


### 1.3.2. Atom Bond Connectivity Entropy

Using equation (4), equation (9) reduced to(11)ENTABCG=logABC−1ABC∑i=1m∑uv∈EiGlogdu+dv−2du×dvdu+dv−2/du×dv.


### 1.3.3. The Geometric Arithmetic Entropy

Using equation (5), equation (9) reduced to(12)ENTGAG=logGA−1GA∑i=1m∑uv∈EiGlog2du×dvdu+dv2du×dv/du+dv.


## 1.3.1. Randic´ Entropy

Using equation (3), equation (9) reduced to(10)ENTRαG=logRα−1Rα∑i=1m∑uv∈EiGdu×dvαdu×dvα.


## 1.3.2. Atom Bond Connectivity Entropy

Using equation (4), equation (9) reduced to(11)ENTABCG=logABC−1ABC∑i=1m∑uv∈EiGlogdu+dv−2du×dvdu+dv−2/du×dv.


## 1.3.3. The Geometric Arithmetic Entropy

Using equation (5), equation (9) reduced to(12)ENTGAG=logGA−1GA∑i=1m∑uv∈EiGlog2du×dvdu+dv2du×dv/du+dv.


## 2. Main Results

Simonraj and George [13] computed the metric dimension of David network and Imran et al. [8] computed the degree-based 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.


### 2.1. Results on Dominating David Derived Network of Type 1

Here, we calculate certain degree-based entropies of dominating David derived network of type 1.D1n is shown in Figure 1. The edge partition of D1n is shown in Table 1. We compute Randic entropy, ABC entropy, and GA entropy for D1n.Figure 1
Dominating David derived network of type 1D12.Table 1
Edge partition ofD1n.
du,dv, where uv∈ED1nCardinality of edges2,24n2,34n−42,428n−163,39n2−13n+53,436n2−56n+244,436n2−52n+20


### 2.2. Randic Entropy ofD1n

IfH≅D1n, then, by using Table 1 and equation (3), we have(13)RαH=4n×4α+4n−4×6α+28n−16×8α+9n2−13n+5×9α+36n2−56n+24×12α+36n2−52n+20×16α,for α=1. Randic index is(14)⟹R1H=1089n2−1357n+501.Forα=−1,(15)⟹R−1H=254n2−15136n+4146.Forα=1/2,(16)⟹R1/2H=295.707658n2−343.995772n+123.085646.Forα=−1/2,(17)⟹R−1/2H=22.392305n2−19.966653n+6.305022.Using equation (10) and Table 1, the Randic entropy is(18)ENTRαH=logRα−1Rα4n×log4α4α+4n−4×log6α6α+28n−16×log8α8α+9n2−13n+5×log9α9α+36n2−56n+24×log12α12α+36n2−52n+20×log16α16α.Forα=1,(19)ENTR1H=logR1−1R14n×log44+4n−4×log66+28n−16×log88+9n2−13n+5×log99+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1H=logR1−1R11237.073052n2−1608.08325n+604.792358.Forα=1/2,(20)ENTR1/2H=logR1/2−1R1/24n×log44+4n−4×log66+28n−16×log88+9n2−13n+5×log99+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1/2H=logR1/2−1R1/2166.869996n2−206.530299n+75.935599.Forα=−1,(21)ENTR−1H=logR−1−1R−14n×log141/4+4n−4×log161/6+28n−16×log181/8+9n2−13n+5×log191/9+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1H=logR−1−1R−1−6.901056n2+6.046277n−1.8687.Forα=−1/2,(22)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+4n−4×log161/6+28n−16×log181/8+9n2−13n+5×log191/9+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−12.457494n2+12.909738n−4.354213,where Rα for α=1,−1,1/2,−1/2 is written in equations (14)–(17), respectively.


### 2.3. The Atom Bond Entropy ofD1n

IfH≅D1n, then, by using equation (4) and Table 1, the ABC index is(23)ABCH=4n×2+2−22×2+4n−4×2+3−22×3+28n−16×2+4−22×4+9n2−13n+5×3+3−23×3+36n2−56n+24×3+4−23×4+36n2−52n+20×4+4−24×4,⟹ABCH=51.283308n2−51.202034n+16.93058.Using equation (11) and Table 1, the ABC entropy is(24)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+4n−4×log2+3−22×32+3−2/2×3+28n−16×log2+4−22×42+4−2/2×4+9n2−13n+5×log3+3−23×33+3−2/3×3+36n2−56n+24×log3+4−23×43+4−23×4+36n2−52n+20×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−10.16953n2+11.348686n−4.011986,where ABC index of D1n is written in equation (23).


