# On Computation of Entropy of Hex-Derived Network

Authors: Pingping Song; Haidar Ali; Muhammad Ahsan Binyamin; Bilal Ali; Jia-Bao Liu
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. 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 asHDN1n and HDN2n, respectively of order n. We also compute some degree-based entropies such as Randić, ABC, and GA entropy of HDN1n and HDN2n.


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## Body

## 1. Introduction and Preliminary Results

A graphG is a tuple GV,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 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(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. The idea of entropy was given by Shannon in 1948 [2]. 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 is 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 [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 HX2 (Figure 1) is obtained from six triangles and HX3 (Figure 1) is obtained from HX2 by adding the layer of triangles around the boundary. Similarly, by adding the layer of triangles, we got HXn.Figure 1
Hexagonal meshes.
(a)(b)(c)The number of arc-wise connected open sets obtained by the partition of plane by a graph are known as faces ofG [6]. In Figure 1, (3) shows the faces of HX2. By combining the faces of HXn, with the vertices, we obtain HDN1n hex-derived network of type 1. Figure 2 shows the HDN14 hex-derived network of type 1 with 4 vertices [7, 8].Figure 2
Hex-derived networkHDN14.The vertex and edge partition ofHDN1n is shown in Tables 1 and 2, respectively.Table 1
Vertex partition ofHDN1n.
du, where u∈VHDN1nCardinality of vertices36n2−12n+65676n−12123n2−9n+7Table 2
Edge partition ofHDN1n.
du,dv, where uv∈EHDN1nCardinality of edges3,5123,718n−363,1218n2−54n+425,7125,1267,76n−187,1212n−2412,129n2−33n+30As discussed before in the formationHDN1n, if we join the vertices of HDN1n with each other, then the new figure formed by this is hex-derived network of type 2 HDN2n. It is clear that HDN1n is a subgraph of HDN2n. The hex-derived network of type 2 with 4 vertices is shown in Figure 3.Figure 3
Hex-derived networkHDN24.The vertex and edge partition ofHDN2n is shown in Tables 3 and 4, respectively.Table 3
Vertex partition ofHDN2n.
du, where u∈VHDN2nCardinality of vertices56n66n2−18n+1276n−12123n2−9n+7Table 4
Edge partition ofHDN2n.
du,dv, where uv∈EHDN2nCardinality of edges5,5185,612n−245,712n−125,126n6,69n2−33n+306,76n−126,1218n2−60n+487,76n−187,1212n−2412,129n2−33n+30


### 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.(3)RαG=∑e=uv∈EGdu×dvα,where α=1,−1,1/2,−1/2.ABC index was introduced in 1998 by Estrada et al. [12]. It has the formulae:(4)ABCG=∑e=uv∈EGdu+dv−2du×dv.Vukičević was the person who studied this index for the first time [13]. It is written as GA index and written as follows:(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, we have∑j=1pduj=2q. So,(8)ENTdG=log2q−12q∑i=1plogduidui.


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

The edge partition entropy of a graphG was introduced in 2014 by Chen et al. [14].(9)ENTdG=−∑u′v′∈EGdu′v′∑uv∈EGduvlogdu′v′∑uv∈EGduv.


### 1.4. Randic Entropy

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


### 1.5. Atom Bond Connectivity Entropy

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


### 1.6. The Geometric Arithmetic Entropy

Using equation (5), equation (9) is reduced as(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 Randić [9] and generalised by Bollobás and Erdos [10], and Amić et al. [11], 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. [12]. It has the formulae:(4)ABCG=∑e=uv∈EGdu+dv−2du×dv.Vukičević was the person who studied this index for the first time [13]. It is written as GA index and written as follows:(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, we have∑j=1pduj=2q. So,(8)ENTdG=log2q−12q∑i=1plogduidui.


## 1.3. Edge Partition-Based Entropy of Graph

The edge partition entropy of a graphG was introduced in 2014 by Chen et al. [14].(9)ENTdG=−∑u′v′∈EGdu′v′∑uv∈EGduvlogdu′v′∑uv∈EGduv.


