On topological aspects of degree based entropy for two carbon nanosheets

Shazia Manzoor; Yu-Ming Chu; Muhammad Kamran Siddiqui; Sarfraz Ahmad

doi:10.1515/mgmc-2020-0025

Open Access Published by De Gruyter December 31, 2020

On topological aspects of degree based entropy for two carbon nanosheets

Shazia Manzoor , Yu-Ming Chu , Muhammad Kamran Siddiqui and Sarfraz Ahmad

From the journal Main Group Metal Chemistry

https://doi.org/10.1515/mgmc-2020-0025

Abstract

The entropy-based procedures from the configuration of chemical graphs and multifaceted networks, several graph properties have been utilized. For computing, the organizational evidence of organic graphs and multifaceted networks, the graph entropies have converted the information-theoretic magnitudes. The graph entropy portion has attracted the research community due to its potential application in chemistry. In this paper, our input is to reconnoiter graph entropies constructed on innovative information function, which is the quantity of different degree vertices along with the quantity of edges between innumerable degree vertices.”In this study, we explore two dissimilar curricula of carbon nanosheets that composed by C₄ and C₈ denoted by T¹C₄C₈(S)[m, n] and T²C₄C₈(R)[m, n]. Additionally, we calculate entropies of these configurations by creating a connection of degree-based topological indices with the advantage of evidence occupation.

Keywords: entropy; Zagreb kind indices; Balaban index; carbon nanosheets; T1C4C8(S)[m, n]; T2C4C8(R)[m, n]

1 Introduction

A branch of mathematical chemistry that uses the tools of graph theory to develop the organic phenomenon mathematically is called chemical graph theory (Ali et al., 2019). Additionally, for resolving molecular problems, chemical graph theory links to the nontrivial solicitations of graph theory. This theory has important applications in the domain of chemical sciences (Gao and Farahani, 2016; Gao et al., 2018). Chem-informatics (containing chemistry and information science) analysis (QSAR) and (QSPR) are used to anticipate the bioactivity and physiochemical possessions of organic mixtures (Wu et al., 2015).

G is a graph in which the vertex and the edge set are represented by V(G) and E(G), respectively. The degree Ξ(r) of a vertex r is the quantity of edges of G contiguous with vertex r. Let G be a graph with m vertices and n edges, where m embodies the order and n refers to the size of the graph. A graph of order m and size n is characterized as (m, n)-graph see: Assaye et al. (2019), Basavanagoud et al. (2017), Hosamani et al. (2017), and Shirakol (2019).

In molecular graph, the vertices designate atoms and edges signpost as the substance bonds. A arithmetic value that is calculated arithmetically by using the molecular graph is characterized as a topological index. It is connected to chemical composition demonstrating for association of chemical structure with plentiful physical, chemical possessions and biological activities. For further details of formulas of topological indices and application points see: Siddiqui et al., (2016a, 2016b), and Gao et al. (2017) and Imran et al. (2018), respectively.

The basic idea of entropy was introduced in the following statement: “The entropy of a possible dissemination is known as a quota of the unpredictability of evidence content or a portion of the uncertainty of a coordination” (Shannon, 2001), which was developed for evaluating the mechanical evidence of graphs and chemical networks.

Afterward, it has been used significantly in graphs and chemical networks. Rashevsky (1955) introduced the graph entropy impression established on the classifications of vertex orbits in 1955. Currently, graph entropies have been widely pragmatic in an extensive assortment of questions, such as chemistry and sociology (Dehmer and Grabner, 2013; Ulanowicz, 2004).

There are many specific categories of aforementioned graph entropy procedures. They associate probability distributions with elements (vertices, edges, etc.) of a graph that can be categorized as core and extrinsic dealings. Dehmer (2008) and Dehmer et al. (2012) presented information based function graph entropies, which imprisonment operational evidence and analyze their properties. For more details, see: Bonchev (2003), Dehmer and Mowshowitz (2011), Morowitz (1955), Rashevsky (1955), Shannon (2001), Solé and Valverde (2004), Quastler (1954), Tan and Wu (2004), and Trucco (1956).

2 Degree based topological indices of graph

Gutman with coauthors (Gutman and Das, 2004; Gutman and Trinajstic, 1972) defined first and second Zagreb index as:

M1=∑rs∈E(G)( Ξ (r)+ Ξ (s)),M2(G)=∑rs∈E(G)( Ξ (r)× Ξ (s))

Shirdel et al. (2013) introduced “hyper Zagreb index” as:

HM=∑rs∈E(G)( Ξ (r)+ Ξ (s))2

Furtula and Gutman (2015) defined F-index as:

F(G)=∑rs∈E(G)( Ξ (r))2+( Ξ (s))2

Furtula et al. (2010) acquainted with a new topological index named “augmented Zagreb index” and is characterized as:

AZI(G)=∑rs∈E(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23

Balaban (1982) and Balaban and Quintas (1983) announced a new topological index for a graph G of order p, size q as:

J(G)=qq−p+2∑rs∈E(G)1Ξ(r)Ξ(s)

For more details about these indices see: Akhter et al. (2016), Ali et al. (2019), Liu et al. (2020a, 2020b), Raza (2020a, 2020b), Raza and Ali (2020), and Raza and Sukaiti (2020).