### 2.4. The Geometric Arithmetic Entropy ofD1n

IfH≅D1n, then, by using equation (5) and Table 1, the GA index is(25)GAH=4n×22×22+2+4n−4×22×32+3+28n−16×22×42+4+9n2−13n+5×23×33+3+36n2−56n+24×23×43+4+36n2−52n+20×24×44+4,⟹GAH=80.630759n2−86.107789n+29.749711.Using equation (12) and Table 1, we have(26)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+4n−4×log22×32+322×3/2+3+28n−16×log22×42+422×4/2+4+9n2−13n+5×log23×33+323×3/3+3+36n2−56n+24×log23×43+423×4/3+4+36n2−52n+20×log24×44+424×4/4+4,⟹ENTGAH=logGA−1GA−0.159534n2−0.461756n+0.314202,where GA index of D1n is written in equation (25).


### 2.5. Results on Dominating David Derived Network of Type 2

Here, we calculate certain degree-based entropies of Dominating David Derived network of type 2. TheD2n shown in Figure 2, and edge partition is shown in Table 2. We compute Randic entropy, ABC entropy and GA entropy for D2n.Figure 2
Dominating david derived network of type 2D24.Table 2
Edge partition ofD2n.
du,dv where uv∈ED2nCardinality of edges2,24n2,318n2−22n+62,428n−163,436n2−56n+244,436n2−52n+20


### 2.6. Randic Entropy ofD2n

IfH≅D2n, then, by using equation (3) and Table 2, we have(27)RαH=4n×4α+18n2−22n+6×6α+28n−16×8α+36n2−56n+24×12α+36n2−52n+20×16α.Forα=1,(28)⟹R1H=1116n2−1396n+516.Forα=−1,(29)⟹R−1H=334n2−8512n+94.Forα=1/2,(30)⟹R1/2H=312.798474n2−368.682505n+132.580543.Forα=−1/2,(31)⟹R−1/2H=26.740774n2−26.247775n+8.720839.Using equation (10) and Table 2, we have(32)ENTRαH=logRα−1Rα4n×log4α4α+18n2−22n+6×log6α6α+28n−16×log8α8α+36n2−56n+24×log12α12α+36n2−52n+20×log16α16α.Forα=1,(33)ENTR1H=logR1−1R14n×log44+18n2−22n+6×log66+28n−16×log88+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1H=logR1−1R11243.819743n2−1617.828471n+608.54052.Forα=1/2,(34)ENTR1/2H=logR1/2−1R1/24n×log44+18n2−22n+6×log66+28n−16×log88+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1/2H=logR1/2−1R1/2171.142383n2−212.701526n+78.309148.Forα=−1,(35)ENTR−1H=logR−1−1R−14n×log141/4+18n2−22n+6×log161/6+28n−16×log181/8+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1H=logR−1−1R−1−8.281267n2+8.039915n−2.635484.Forα=−1/2,(36)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+18n2−22n+6×log161/6+28n−16×log181/8+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−13.88524n2+14.972039n−5.147406,where Rα for α=1,−1,1/2,−1/2 is written in equations (28)–(31), respectively.


### 2.7. The Atom Bond Entropy ofD2n

IfH≅D2n, then, by using equation (4) and Table 2, we have(37)ABCH=4n×2+2−22×2+18n2−22n+6×2+3−22×3+28n−16×2+4−22×4+36n2−56n+24×3+4−23×4+36n2−52n+20×4+4−24×4,⟹ABCH=58.01123n2−60.920143n+20.668314.Using equation (11) and Table 2, we have(38)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+18n2−22n+6×log2+3−22×32+3−2/2×3+28n−16×log2+4−22×42+4−2/2×4+36n2−56n+24×log3+4−23×43+4−2/3×4+36n2−52n+20×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−11.028726n2+12.589746n−4.489317,where ABC index of D2n is written in equation (37).