## 1.4. Randic Entropy

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


## 1.5. Atom Bond Connectivity Entropy

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


## 1.6. The Geometric Arithmetic Entropy

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


## 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 ic´ entropy, ABC entropy, and GA entropy for hex-derived network HDN1n.


### 2.2. Degree-Based Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (8) and Table 1, we get(13)ENTdH1=log227n2−51n+24−1227n2−51n+24∑i=1plogdridri,⟹ENTdH1=log227n2−51n+24−1227n2−51n+246n2−12n+6×log33+6×log55+6n−12×log77+3n2−9n+7×log1212,⟹ENTdH1=log54n2−102n+48−154n2−102n+486n2−12n+6×log33+6×log55+6n−12×log77+3n2−9n+7×log1212,⟹ENTdH1=log54n2−102n+48−154n2−102n+4847.438707n2−98.233822n+49.220272.


### 2.3. Randic Entropy

IfH1≅HDN1n, then by using Table 2 and equation (3), we have(14)RαH1=12×15α+18n−36×21α+18n2−54n+42×36α+12×35α+6×60α+6n−18×49α+12n−24×84α+9n2−33n+30×144α,and for α=1, Randic index is(15)R1H1=1944n2−5016n+3138.Forα=−1,(16)R−1H1=916n2−3548n+173280.Forα=1/2,(17)R1/2H1=216n2+4221−720n+2415−8421+1235+612.Forα=−1/2,(18)R−1/2H1=154n2+2421−14112n+15−4821+1235+12514.Using equation (10) and Table 2, the Randic entropy is(19)ENTRαH1=logRα−1Rα12×log15α15α+18n−36×log21α21α+18n2−54n+42×log36α36α+12×log35α35α+6×log60α60α+6n−18×log49α49α+12n−24×log84α84α+9n2−33n+30×log144α144α.Forα=1,(20)ENTR1H1=logR1−1R112×log1515+18n−36×log2121+18n2−54n+42×log3636+12×log3535+6×log6060+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1H1=logR1−1R13805.72181n2−10345.600562n+6807.897032.Forα=1/2,(21)ENTR1/2H1=logR1/2−1R1/212×log1515+18n−36×log2121+18n2−54n+42×log3636+12×log3535+6×log6060+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1/2H1=logR1/2−1R1/2200.59191n2−483.632264n+280.875589.Forα=−1,(22)ENTR−1H1=logR−1−1R−112×log1151/15+18n−36×log1211/21+18n2−54n+42×log1361/36+12×log1351/35+6×log1601/60+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1H1=logR−1−1R−1−0.913049n2+1.213888n−0.476083.Forα=−1/2,(23)ENTR−1/2H1=logR−1/2−1R−1/212×log1151/15+18n−36×log1211/21+12×log1351/35+18n2−54n+42×log1361/36+6×log1601/60+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1/2H1=logR−12−1R−1/2−3.14384n2+5.390216n−2.335494,where Rα for α=1,−1,1/2,−1/2 is written in equations (15)–(18), respectively.


### 2.4. The Atom Bond Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (4) and Table 2, the ABC index is(24)ABCH1=12×3+5−23×5+18n−36×3+7−23×7+18n2−54n+42×3+12−23×12+12×5+7−25×7+6×5+12−25×12+6n−18×7+7−27×7+12n−24×7+12−27×12+9n2−51n+24×12+12−212×12,⟹ABCH1=3224+313n2+12427−913−23577−332212n+12105−24427+713+12147−43577+5222.Using equation (11) and Table 2, the ABC entropy is(25)ENTABCH1=logABC−1ABC12×log3+5−23×53+5−2/3×5+18n−36×log3+7−23×73+7−2/3×7+18n2−54n+42×log3+12−23×123+12−2/3×12+12×log5+7−25×75+7−2/5×7+6×log5+12−25×125+12−2/5×12+6n−18×log7+7−27×77+7−2/7×7+12n−24×log7+12−27×127+12−2/7×12+9n2−51n+24×log12+12−212×1212+12−2/12×12,ENTABCH1=logABC−1ABC12×log6156/15+18n−36×log8218/21+18n2−54n+42×log133613/36+12×log103510/35+6×log156015/60+6n−18×log124912/49+12n−24×log178417/84+9n2−51n+24×log2214422/144,⟹ENTABCH1=logABC−1ABC−3.827584n2+10.201707n−2.444075,where ABC index of HDN1n is written in equation (24).