3 General entropy of graph

In 2014, Chen et al. (2014) familiarized the definition of the entropy of edge partisan graph. The entropy of edge partisan graph is characterized in Eq. 1:

(1)ENTψ(G)=∑r′s′∈E(G)ψr′s′∑rs∈E(G)ψ(rs)logψr′s′∑rs∈E(G)ψ(rs)

3.1 The first Zagreb entropy

If Ψ(rs) = Ξ(r)+Ξ(s), then:

∑rs∈E(G)ψ(rs)=∑rs∈E(G)(Ξ(r)+Ξ(s))=M1(G)

Now Eq. 1 is converted and called the first Zagreb entropy:

(2)ENTM1(G)=logM1(G)−1M1(G)log∏rs∈E1(G)[ Ξ (r)+ Ξ (s)][E(r)+E(s)]

3.2 The second Zagreb entropy

If Ψ(rs) =Ξ(r)xΞ(s), then:

∑rs∈E(G) ψ (rs)=∑rs∈E(G)( Ξ (r)× Ξ (s))=M2(G)

Now Eq. 1 is converted and called the second Zagreb entropy:

(3)ENTM2(G)=log(M2(G)) −1M2(G)log[∏rs∈E(G)[Ξ(r)×Ξ(s)][Ξ(r)×Ξ(s)]]

3.3 The hyper Zagreb entropy

If ψ(rs)=[Ξ(r)+Ξ(s)]2,then :

∑rs∈E(G) ψ (rs)=∑rs∈E(G)[ Ξ (r)+ Ξ (s)]2=HM(G)

Now Eq. 1 is converted and called the hyper Zagreb entropy:

(4)ENTHM(G)=log⁡(HM(G))−1HM(G)log∏rs∈E(G)( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2

3.4 The forgotten entropy

If ψ(rs ) = [Ξ(r)²)+(Ξ(s)²)] , then:

∑rs∈E(G) ψ (rs)=∑rs∈E(G) Ξ (r)2+ Ξ (s)2=F(G)

Now Eq. 1 is converted and called the forgotten entropy:

ENT _F (G) =log(F(G))

(5)−1F(G)log∏rs∈E(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2

3.5 The augmented Zagreb entropy

If ψ(rs)=Ξ(r)Ξ(s)Ξ(r)+Ξ(s)−23, then: ∑rs∈E(G)ψ(rs)=∑rs∈E(G)Ξ(r)Ξ(s)Ξ(r)+Ξ(s)−23=AZI(G)

Now Eq. 1 is converted and called the augmented Zagreb entropy:

(6)ENTAZI(G)=log⁡(AZI(G))−1AZI(G)log∏rs∈E(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23 Ξ (r) Ξ (s) Ξ (r)+(s)−23

3.6 The Balaban entropy

If ψ(rs)=qq−p+2×1Ξ(r)Ξ(s),then :

∑rs∈E(G) ψ (rs)=qq−p+2∑rs∈E(G)1 Ξ (r) Ξ (s)=J(G)

Now Eq. 1 is converted and called the Balaban entropy:

ENT _J (G) = log ( J(G) )

(7)−1J(G)log∏rs∈E(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1Ξ(r)Ξ(s)

4 Crystallographic structure of first carbon nanosheet T¹C₄C₈(S)[m, n]

A C₄C₈(S) nanosheet is a trivalent ornamentation prepared by blinking tetragons C₄ and octagons C₈. There are two categories of nanosheets which canister be completed by C₄ and C₈ following the trivalent ornamentation which we mention to as T¹C₄C₈(S)[m, n] and T²C₄C₈ (R)[m, n]. T¹C₄C₈ (S)[m, n] nanosheet is the two-dimensional lattice of TUC₄C₈(S)[m, n], where m and n are significant restrictions in Figure 1. In this segment, we deliberate the first kind of nanosheet i.e. T¹C₄C₈ (S)[m, n] in which C₄ acts as a tetragonal and m and n are the quantity of octagons in any pillar and racket individually. Figure 2 portrays the Type-I C₄C₈(S) nanosheet T¹C₄C₈ (S)[m, n]. The vertex and edge cardinalities of this organic graph are 8mn and 12mn - 2(m + n) correspondingly.