### 2.8. The Geometric Arithmetic Entropy ofD2n

IfH≅D2n, then, by using equation (5) and Table 2, we have(39)GAH=4n×22×22+2+18n2−22n+6×22×32+3+28n−16×22×42+4+36n2−56n+24×23×43+4+36n2−52n+20×24×44+4,⟹GAH=89.267086n2−98.582482n+34.54767.Using equation (12) and Table 2, we have(40)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+18n2−22n+6×log22×32+322×3/2+3+28n−16×log22×42+422×4/2+4+36n2−56n+24×log23×43+423×4/3+4+36n2−52n+20×log24×44+424×4/4+4,ENTGAH=logGA−1GA−0.315869n2−0.235939n+0.227349,where the GA index of D2n is written in equation (39).


### 2.9. Results on Dominating David Derived Network of Type 3

Here, we calculate certain degree-based entropies of dominating David derived network of type 3.D3n is shown in Figure 3, and edge partition is shown in Table 3. We compute Randic entropy, ABC entropy, and GA entropy for D3n.Figure 3
Dominating David derived network of type 3D32.Table 3
Edge partition ofD3n.
du,dv, where uv∈ED3nCardinality of edges2,24n2,436n2−20n4,472n2−108n+44


### 2.10. Randic Entropy ofD3n

IfH≅D3n, then, by using equation (3) and Table 3, we have(41)RαH=4n×4α+36n2−20n×8α+72n2−108n+44×16α.Forα=1,(42)⟹R1H=1440n2−1872n+704.Forα=−1,(43)⟹R−1H=9n2−334n+114.Forα=1/2,(44)⟹R1/2H=389.823376n2−480.568542n+176.Forα=−1/2,(45)⟹R−1/2H=30.727922n2−32.071068n+11.Using equation (10) and Table 2, we have(46)ENTRαH=logRα−1Rα4n×log4α4α+36n2−20n×log8α8α+72n2−108n+44×log16α16α.Forα=1,(47)ENTR1H=logR1−1R14n×log44+36n2−20n×log88+72n2−108n+44×log1616,⟹ENTR1H=logR1−1R11647.236136n2−2215.580768n+847.700468.Forα=1/2,(48)ENTR1/2H=logR1/2−1R1/24n×log44+36n2−20n×log88+72n2−108n+44×log1616,⟹ENTR1/2H=logR1/2−1R1/2219.371113n2−283.224918n+105.962558.Forα=−1,(49)ENTR−1H=logR−1−1R−14n×log141/4+36n2−20n×log181/8+72n2−108n+44×log1161/16,⟹ENTR−1H=logR−1−1R−1−9.482445n2+9.783475n−3.31133.Forα=−1/2,(50)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+36n2−20n×log181/8+72n2−108n+44×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−16.584309n2+18.846465n−6.62266,where Rα for α=1,−1,1/2,−1/2 is written in equations (42)–(45), respectively.


### 2.11. The Atom Bond Entropy ofD3n

IfH≅D2n, then, by using equation (4) and Table 3, we have(51)ABCH=4n×2+2−22×2+36n2−20n×2+4−22×4+72n2−108n+44×4+4−24×4,⟹ABCH=69.546659n2−77.449932n+26.944387.Using equation (11) and Table 3, we have(52)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+36n2−20n×log2+4−22×42+4−2/2×4+72n2−108n+44×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−13.222141n2+15.788864n−5.738733,where the ABC index of D3n is written in equation (51).


### 2.12. The Geometric Arithmetic Entropy ofD3n

IfH≅D3n, then, by using equation (5) and Table 3, we have(53)GAH=4n×22×22+2+36n2−20n×22×42+4+72n2−108n+44×24×44+4,⟹GAH=105.941125n2−122.856181n+44.Using equation (12) and Table 2, we have(54)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+36n2−20n×log22×42+422×4/2+4+72n2−108n+44×log24×44+424×4/4+4,ENTGAH=logGA−1GA−0.868087n2+0.482271n,where the GA index of D3n is written in equation (53).


## 2.1. Results on Dominating David Derived Network of Type 1

Here, we calculate certain degree-based entropies of dominating David derived network of type 1.D1n is shown in Figure 1. The edge partition of D1n is shown in Table 1. We compute Randic entropy, ABC entropy, and GA entropy for D1n.Figure 1
Dominating David derived network of type 1D12.Table 1
Edge partition ofD1n.
du,dv, where uv∈ED1nCardinality of edges2,24n2,34n−42,428n−163,39n2−13n+53,436n2−56n+244,436n2−52n+20