### 2.5. The Geometric Arithmetic Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (5) and Table 2, GA index is(26)GAH1=12×23×53+5+18n−36×23×73+7+18n2−54n+42×23×123+12+12×25×75+7+6×25×125+12+6n−18×27×77+7+12n−24×27×127+12+9n2−33n+30×212×1212+12,⟹GAH1=1175n2+5822195−3815n+753517−11282195+235+3185.Using equation (10) and Table 2, we have(27)ENTGAH1=logGA−1GA12×log2158215/8+18n−36×log22110221/10+18n2−54n+42×log23615236/15+12×log23512235/12+6×log26017260/17+6n−18×log24914249/14+12n−24×log28419284/19+9n2−33n+30×log2144242144/24,⟹ENTGAH1=logGA−1GA−1.395504n2+3.381504n−2.101958,where GA index of HDN1n 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 ic´ entropy, ABC entropy, and GA entropy for hex-derived network HDN2n.


### 2.7. Degree-Based Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (8) and Table 3, we have(28)ENTdH2=log236n2−72n+36−1236n2−72n+366n×log55+6n2−18n+12×log66+6n−12×log77+3n2−9n+7×log1212⟹ENTdH2=log72n2−144n+72−172n2−144n+726n×log55+6n2−18n+12×log66+6n−12×log77+3n2−9n+7×log1212⟹ENTdH2=log72n2−144n+72−172n2−144n+7266.86397n2−144.128692n+75.689879.


### 2.8. Randic Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (3) and Table 4, we have(29)RαH2=18×25α+12n−24×30α+12n−12×35α+6n×60α+9n2−33n+30×36α+6n−12×42α+18n2−60n+48×72α+6n−18×49α+12n−24×84α+9n2−33n+30×144α.Forα=1,(30)R1H2=6486n2−1261n+794.Forα=−1,(31)R−1H2=33075n2+42815n−2038658800.Forα=1/2,(32)R1/2H2=6105+7−3+n+421−2+n+230−2+n+42−2+n+235−1+n−99n+215n+27n2+628−10n+3n2.Forα=−1/2,(33)R−1/2H2=185+67−3+n+237−2+n+67−2+n+265−2+na+12−1+n35+35n+3410−11n+3n2+8−10n+3n22.Using equation (10) and Table 4, we have(34)ENTRαH2=logRα−1Rα18×log25α25α+12n−24×log30α30α+12n−12×log35α35α+6n×log60α60α+9n2−33n+30×log36α36α+6n−12×log42α42α+18n2−60n+48×log72α72α+6n−18×log49α49α+12n−24×log84α84α+9n2−33n+30×log144α144α.Forα=1,(35)ENTR1H2=logR1−1R118×log2525+12n−24×log3030+12n−12×log3535+6n×log6060+9n2−33n+30×log3636+6n−12×log4242+18n2−60n+48×log7272+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144⟹ENTR1H2=logR1−1R15708.582715n2−15463.045648n+10152.693271.Forα=1/2,(36)ENTR1/2H2=logR1/2−1R1/218×log2525+12n−24×log3030+12n−12×log3535+6n×log6060+9n2−33n+30×log3636+6n−12×log4242+18n2−60n+48×log7272+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1/2H2=logR1/2−1R1/2258.391474n2−614.170818n+359.638902.Forα=−1,(37)ENTR−1H2=logR−1−1R−118×log1251/25+12n−24×log1301/30+12n−12×log1351/35+6n×log1601/30+9n2−33n+30×log1361/36+6n−12×log1421/42+18n2−60n+48×log1721/72+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1H2=logR−1−1R−1−0.599231n2+0.262484n+0.187377.Forα=−1/2,(38)ENTR−1/2H2=logR−1/2−1R−1/218×log1251/25+12n−24×log1301/30+12n−12×log1351/35+6n×log1601/60+9n2−33n+30×log1361/36+6n−12×log1421/42+18n2−60n+48×log1721/72+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1/2H2=logR−1/2−1R−1/2−2.779385n2+3.677553n−0.972811,where Rα for α=1,−1,1/2,−1/2 is written in equations (30)–(33), respectively.