Figure 1

A TUC₄C₈(S)[m, n] nanotube.

Figure 2

Type-I C₄C₈ (S) nanosheet T¹C₄C₈ (S)[m; n].

The vertex barrier of T¹C₄C₈ (S)[m, n] established on degrees of respectively vertex is portrayed in Table 1. Also the edge barrier of T¹C₄C₈ (S)[m; n] centered on degrees of end vertices of apiece edge are depicted in Table 2.

Table 1

Vertex partition of T¹C₄C₈ (S)[m, n].

Ξ(r)	Frequency	Set of vertices
2	4m + 4n	V₁
3	8mn - 4m - 4n	V₂

Table 2

Edge partition of T¹C₄C₈ (S)[m, n].

(Ξ(r), Ξ(r))	Frequency	Set of edges
(2, 2)	2m + 2n + 4	E₁
(3, 2)	4m + 4n - 8	E₂
(3, 3)	12mn - 8m - 8n + 4	E₃

4.1 Results for carbon nanosheet T¹C₄C₈(S)[m, n]

4.1.1 The first Zagreb entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 2 and Table 2, we computed following results. The first Zagreb index by using Table 2 is:

M1(G)=72mn−20(m+n)

Now Eq. 2 (with Table 2) can take the following form:

ENTM1T1C4C8(S)=logM1(G)−1M1(G)log∏rs∈E1(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]×∏rs∈E2(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]×∏rs∈E3(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]=logM1−1M1log⁡[[(2m+2n+4)×(256)]×[(4m+4n−8)×(3125)]×[(12mn−8m−8n+4)×(46656)]]=log⁡(72mn−20(m+n))−log⁡[(2m+2n+4)×(256)](72mn−20(m+n))−log⁡[(4m+4n−8)×(3125)](72mn−20(m+n))−log⁡[(12mn−8m−8n+4)×(46656)](72mn−20(m+n))

4.1.2 The second Zagreb entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 3 and Table 2, we computed following results. The second Zagreb index by using Table 2 is:

M2(G)=108mn−40(m+n)+4

Now Eq. 3 (with Table 2) can take the following form:

ENTM2T1C4C8(S)=logM2(G)−1M2(G)log∏rs∈E1G Ξ (r)× Ξ (s) Ξ (r)× Ξ (s)×∏rs∈E2G Ξ (r)× Ξ (s) Ξ (r)× Ξ (s)×∏rs∈E3G Ξ (r)× Ξ (s) Ξ (r)× Ξ (s)=logM2−1M2log[(2m+2n+4)×(256)]×[(4m+4n−8)×(46656)]×(12mn−8m−8n+4)×99=log⁡(108mn−40(m+n)+4)−log⁡[(2m+2n+4)×(256)](108mn−40(m+n)+4)−log⁡[(4m+4n−8)×(46656)](108mn−40(m+n)+4)−log(12mn−8m−8n+4)×99(108mn−40(m+n)+4)

4.1.3 The hyper Zagreb entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 4 and Table 2, we computed following result. By using Table 2, we have:

HM(G)=432mn−156(m+n)+8

Now Eq. 4 (with Table 2) is reduced to the following form:

ENTHMT1C4C8(S)=log⁡(HM(G))−1HM(G)log∏rs∈E1G( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2×∏rs∈E2G( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2×∏rs∈E3G( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2=log⁡(HM(G))−1(HM(G))log(2 m+2n+4)×(16)(16)×(4 m+4n−8)×(25)(25)×(12mn−8 m−8n+4)×(36)(36)=log⁡(432mn−156(m+n)+8)−log(2m+2n+4)×(16)(16)(432mn−156(m+n)+8)−log(4m+4n−8)×(25)(25)(432mn−156(m+n)+8)−log(12mn−8m−8n+4)×(36)(36)(432mn−156(m+n)+8)

4.1.4 The forgotten Zagreb entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 5 and Table 2, we got following expressions. By using Table 2, we have:

F(G)=216mn−76(m+n)

Now Eq. 4 (with Table 2) is reduced to the following form:

ENTFT1C4C8(S)=log⁡(F(G))−1F(G)log∏rs∈E1(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+ Ξ s2×∏rs∈E2(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2×∏rs∈E3(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2=log⁡(F(G))−1(F(G))log(2m+2n+4)×(8)(8)×(4m+4n−8)×(13)(13)×(12mn−8m−8n+4)×(18)(18)=log⁡(216mn−76(m+n))−log(2m+2n+4)×(16)(16)(216mn−76(m+n))−log(4m+4n−8)×(25)(25)(216mn−76(m+n))−log(12mn−8m−8n+4)×(36)(36)(216mn−76(m+n))