## 2.2. Randic Entropy ofD1n

IfH≅D1n, then, by using Table 1 and equation (3), we have(13)RαH=4n×4α+4n−4×6α+28n−16×8α+9n2−13n+5×9α+36n2−56n+24×12α+36n2−52n+20×16α,for α=1. Randic index is(14)⟹R1H=1089n2−1357n+501.Forα=−1,(15)⟹R−1H=254n2−15136n+4146.Forα=1/2,(16)⟹R1/2H=295.707658n2−343.995772n+123.085646.Forα=−1/2,(17)⟹R−1/2H=22.392305n2−19.966653n+6.305022.Using equation (10) and Table 1, the Randic entropy is(18)ENTRαH=logRα−1Rα4n×log4α4α+4n−4×log6α6α+28n−16×log8α8α+9n2−13n+5×log9α9α+36n2−56n+24×log12α12α+36n2−52n+20×log16α16α.Forα=1,(19)ENTR1H=logR1−1R14n×log44+4n−4×log66+28n−16×log88+9n2−13n+5×log99+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1H=logR1−1R11237.073052n2−1608.08325n+604.792358.Forα=1/2,(20)ENTR1/2H=logR1/2−1R1/24n×log44+4n−4×log66+28n−16×log88+9n2−13n+5×log99+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1/2H=logR1/2−1R1/2166.869996n2−206.530299n+75.935599.Forα=−1,(21)ENTR−1H=logR−1−1R−14n×log141/4+4n−4×log161/6+28n−16×log181/8+9n2−13n+5×log191/9+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1H=logR−1−1R−1−6.901056n2+6.046277n−1.8687.Forα=−1/2,(22)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+4n−4×log161/6+28n−16×log181/8+9n2−13n+5×log191/9+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−12.457494n2+12.909738n−4.354213,where Rα for α=1,−1,1/2,−1/2 is written in equations (14)–(17), respectively.


## 2.3. The Atom Bond Entropy ofD1n

IfH≅D1n, then, by using equation (4) and Table 1, the ABC index is(23)ABCH=4n×2+2−22×2+4n−4×2+3−22×3+28n−16×2+4−22×4+9n2−13n+5×3+3−23×3+36n2−56n+24×3+4−23×4+36n2−52n+20×4+4−24×4,⟹ABCH=51.283308n2−51.202034n+16.93058.Using equation (11) and Table 1, the ABC entropy is(24)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+4n−4×log2+3−22×32+3−2/2×3+28n−16×log2+4−22×42+4−2/2×4+9n2−13n+5×log3+3−23×33+3−2/3×3+36n2−56n+24×log3+4−23×43+4−23×4+36n2−52n+20×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−10.16953n2+11.348686n−4.011986,where ABC index of D1n is written in equation (23).


## 2.4. The Geometric Arithmetic Entropy ofD1n

IfH≅D1n, then, by using equation (5) and Table 1, the GA index is(25)GAH=4n×22×22+2+4n−4×22×32+3+28n−16×22×42+4+9n2−13n+5×23×33+3+36n2−56n+24×23×43+4+36n2−52n+20×24×44+4,⟹GAH=80.630759n2−86.107789n+29.749711.Using equation (12) and Table 1, we have(26)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+4n−4×log22×32+322×3/2+3+28n−16×log22×42+422×4/2+4+9n2−13n+5×log23×33+323×3/3+3+36n2−56n+24×log23×43+423×4/3+4+36n2−52n+20×log24×44+424×4/4+4,⟹ENTGAH=logGA−1GA−0.159534n2−0.461756n+0.314202,where GA index of D1n is written in equation (25).


## 2.5. Results on Dominating David Derived Network of Type 2

Here, we calculate certain degree-based entropies of Dominating David Derived network of type 2. TheD2n shown in Figure 2, and edge partition is shown in Table 2. We compute Randic entropy, ABC entropy and GA entropy for D2n.Figure 2
Dominating david derived network of type 2D24.Table 2
Edge partition ofD2n.
du,dv where uv∈ED2nCardinality of edges2,24n2,318n2−22n+62,428n−163,436n2−56n+244,436n2−52n+20