### 2.9. The Atom Bond Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (4) and Table 4, we have(39)ABCH2=18×5+5−25×5+12n−24×5+6−25×6+12n−12×5+7−25×7+6n×5+12−25×12+9n2−33n+30×6+6−26×6+6n−12×6+7−26×7+18n2−60n+48×6+12−26×12+6n−18×7+7−27×7+12n−24×7+12−27×12+9n2−33n+30×12+12−212×12,⟹ABCH2=3625+1273−3+n+665−2+n+2517−2+n+667−2+n,+1227−1+n+3n+5210−11n+3n2+1211210−11n+3n2+228−10n+3n2.Using equation (11) and Table 4, we have(40)ENTABCH2=logABC−1ABC18×log5+5−25×55+5−2/5×5+12n−24×log5+6−25×65+6−2/5×6+12n−12×log5+7−25×75+7−2/5×7+6n×log5+12−25×125+12−2/5×12+9n2−33n+30×log6+6−26×66+6−2/6×6+6n−12×log6+7−26×76+7−2/6×7+18n2−60n+48×log6+12−26×126+12−2/6×12+6n−18×log7+7−27×77+7−2/7×7+12n−24×log7+12−27×127+12−2/7×12+9n2−33n+30×log12+12−212×1212+12−2/12×12,ENTABCH2=logABC−1ABC18×log8258/25+12n−24×log9309/30+12n−12×log103510/35+6n×log156015/60+9n2−33n+30×log103610/36+6n−12×log114211/42+18n2−60n+48×log167216/72+6n−18×log124912/49+12n−24×log178417/84+9n2−33n+30×log2214422/144,⟹ENTABCH2=logABC−1ABC−5.525895n2+11.29825n−5.656245,where ABC index of HDN2n is written in equation (39).