4.1.5 The augmented Zagreb entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 5 and Table 2, we got following expressions. By using Table 2, we have:

AZI(G)=218716mn−3455(m+n)+21716

Now Eq. 6 (with Table 2) becomes:

ENTAZIT1C4C8(S)=log⁡(AZI(G))−1AZI(G)log∏rs∈E1(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3×∏rs∈E2(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3∏rs∈E3(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−2)3=log⁡(AZI(G))−1(AZI(G))log(2m+2n+4)×(8)(8)×(4m+4n−8)×(8)(8)×(12mn−8m−8n+4)×93439343=log218716mn−3455(m+n)+21716=−log(2m+2n+4)×(8)(8)218716mn−3455(m+n)+21716−log(4m+4n−8)×(8)(8)218716mn−3455(m+n)+21716log(12mn−8m−8n+4)×93439343218716mn−3455(m+n)+21716

4.1.6 The Balaban entropy of T¹C₄C₈(S)[m, n]

Now using Eq. 7 and Table 2, we move in the following way. By using Table 2, we have:

J(G)=12mn−2m−2n4mn−2m−2n+2(4mn+(46−53)(m+n)+103−86)

Equation 7 (with Table 2) takes the following form:

ENTJT1C4C8(S)=log⁡(J(G))−1J(G)log∏rs∈E1(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)×∏rs∈E2(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)×∏rs∈E3(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)=log⁡(J(G))−1(J(G))log(2m+2n+4)×q2(q−p+2)q2(q−p+2)×(4m+4n−8)×q6(q−p+2)q6(q−p+2)×(12mn−2n−8n+4)×q3(q−p+2)q3(q−p+2)=logJG−log(2m+2n+4)×q2(q−p+2)q2(q−p+2)JG−log(4m+4n−8)×q6(q−p+2)q6(q−p+2)(J(G))log(12mn−8m−8n+4)×q3(q−p+2)q3(q−p+2)(J(G))

5 Carbon nanosheet T²C₄C₈(R)[m, n]

A C₄C₈(R) nanosheet is a trivalent decoration prepared by ashing rhombus C₄ and octagons C₈. We talk about this nanosheet as T₂C₄C₈(R)[m, n]. This nanosheet is the two-dimensional lattice of TUC₄C₈(R)[m, n], where m and n are essential limitations in Figure 3. In this section we discourse T²C₄C₈(R)[m, n] in which C₄ entertainments as a rhombus and m and n are quantity of octagons in any pillar and row correspondingly. Figure 4 depicts the type-II C₄C₈(R) nanosheet T²C₄C₈(R)[m, n]. The vertex and edge cardinalities of this organic graph are 4mn + 4(m + n) + 4 and 6mn + 5(m + n) + 4 correspondingly.

Figure 3

A TUC₄C₈ (R)[m, n] nanotube.

Figure 4

Type-II C₄C₈(R) nanosheet T²C₄C₈(R)[m, n].

The vertex barrier of T²C₄C₈ (R)[m, n] grounded on degrees of each vertex is portrayed in Table 3. Also the edge dividing wall of T²C₄C₈ (R)[m, n] centered on degrees of expiration vertices of respectively edge are showed in Table 4.

Table 3

Vertex partition of T²C₄C₈(R)[m, n].

Ξ(r)	Frequency	Set of vertices
2	2m + 2n + 4	V₁
3	4mn + 2m + 2n	V₂

Table 4

Edge partition of T²C₄C₈ (R)[m, n].

(Ξ(r), Ξ(r))	Frequency	Set of edges
(2, 2)	4	E₁
(3, 2)	4m + 4n	E₂
(3, 3)	6mn + m + n	E₃

5.1 Results for carbon nanosheet T²C₄C₈(R)[m, n]

5.1.1 The first Zagreb entropy of T²C₄C₈(R)[m, n]

Now using Eq. 2 and Table 4, we have:

M ₁ (G) = 36mn + 26(m + n) + 16

Equation 2 (with Table 4) is converted into the following form:

ENTM1T2C4C8(R)=logM1−1M1log∏rs=E1(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]+∏rs=E2(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]+∏rs=E3(G)[ Ξ (r)+ Ξ (s)][ Ξ (r)+ Ξ (s)]

=log(M1)−1(M1)log[[(4)×(256)]×[(4m+4n)×(3125)]×[(6mn+m+n)×(46656)]]=log(36mn+26(m+n)+16)−log[(4)×(256)](36mn+26(m+n)+16)−log[(4m+4n)×(3125)](36mn+26(m+n)+16)−log[(6mn+m+n)×(46656)](36mn+26(m+n)+16)

5.1.2 The second Zagreb entropy of T²C₄C₈(S)[m, n]

Now using Eq. 3 and Table 4, we computed the first Zagreb entropy in the following way:

By using Table 4, we have:

M2(G)=54mn+33(m+n)+16"

Equation 3 (with Table 4) becomes:

ENTM2T2C4C8(R)=logM2−1M2log∏rs∈E1(G)[ Ξ (r)× Ξ (s)][ Ξ (r)× Ξ (s)]+∏rs∈E2G[ Ξ (r)× Ξ (s)][ Ξ (r)× Ξ (s)]+∏rs∈E3G[ Ξ (r)× Ξ (s)][ Ξ (r)× Ξ (s)]=logM2−1M2log⁡[[(4)×(256)]×[(4m+4n)×(46656)]×[(6mn+m+n)×(387420489)]]=log⁡(54mn+33(m+n)+16)−log⁡[(4)×(256)](54mn+33(m+n)+16)−log⁡[(4m+4n)×(46656)](54mn+33(m+n)+16)−log⁡[(6mn+m+n)×(387420489)](54mn+33(m+n)+16)

5.1.3 The hyper Zagreb entropy of T²C₄C₈(R)[m, n]

Now using Eq. 4 and Table 4, we computed the following result:

HM(G)=216mn+136(m+n)+64

ENTHMT2C4C8(R)=log⁡(HM)−1(HM)log∏rs∈E1(G)( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2+∏rs∈E2(G)( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2+∏rs∈E3(G)( Ξ (r)+ Ξ (s))2( Ξ (r)+ Ξ (s))2=log⁡(HM)−1(HM)log(4)×(16)(16)×(4m+4n)×(25)(25)×6mn+m+n)×(36)(36)=log⁡(216mn+136(m+n)+64)−log(4)×(16)(16)(216mn+136(m+n)+64)−log(4m+4n)×(25)(25)(216mn+136(m+n)+64)−log(6mn+m+n)×(36)(36)(216mn+136(m+n)+64)

5.1.4 The forgotten Zagreb entropy of T²C₄C₈(R)[m, n]

Using Table 4, we obtained:

ENTFT2C4C8(R)=log⁡(F(G))−1F(G)log∏rs∈E1(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2×∏rs∈E2(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2×∏rs∈E3(G)( Ξ (r))2+( Ξ (s))2( Ξ (r))2+( Ξ (s))2=log⁡(F(G))−1(F(G))log4×88×4m+4n×1313×6mn+m+n×1818=log⁡(108mn+70(m+n)+32)−log(2m+2n+4)×(16)(16)(108mn+70(m+n)+32)−log(4m+4n+4)×(25)(25)(108mn+70(m+n)+32)−log(12mn−8m−8m+4)×(36)(36)(108mn+70(m+n)+32)

Equation 5 (with Table 4) becomes:

F(G)=108mn+70(m+n)+32

5.1.5 The augmented Zagreb entropy of T²C₄C₈(S)[m, n]

Using Table 2, we have:

AZI(G)=218732mn−277764(m+n)+32

ENTAZIT2C4C8(R)=log⁡(AZI(G))−1AZI(G)log∏rs∈E1(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23×∏rs∈E2(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23×∏rs∈E3(G) Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23 Ξ (r) Ξ (s) Ξ (r)+ Ξ (s)−23=log⁡(AZI(G))−1(AZI(G))log(4)×(8)(8)×(4m+4n)×(8)(8)×(6mn+m+n)×7296472964=log218732mn−277764(m+n)+32−log(4)×(8)(8)218732mn−277764(m+n)+32−log(4m+4n)×(8)(8)218732mn−277764(m+n)+32−log(6mn+m+n)×7296472964218732mn−277764(m+n)+32

5.1.6 The Balaban entropy of T²C₄C₈(R)[m, n]

By using Table 4, we obtained:

J(G)=6mn+5(m+n)+42mn+(m+n)+2[2mn+(13+46)(m+n)+2]

Equation 7 (with Table 4) takes the following form:

ENTJT2C4C8(R)=log⁡(J(G))−1J(G)log∏rs∈E1(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)×∏rs∈E2(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)×∏rs∈E3(G)qq−p+2×1 Ξ (r) Ξ (s)qq−p+2×1 Ξ (r) Ξ (s)=log⁡(J(G))−1(J(G))log(4)×q2(q−p+2)q2(q−p+2)×(4m+4n)×q6(q−p+2)q6(q−p+2)×(6mn+m+n)×q3(q−p+2)q3(q−p+2)=log⁡(J(G))−log(4)×q2(q−p+2)q2(q−p+2)(J(G))−log(4m+4n)×q6(q−p+2)q6(q−p+2)(J(G))−log(6mn+m+n)q3(q−p+2)q3(q−p+2)(J(G))

6 Comparisons and discussion for T^₁C₄C₈(S)[m, n]

Since the degree based entropy has part of utilization in various parts of science, in particular pharmaceutical,

science, organic medications and software engineering. So the numerical and graphical portrayal of these determined outcomes are useful to researcher. So in this area, we have registered numerically all degree based entropies for various estimations of m, n for T¹C₄C₈(S)[m, n]. Furthermore, we develop Tables 5 and 6

Table 5

Comparison of first Zagreb entropy and second Zagreb entropy for T¹C₄C₈(S)[m, n].