## 2.6. Randic Entropy ofD2n

IfH≅D2n, then, by using equation (3) and Table 2, we have(27)RαH=4n×4α+18n2−22n+6×6α+28n−16×8α+36n2−56n+24×12α+36n2−52n+20×16α.Forα=1,(28)⟹R1H=1116n2−1396n+516.Forα=−1,(29)⟹R−1H=334n2−8512n+94.Forα=1/2,(30)⟹R1/2H=312.798474n2−368.682505n+132.580543.Forα=−1/2,(31)⟹R−1/2H=26.740774n2−26.247775n+8.720839.Using equation (10) and Table 2, we have(32)ENTRαH=logRα−1Rα4n×log4α4α+18n2−22n+6×log6α6α+28n−16×log8α8α+36n2−56n+24×log12α12α+36n2−52n+20×log16α16α.Forα=1,(33)ENTR1H=logR1−1R14n×log44+18n2−22n+6×log66+28n−16×log88+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1H=logR1−1R11243.819743n2−1617.828471n+608.54052.Forα=1/2,(34)ENTR1/2H=logR1/2−1R1/24n×log44+18n2−22n+6×log66+28n−16×log88+36n2−56n+24×log1212+36n2−52n+20×log1616,⟹ENTR1/2H=logR1/2−1R1/2171.142383n2−212.701526n+78.309148.Forα=−1,(35)ENTR−1H=logR−1−1R−14n×log141/4+18n2−22n+6×log161/6+28n−16×log181/8+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1H=logR−1−1R−1−8.281267n2+8.039915n−2.635484.Forα=−1/2,(36)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+18n2−22n+6×log161/6+28n−16×log181/8+36n2−56n+24×log1121/12+36n2−52n+20×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−13.88524n2+14.972039n−5.147406,where Rα for α=1,−1,1/2,−1/2 is written in equations (28)–(31), respectively.


## 2.7. The Atom Bond Entropy ofD2n

IfH≅D2n, then, by using equation (4) and Table 2, we have(37)ABCH=4n×2+2−22×2+18n2−22n+6×2+3−22×3+28n−16×2+4−22×4+36n2−56n+24×3+4−23×4+36n2−52n+20×4+4−24×4,⟹ABCH=58.01123n2−60.920143n+20.668314.Using equation (11) and Table 2, we have(38)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+18n2−22n+6×log2+3−22×32+3−2/2×3+28n−16×log2+4−22×42+4−2/2×4+36n2−56n+24×log3+4−23×43+4−2/3×4+36n2−52n+20×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−11.028726n2+12.589746n−4.489317,where ABC index of D2n is written in equation (37).


## 2.8. The Geometric Arithmetic Entropy ofD2n

IfH≅D2n, then, by using equation (5) and Table 2, we have(39)GAH=4n×22×22+2+18n2−22n+6×22×32+3+28n−16×22×42+4+36n2−56n+24×23×43+4+36n2−52n+20×24×44+4,⟹GAH=89.267086n2−98.582482n+34.54767.Using equation (12) and Table 2, we have(40)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+18n2−22n+6×log22×32+322×3/2+3+28n−16×log22×42+422×4/2+4+36n2−56n+24×log23×43+423×4/3+4+36n2−52n+20×log24×44+424×4/4+4,ENTGAH=logGA−1GA−0.315869n2−0.235939n+0.227349,where the GA index of D2n is written in equation (39).


## 2.9. Results on Dominating David Derived Network of Type 3

Here, we calculate certain degree-based entropies of dominating David derived network of type 3.D3n is shown in Figure 3, and edge partition is shown in Table 3. We compute Randic entropy, ABC entropy, and GA entropy for D3n.Figure 3
Dominating David derived network of type 3D32.Table 3
Edge partition ofD3n.
du,dv, where uv∈ED3nCardinality of edges2,24n2,436n2−20n4,472n2−108n+44


## 2.10. Randic Entropy ofD3n

IfH≅D3n, then, by using equation (3) and Table 3, we have(41)RαH=4n×4α+36n2−20n×8α+72n2−108n+44×16α.Forα=1,(42)⟹R1H=1440n2−1872n+704.Forα=−1,(43)⟹R−1H=9n2−334n+114.Forα=1/2,(44)⟹R1/2H=389.823376n2−480.568542n+176.Forα=−1/2,(45)⟹R−1/2H=30.727922n2−32.071068n+11.Using equation (10) and Table 2, we have(46)ENTRαH=logRα−1Rα4n×log4α4α+36n2−20n×log8α8α+72n2−108n+44×log16α16α.Forα=1,(47)ENTR1H=logR1−1R14n×log44+36n2−20n×log88+72n2−108n+44×log1616,⟹ENTR1H=logR1−1R11647.236136n2−2215.580768n+847.700468.Forα=1/2,(48)ENTR1/2H=logR1/2−1R1/24n×log44+36n2−20n×log88+72n2−108n+44×log1616,⟹ENTR1/2H=logR1/2−1R1/2219.371113n2−283.224918n+105.962558.Forα=−1,(49)ENTR−1H=logR−1−1R−14n×log141/4+36n2−20n×log181/8+72n2−108n+44×log1161/16,⟹ENTR−1H=logR−1−1R−1−9.482445n2+9.783475n−3.31133.Forα=−1/2,(50)ENTR−1/2H=logR−1/2−1R−1/24n×log141/4+36n2−20n×log181/8+72n2−108n+44×log1161/16,⟹ENTR−1/2H=logR−1/2−1R−1/2−16.584309n2+18.846465n−6.62266,where Rα for α=1,−1,1/2,−1/2 is written in equations (42)–(45), respectively.