### 2.10. The Geometric Arithmetic Entropy ofHDN2n

IfH2≅HDN2n, by using equation (5) and Table 4, we have(41)GAH2=18×25×55+5+12n−24×25×65+6+12n−12×25×75+7+6n×25×125+12+9n2−33n+30×26×66+6+6n−12×26×76+7+6n−18×27×77+7+18n2−60n+48×26×126+12+12n−24×27×127+12+9n2−33n+30×212×1212+12,⟹GAH2=60+481921−2+n+241130−2+n+121342−2+n+235−1+n−60n+2415n17+18n2+428−10n+3n2.Using equation (12) and Table 4, we have(42)ENTGAH2=logGA−1GA18×log25×55+525×5/5+5+12n−24×log25×65+625×6/5+6+12n−12×log25×75+725×7/5+7+6n×log25×125+1225×12/5+12+9n2−33n+30×log26×66+626×6/6+6+6n−12×log26×76+726×7/6+7+18n2−60n+48×log26×126+1226×12/6+12+6n−18×log27×77+727×7/7+7+12n−24×log27×127+1227×12/7+12+9n2−33n+30×log212×1212+12212×12/12+12,ENTGAH2=logGA−1GA18×log22510225/10+12n−24×log23011230/11+12n−12×log23512235/12+6n×log26017260/17+9n2−33n+30×log23612236/12+6n−12×log24213242/13+18n2−60n+48×log27218272/18+6n−18×log24914249/14+12n−24×log28419284/19+9n2−33n+30×log2144242144/24,⟹ENTGAH2=logGA−1GA−0.434044n2+0.944184n−0.665748,where GA index of HDN2n 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 HDN1n and HDN2n. 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.Table 5
Comparison table of entropies forHDN1n.
NENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA21.4266671.7208561.347411.453922—1.66876831.9528672.0320981.9895212.0475081.6720822.09424842.2954262.3440172.3526442.3945412.2059652.40748152.5418552.5862462.6044772.6401222.515292.64401962.7339432.7793842.7975272.8306212.7383712.83205672.8912732.9388692.9543392.9864172.9140422.98753283.0244853.0743173.0865413.1182923.0593373.11984993.1399853.1918693.2009073.2326553.1833813.234919103.2419313.2956283.3017323.3336363.2916793.336674Table 6
Comparison table of entropies forHDN2n.
nENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA21.5223791.5483621.0277111.5499441.555931.55594632.0976032.2263591.4791682.0351142.1477752.15813642.4518362.5982161.7851632.3401082.4964132.51034252.7035822.8570862.0161722.564592.7448162.76022762.898723.0560512.2012712.7425652.9379182.9540573.0580313.2177442.3554032.8901043.0958923.11241583.1926313.3539682.4872753.0161353.229563.24630993.3091613.4716752.6023993.1261523.3454063.362293103.4118993.5753052.7044833.22377263.4476263.464599Figure 4
Randic entropy ofHDN1n for α=1.Figure 5
Randić entropy ofHDN1n for α=1/2.Figure 6
Randić entropy ofHDN1n for α=−1.Figure 7
Randić entropy ofHDN1n for α=−1/2.Figure 8
ABC entropy of HDN1n.Figure 9
GA entropy of HDN1n.Figure 10
Randic entropy ofHDN2n for α=1.Figure 11
Randić entropy ofHDN2n for α=1/2.Figure 12
Randić entropy ofHDN2n for α=−1.Figure 13
Randić entropy ofHDN2n for α=−1/2.Figure 14
ABC entropy of HDN2n.Figure 15
GA entropy of HDN2n.


## 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 HDN1n.


## 2.2. Degree-Based Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (8) and Table 1, we get(13)ENTdH1=log227n2−51n+24−1227n2−51n+24∑i=1plogdridri,⟹ENTdH1=log227n2−51n+24−1227n2−51n+246n2−12n+6×log33+6×log55+6n−12×log77+3n2−9n+7×log1212,⟹ENTdH1=log54n2−102n+48−154n2−102n+486n2−12n+6×log33+6×log55+6n−12×log77+3n2−9n+7×log1212,⟹ENTdH1=log54n2−102n+48−154n2−102n+4847.438707n2−98.233822n+49.220272.


## 2.3. Randic Entropy

IfH1≅HDN1n, then by using Table 2 and equation (3), we have(14)RαH1=12×15α+18n−36×21α+18n2−54n+42×36α+12×35α+6×60α+6n−18×49α+12n−24×84α+9n2−33n+30×144α,and for α=1, Randic index is(15)R1H1=1944n2−5016n+3138.Forα=−1,(16)R−1H1=916n2−3548n+173280.Forα=1/2,(17)R1/2H1=216n2+4221−720n+2415−8421+1235+612.Forα=−1/2,(18)R−1/2H1=154n2+2421−14112n+15−4821+1235+12514.Using equation (10) and Table 2, the Randic entropy is(19)ENTRαH1=logRα−1Rα12×log15α15α+18n−36×log21α21α+18n2−54n+42×log36α36α+12×log35α35α+6×log60α60α+6n−18×log49α49α+12n−24×log84α84α+9n2−33n+30×log144α144α.Forα=1,(20)ENTR1H1=logR1−1R112×log1515+18n−36×log2121+18n2−54n+42×log3636+12×log3535+6×log6060+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1H1=logR1−1R13805.72181n2−10345.600562n+6807.897032.Forα=1/2,(21)ENTR1/2H1=logR1/2−1R1/212×log1515+18n−36×log2121+18n2−54n+42×log3636+12×log3535+6×log6060+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1/2H1=logR1/2−1R1/2200.59191n2−483.632264n+280.875589.Forα=−1,(22)ENTR−1H1=logR−1−1R−112×log1151/15+18n−36×log1211/21+18n2−54n+42×log1361/36+12×log1351/35+6×log1601/60+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1H1=logR−1−1R−1−0.913049n2+1.213888n−0.476083.Forα=−1/2,(23)ENTR−1/2H1=logR−1/2−1R−1/212×log1151/15+18n−36×log1211/21+12×log1351/35+18n2−54n+42×log1361/36+6×log1601/60+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1/2H1=logR−12−1R−1/2−3.14384n2+5.390216n−2.335494,where Rα for α=1,−1,1/2,−1/2 is written in equations (15)–(18), respectively.