[m, n]	ENT_M₁	ENT_M₂
[2, 2]	2.25	2.37
[3, 3]	2.69	2.84
[4, 4]	2.98	3.14
[5, 5]	3.19	3.35
[6, 6]	3.36	3.53
[7, 7]	3.51	3.67
[8, 8]	3.63	3.79
[9, 9]	3.74	3.90
[10, 10]	3.83	4.01

Table 6

Comparison of ENT_HM, ENT_F, ENT_AZI, and ENT_J entropies for T¹C₄C₈(S)[m, n].

[m, n]	ENT_HM	ENT_F	ENT_AZI	ENT_J
[2, 2]	2.94	2.66	2.51	1.74
[3, 3]	3.43	3.14	2.96	2.08
[4, 4]	3.73	3.44	3.25	2.33
[5, 5]	3.95	3.36	3.47	2.51
[6, 6]	4.13	3.83	3.64	2.67
[7, 7]	4.27	3.97	3.78	2.80
[8, 8]	4.40	4.10	3.90	2.91
[9, 9]	4.50	4.20	4.01	3.01
[10, 10]	4.60	4.30	4.11	3.10

for little estimations of m, n for degree based entropy to numerical correlation for the structure of T¹C₄C₈ (S) [m, n]. Presently, from Tables 5 and 6, we can without much of a stretch see that all the estimations of entropy are in expanding request as the estimations of m; n are increments. The graphical portrayals of registered outcomes are delineated in Figures 5-7 for specific estimations of m, n.

Figure 5

(a) The first Zagreb entropy, (b) the second Zagreb entropy.

Figure 6

(a) The hyper Zagreb entropy, (b) the forgotten entropy.

Figure 7

(a) The augmented Zagreb entropy, (b) the Balaban entropy.

7 Comparisons and discussion for T²C₄C₈(R)[m, n]

Since the degree based entropy has parcel of utilization in various parts of science, in particular pharmaceutical, science, natural medications and software engineering. So the numerical and graphical portrayal of these determined outcomes are useful to researcher.

So in this area, we have figured numerically all degree based entropies for various estimations of m; n for T²C₄C₈(R)[m, n]. Moreover, we build Tables 7 and 8 for little estimations of m, n for degree based entropy to numerical correlation for the structure of T²C₄C₈ (R)[m, n]. Presently, from Tables 7 and 8, we can without much of a stretch see that all the estimations of entropy are in expanding request as the estimations of m; n are increments. The graphical portrayals of processed outcomes are delineated in Figures 8-10 for specific estimations of m, n.

Table 7

Comparison of first Zagreb entropy and second Zagreb entropy for T²C₄C₈(S)[m, n].

[m, n]	ENT_M₁	ENT_M₂
[2, 2]	2.37	2.51
[3, 3]	2.67	2.82
[4, 4]	2.88	3.04
[5, 5]	3.06	3.22
[6, 6]	3.20	3.36
[7, 7]	3.32	3.49
[8, 8]	3.43	3.60
[9, 9]	3.53	3.69
[10, 10]	3.61	4.78

Table 8

Comparison of ENT_HM, ENT_F, ENT_AZI, and ENT_J entropies for T²C₄C₈(S)[m, n].

[m, n]	ENT_HM	ENT_F	ENT_AZI	ENT_J
[2, 2]	3.09	2.81	2.62	1.72
[3, 3]	3.41	3.12	2.92	1.99
[4, 4]	3.64	3.34	3.15	2.19
[5, 5]	3.82	3.52	3.32	2.35
[6, 6]	3.96	3.67	3.47	2.49
[7, 7]	4.09	3.79	3.59	2.60
[8, 8]	4.20	3.90	3.70	2.70
[9, 9]	4.30	3.99	3.80	2.79
[10, 10]	4.38	4.08	3.88	2.88

Figure 8

(a) The first Zagreb entropy, (b) the second Zagreb entropy.

Figure 9

(a) The hyper Zagreb entropy, (b) the forgotten entropy.

Figure 10

(a) The augmented Zagreb entropy, (b) the Balaban entropy.