## 2.11. The Atom Bond Entropy ofD3n

IfH≅D2n, then, by using equation (4) and Table 3, we have(51)ABCH=4n×2+2−22×2+36n2−20n×2+4−22×4+72n2−108n+44×4+4−24×4,⟹ABCH=69.546659n2−77.449932n+26.944387.Using equation (11) and Table 3, we have(52)ENTABCH=logABC−1ABC4n×log2+2−22×22+2−2/2×2+36n2−20n×log2+4−22×42+4−2/2×4+72n2−108n+44×log4+4−24×44+4−2/4×4,⟹ENTABCH=logABC−1ABC−13.222141n2+15.788864n−5.738733,where the ABC index of D3n is written in equation (51).


## 2.12. The Geometric Arithmetic Entropy ofD3n

IfH≅D3n, then, by using equation (5) and Table 3, we have(53)GAH=4n×22×22+2+36n2−20n×22×42+4+72n2−108n+44×24×44+4,⟹GAH=105.941125n2−122.856181n+44.Using equation (12) and Table 2, we have(54)ENTGAH=logGA−1GA4n×log22×22+222×2/2+2+36n2−20n×log22×42+422×4/2+4+72n2−108n+44×log24×44+424×4/4+4,ENTGAH=logGA−1GA−0.868087n2+0.482271n,where the GA index of D3n is written in equation (53).


## 3. Discussion

Since entropy plays a vital role in various fields of science such as software engineering, medication, and pharmaceutical, its numerical values and graphical representation is very much important for researchers. Here, we calculate some exact values of degree-based entropies of dominating David derived networksD1n, D2n, and D3n. Furthermore, we construct Tables 4–6 to estimate the degree-based entropies for various values of n. From Tables 4–6, we can see that, as n increases, the degree-based entropies of these networks also increase.Table 4
Comparison table of entropies forD1n.
nENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA11.36401.38891.39351.38791.39731.397722.24052.25632.23612.25472.26172.262332.68492.69692.67772.69562.70092.701542.97932.98952.97232.98842.99292.993453.19923.20843.19283.20753.21153.211863.37473.38333.36893.38243.38613.386573.52073.52883.51543.52813.53153.531883.64583.65353.64083.65283.65603.656493.75503.76253.75053.76193.76493.7653103.85213.85943.84783.85883.86173.8620Table 5
Comparison table of entropies forD2n.
nENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA11.37911.40561.37561.40511.41431.414822.26682.28842.25742.28722.29582.296632.71362.73282.70252.73142.73942.740343.00903.02712.99723.02553.03353.034253.22953.24693.21743.24533.25313.253763.40533.42233.39303.42073.42843.429173.55163.56833.53923.56663.57433.574983.67673.69323.66433.69163.69923.699893.78623.80253.77363.80083.80833.8089103.88343.89953.87073.89783.90533.9059Table 6
Comparison table of entropies forD3n.
nENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA11.40751.43681.40421.43641.44631.446922.32962.35022.32152.34912.35682.357732.78422.80202.77592.80102.80772.808743.08273.09943.07483.09843.10473.105753.30493.32093.29723.31993.32613.326963.48193.49753.47443.49653.50243.503373.62893.64423.62153.64323.64903.649983.75463.76973.74733.76873.77453.775493.86453.87943.85723.87843.88413.8849103.96193.97673.95473.97583.98143.9823


## 4. Conclusion

In this study, taking into account the definition of Shannon and Chen entropy, we studied the classifications ofDDDn and also computed 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 dominating David derived network. We, in like manner, enlisted the mathematical assessments of these entropies in Tables 4–6. We gave the relation of these entropies which help us to know the physio-chemical activity of these networks [10, 3, 4].



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