## 2.4. The Atom Bond Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (4) and Table 2, the ABC index is(24)ABCH1=12×3+5−23×5+18n−36×3+7−23×7+18n2−54n+42×3+12−23×12+12×5+7−25×7+6×5+12−25×12+6n−18×7+7−27×7+12n−24×7+12−27×12+9n2−51n+24×12+12−212×12,⟹ABCH1=3224+313n2+12427−913−23577−332212n+12105−24427+713+12147−43577+5222.Using equation (11) and Table 2, the ABC entropy is(25)ENTABCH1=logABC−1ABC12×log3+5−23×53+5−2/3×5+18n−36×log3+7−23×73+7−2/3×7+18n2−54n+42×log3+12−23×123+12−2/3×12+12×log5+7−25×75+7−2/5×7+6×log5+12−25×125+12−2/5×12+6n−18×log7+7−27×77+7−2/7×7+12n−24×log7+12−27×127+12−2/7×12+9n2−51n+24×log12+12−212×1212+12−2/12×12,ENTABCH1=logABC−1ABC12×log6156/15+18n−36×log8218/21+18n2−54n+42×log133613/36+12×log103510/35+6×log156015/60+6n−18×log124912/49+12n−24×log178417/84+9n2−51n+24×log2214422/144,⟹ENTABCH1=logABC−1ABC−3.827584n2+10.201707n−2.444075,where ABC index of HDN1n is written in equation (24).


## 2.5. The Geometric Arithmetic Entropy ofHDN1n

IfH1≅HDN1n, then by using equation (5) and Table 2, GA index is(26)GAH1=12×23×53+5+18n−36×23×73+7+18n2−54n+42×23×123+12+12×25×75+7+6×25×125+12+6n−18×27×77+7+12n−24×27×127+12+9n2−33n+30×212×1212+12,⟹GAH1=1175n2+5822195−3815n+753517−11282195+235+3185.Using equation (10) and Table 2, we have(27)ENTGAH1=logGA−1GA12×log2158215/8+18n−36×log22110221/10+18n2−54n+42×log23615236/15+12×log23512235/12+6×log26017260/17+6n−18×log24914249/14+12n−24×log28419284/19+9n2−33n+30×log2144242144/24,⟹ENTGAH1=logGA−1GA−1.395504n2+3.381504n−2.101958,where GA index of HDN1n 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 ic´ entropy, ABC entropy, and GA entropy for hex-derived network HDN2n.


## 2.7. Degree-Based Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (8) and Table 3, we have(28)ENTdH2=log236n2−72n+36−1236n2−72n+366n×log55+6n2−18n+12×log66+6n−12×log77+3n2−9n+7×log1212⟹ENTdH2=log72n2−144n+72−172n2−144n+726n×log55+6n2−18n+12×log66+6n−12×log77+3n2−9n+7×log1212⟹ENTdH2=log72n2−144n+72−172n2−144n+7266.86397n2−144.128692n+75.689879.