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11971142

Award Identifier / Grant number: 11871202

Award Identifier / Grant number: 61673169

Award Identifier / Grant number: 11701176

Award Identifier / Grant number: 11626101

Award Identifier / Grant number: 11601485

Funding statement: The research was supported by the National Natural Science Foundation of China (Grant Nos. 11971142, 11871202, 61673169, 11701176, 11626101, 11601485).

References

Akhter S., Imran M., Raza Z., On the general sum-connectivity index and general Randic index of cacti. J. Ineq. Applic., 2016, 1, 300-310.10.1186/s13660-016-1250-6Search in Google Scholar

Ali A., Raza Z., Bhatti A.A., Some vertex-degree-based topological indices of cacti. Ars Combinatoria, 2019, 144, 195-206.Search in Google Scholar

Assaye B., Alamneh M., Mishra L.N., Mebrat Y., Dual skew Heyting almost distributive lattices. Appl. Math. Nonli. Sci., 2019, 4(1), 151-162.10.2478/AMNS.2019.1.00015Search in Google Scholar

Balaban A.T., Highly discriminating distance-based topological index. Chem. Phys. Lett., 1982, 89(5), 399-404.10.1016/0009-2614(82)80009-2Search in Google Scholar

Balaban A.T., Quintas L.V., The smallest graphs, trees, and 4-trees with degenerate topological index. J. Math. Chem., 1983, 14, 213-233.Search in Google Scholar

Basavanagoud B., Gao W., Patil S., Desai V.R., Mirajkar K.G., Pooja B., Computing First Zagreb index and F-index of New C-products of Graphs. Appl. Math. Nonlinear Sci., 2017, 2(1), 285-298.10.21042/AMNS.2017.1.00024Search in Google Scholar

Bonchev D., Complexity in Chemistry, Introduction and Fundamentals. Taylor and Francis, Boca Raton, FL, USA, 2003, 7, 157-185.10.1201/9781482288247Search in Google Scholar

Chen Z., Dehmer M., Shi Y., A note on distance-based graph entropies. Entropy, 2014, 16(10), 5416-5427.10.3390/e16105416Search in Google Scholar

Dehmer M., Information processing in complex networks: Graph entropy and information functionals. Appl. Math. Comput., 2008, 201(1-2), 82-94.10.1016/j.amc.2007.12.010Search in Google Scholar

Dehmer M., Grabner M., The discrimination power of molecular identification numbers revisited. Match Commun. Math. Comput. Chem., 2013, 69(3), 785-794.Search in Google Scholar

Dehmer M., Mowshowitz A., A history of graph entropy measures. Info. Sci., 2011, 181(1), 57-78.10.1016/j.ins.2010.08.041Search in Google Scholar

Dehmer M., Sivakumar L., Varmuza K., Uniquely discriminating molecular structures using novel eigen-value-based descriptors. Match. Commun. Math. Comput. Chem., 2012, 67(1), 147.Search in Google Scholar

Furtula B., Graovac A., Vukicevic D., Augmented Zagreb index. J. Math. Chem., 2010, 48(2), 370-380.10.1007/s10910-010-9677-3Search in Google Scholar

Furtula B., Gutman I., A forgotten topological index. J. Math. Chem., 2015, 53(4), 1184-1190.10.1007/s10910-015-0480-zSearch in Google Scholar

Gao W., Farahani M.R., Degree-based indices computation for special chemical molecular structures using edge dividing method. Appl. Math. Nonlinear Sci., 2016, 1(1), 99-122.10.21042/AMNS.2016.1.00009Search in Google Scholar

Gao W., Siddiqui M.K., Naeem M., Rehman N.A., Topological characterization of carbon graphite and crystal cubic carbon structures. Molecules, 2017, 22(9), 1496.10.3390/molecules22091496Search in Google Scholar

Gao W., Wu H., Siddiqui M.K., Baig A. Q., Study of biological networks using graph theory. Saudi J. Biol. Sci., 2018, 25(6), 1212-1219.10.1016/j.sjbs.2017.11.022Search in Google Scholar

Gutman I., Das K.C., The first Zagreb index 30 years after. Match Commun. Math. Comput. Chem., 2004, 50(1), 83-92.Search in Google Scholar

Gutman I., Trinajstic N., Graph theory and molecular orbitals. Total ϕ – electron energy of alternant hydrocarbons. Chem. Phys. Lett., 1972, 17(4), 535-538.10.1016/0009-2614(72)85099-1Search in Google Scholar

Hosamani S.M., Kulkarni B.B., Boli R.G., Gadag V.M., QSPR analysis of certain graph theocratical matrices and their corresponding energy. Appl. Math. Nonli. Sci., 2017, 2(1), 131-150.10.21042/AMNS.2017.1.00011Search in Google Scholar