## 2.8. Randic Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (3) and Table 4, we have(29)RαH2=18×25α+12n−24×30α+12n−12×35α+6n×60α+9n2−33n+30×36α+6n−12×42α+18n2−60n+48×72α+6n−18×49α+12n−24×84α+9n2−33n+30×144α.Forα=1,(30)R1H2=6486n2−1261n+794.Forα=−1,(31)R−1H2=33075n2+42815n−2038658800.Forα=1/2,(32)R1/2H2=6105+7−3+n+421−2+n+230−2+n+42−2+n+235−1+n−99n+215n+27n2+628−10n+3n2.Forα=−1/2,(33)R−1/2H2=185+67−3+n+237−2+n+67−2+n+265−2+na+12−1+n35+35n+3410−11n+3n2+8−10n+3n22.Using equation (10) and Table 4, we have(34)ENTRαH2=logRα−1Rα18×log25α25α+12n−24×log30α30α+12n−12×log35α35α+6n×log60α60α+9n2−33n+30×log36α36α+6n−12×log42α42α+18n2−60n+48×log72α72α+6n−18×log49α49α+12n−24×log84α84α+9n2−33n+30×log144α144α.Forα=1,(35)ENTR1H2=logR1−1R118×log2525+12n−24×log3030+12n−12×log3535+6n×log6060+9n2−33n+30×log3636+6n−12×log4242+18n2−60n+48×log7272+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144⟹ENTR1H2=logR1−1R15708.582715n2−15463.045648n+10152.693271.Forα=1/2,(36)ENTR1/2H2=logR1/2−1R1/218×log2525+12n−24×log3030+12n−12×log3535+6n×log6060+9n2−33n+30×log3636+6n−12×log4242+18n2−60n+48×log7272+6n−18×log4949+12n−24×log8484+9n2−33n+30×log144144,⟹ENTR1/2H2=logR1/2−1R1/2258.391474n2−614.170818n+359.638902.Forα=−1,(37)ENTR−1H2=logR−1−1R−118×log1251/25+12n−24×log1301/30+12n−12×log1351/35+6n×log1601/30+9n2−33n+30×log1361/36+6n−12×log1421/42+18n2−60n+48×log1721/72+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1H2=logR−1−1R−1−0.599231n2+0.262484n+0.187377.Forα=−1/2,(38)ENTR−1/2H2=logR−1/2−1R−1/218×log1251/25+12n−24×log1301/30+12n−12×log1351/35+6n×log1601/60+9n2−33n+30×log1361/36+6n−12×log1421/42+18n2−60n+48×log1721/72+6n−18×log1491/49+12n−24×log1841/84+9n2−33n+30×log11441/144,⟹ENTR−1/2H2=logR−1/2−1R−1/2−2.779385n2+3.677553n−0.972811,where Rα for α=1,−1,1/2,−1/2 is written in equations (30)–(33), respectively.


## 2.9. The Atom Bond Entropy ofHDN2n

IfH2≅HDN2n, then by using equation (4) and Table 4, we have(39)ABCH2=18×5+5−25×5+12n−24×5+6−25×6+12n−12×5+7−25×7+6n×5+12−25×12+9n2−33n+30×6+6−26×6+6n−12×6+7−26×7+18n2−60n+48×6+12−26×12+6n−18×7+7−27×7+12n−24×7+12−27×12+9n2−33n+30×12+12−212×12,⟹ABCH2=3625+1273−3+n+665−2+n+2517−2+n+667−2+n,+1227−1+n+3n+5210−11n+3n2+1211210−11n+3n2+228−10n+3n2.Using equation (11) and Table 4, we have(40)ENTABCH2=logABC−1ABC18×log5+5−25×55+5−2/5×5+12n−24×log5+6−25×65+6−2/5×6+12n−12×log5+7−25×75+7−2/5×7+6n×log5+12−25×125+12−2/5×12+9n2−33n+30×log6+6−26×66+6−2/6×6+6n−12×log6+7−26×76+7−2/6×7+18n2−60n+48×log6+12−26×126+12−2/6×12+6n−18×log7+7−27×77+7−2/7×7+12n−24×log7+12−27×127+12−2/7×12+9n2−33n+30×log12+12−212×1212+12−2/12×12,ENTABCH2=logABC−1ABC18×log8258/25+12n−24×log9309/30+12n−12×log103510/35+6n×log156015/60+9n2−33n+30×log103610/36+6n−12×log114211/42+18n2−60n+48×log167216/72+6n−18×log124912/49+12n−24×log178417/84+9n2−33n+30×log2214422/144,⟹ENTABCH2=logABC−1ABC−5.525895n2+11.29825n−5.656245,where ABC index of HDN2n is written in equation (39).


## 2.10. The Geometric Arithmetic Entropy ofHDN2n

IfH2≅HDN2n, by using equation (5) and Table 4, we have(41)GAH2=18×25×55+5+12n−24×25×65+6+12n−12×25×75+7+6n×25×125+12+9n2−33n+30×26×66+6+6n−12×26×76+7+6n−18×27×77+7+18n2−60n+48×26×126+12+12n−24×27×127+12+9n2−33n+30×212×1212+12,⟹GAH2=60+481921−2+n+241130−2+n+121342−2+n+235−1+n−60n+2415n17+18n2+428−10n+3n2.Using equation (12) and Table 4, we have(42)ENTGAH2=logGA−1GA18×log25×55+525×5/5+5+12n−24×log25×65+625×6/5+6+12n−12×log25×75+725×7/5+7+6n×log25×125+1225×12/5+12+9n2−33n+30×log26×66+626×6/6+6+6n−12×log26×76+726×7/6+7+18n2−60n+48×log26×126+1226×12/6+12+6n−18×log27×77+727×7/7+7+12n−24×log27×127+1227×12/7+12+9n2−33n+30×log212×1212+12212×12/12+12,ENTGAH2=logGA−1GA18×log22510225/10+12n−24×log23011230/11+12n−12×log23512235/12+6n×log26017260/17+9n2−33n+30×log23612236/12+6n−12×log24213242/13+18n2−60n+48×log27218272/18+6n−18×log24914249/14+12n−24×log28419284/19+9n2−33n+30×log2144242144/24,⟹ENTGAH2=logGA−1GA−0.434044n2+0.944184n−0.665748,where GA index of HDN2n 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 HDN1n and HDN2n. 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.Table 5
Comparison table of entropies forHDN1n.
NENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA21.4266671.7208561.347411.453922—1.66876831.9528672.0320981.9895212.0475081.6720822.09424842.2954262.3440172.3526442.3945412.2059652.40748152.5418552.5862462.6044772.6401222.515292.64401962.7339432.7793842.7975272.8306212.7383712.83205672.8912732.9388692.9543392.9864172.9140422.98753283.0244853.0743173.0865413.1182923.0593373.11984993.1399853.1918693.2009073.2326553.1833813.234919103.2419313.2956283.3017323.3336363.2916793.336674Table 6
Comparison table of entropies forHDN2n.
nENTR1ENTR1/2ENTR−1ENTR−1/2ENTABCENTGA21.5223791.5483621.0277111.5499441.555931.55594632.0976032.2263591.4791682.0351142.1477752.15813642.4518362.5982161.7851632.3401082.4964132.51034252.7035822.8570862.0161722.564592.7448162.76022762.898723.0560512.2012712.7425652.9379182.9540573.0580313.2177442.3554032.8901043.0958923.11241583.1926313.3539682.4872753.0161353.229563.24630993.3091613.4716752.6023993.1261523.3454063.362293103.4118993.5753052.7044833.22377263.4476263.464599Figure 4
Randic entropy ofHDN1n for α=1.Figure 5
Randić entropy ofHDN1n for α=1/2.Figure 6
Randić entropy ofHDN1n for α=−1.Figure 7
Randić entropy ofHDN1n for α=−1/2.Figure 8
ABC entropy of HDN1n.Figure 9
GA entropy of HDN1n.Figure 10
Randic entropy ofHDN2n for α=1.Figure 11
Randić entropy ofHDN2n for α=1/2.Figure 12
Randić entropy ofHDN2n for α=−1.Figure 13
Randić entropy ofHDN2n for α=−1/2.Figure 14
ABC entropy of HDN2n.Figure 15
GA entropy of HDN2n.


## 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,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.



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*Source: 9993504-2021-04-15.xml*