Imran M., Siddiqui M.K., Naeem M., Iqbal M.A., On topological properties of symmetric chemical structures. Symmetry, 2018, 10(5), 173-183.10.3390/sym10050173Search in Google Scholar

Liu J.B., Baig A.Q., Imran M., Khalid W., Saeed M., Farahani M.R., Computation of bond incident degree (BID) indices of complex structures in drugs. Eurasian Chem. Commun., 2020a, 2(6), 672-679.10.33945/SAMI/ECC.2020.6.4Search in Google Scholar

Liu J.B., Raza Z., Javaid M., Zagreb Connection Numbers for Cellular Neural Networks. Disc. Dynamics Nat. Society, 2020b, 1-15.10.1155/2020/8038304Search in Google Scholar

Morowitz H.J., Some order-disorder considerations in living systems. Bull. Math. Biophys., 1955, 17(2), 81-86.10.1007/BF02477985Search in Google Scholar

Quastler H., Information theory in biology. Bull. Math. Biol., 1954, 8, 183-185.10.1007/BF02478375Search in Google Scholar

Rashevsky N., Life, information theory and topology. Bull. Math. Biophys., 1955, 17(3), 229-235.10.1007/BF02477860Search in Google Scholar

Raza Z., The expected values of arithmetic bond connectivity and geometric indices in random phenylene chains. Heliyon, 2020a, 6(7), 55-69.10.1016/j.heliyon.2020.e04479Search in Google Scholar PubMed PubMed Central

Raza Z., The harmonic and second Zagreb indices in random polyphenyl and spiro chains. Polycyclic Aromatic Compounds, 2020b, 1-10.10.1080/10406638.2020.1749089Search in Google Scholar

Raza Z., Ali A., Bounds on the Zagreb indices for molecular (n, m)-graphs. Int. J. Quantum Chem., 2020, 120(18), 45-55.10.1002/qua.26333Search in Google Scholar

Raza Z., Sukaiti E.K., M-Polynomial and Degree Based Topological Indices of Some Nanostructures. Symmetry, 2020, 12(5), 831-841.10.3390/sym12050831Search in Google Scholar

Shannon C.E., A mathematical theory of communication. ACM SIGMOBILE Mob. Comput. Comm. Rev., 2001, 5(1), 3-55.10.1145/584091.584093Search in Google Scholar

Shirakol S., Kalyanshetti M., Hosamani S.M., QSPR analysis of certain distance based topological indices. Appl. Math. Nonlinear Sci., 2019, 4(2), 371-386.10.2478/AMNS.2019.2.00032Search in Google Scholar

Shirdel G.H., Rezapour H., Sayadi A.M., The hyper-Zagreb index of graph operations. Iran. J. Math. Chem., 2013, 4(2), 213-220.Search in Google Scholar

Siddiqui M.K., Imran M., Ahmad A., On Zagreb indices, Zagreb polynomials of some nanostar dendrimers. Appl. Math. Comput., 2016a, 280, 132-139.10.1016/j.amc.2016.01.041Search in Google Scholar

Siddiqui M.K., Naeem M., Rahman N.A., Imran M., Computing topological indices of certain networks. J. Optoel. Adv. Mater., 2016b, 18(September-October 2016), 884-892.Search in Google Scholar

Solé R.V., Valverde S. Information Theory of Complex Networks: On Evolution and Architectural Constraints. In: Ben-Naim E., Frauenfelder H., Toroczkai Z. (Eds.) Complex Networks. Lecture Notes in Physics, Vol. 650. Springer, Berlin, Heidelberg, 2004.10.1007/978-3-540-44485-5_9Search in Google Scholar

Tan Y.J., Wu J., Network structure entropy and its application to scale-free networks. Syst. Eng. Theory Practice, 2004, 6(001).Search in Google Scholar

Trucco E., A note on the information content of graphs. Bull. Math. Biophys., 1956, 18(2), 129-135.10.1007/BF02477836Search in Google Scholar

Ulanowicz R.E., Quantitative methods for ecological network analysis. Comput. Biol. Chem., 2004, 28(5-6), 321-339.10.1016/j.compbiolchem.2004.09.001Search in Google Scholar PubMed

Wu W., Zhang C., Lin W., Chen Q., Guo X., Qian Y., et al., Quantitative structure-property relationship (QSPR) modeling of drug-loaded polymeric micelles via genetic function approximation. PloS One, 2015 10(3), 51-69.10.1371/journal.pone.0119575Search in Google Scholar PubMed PubMed Central

Received: 2020-07-21

Accepted: 2020-11-10

Published Online: 2020-12-31

This work is licensed under the Creative Commons Attribution 4.0 International License.

On topological aspects of degree based entropy for two carbon nanosheets

Abstract

1 Introduction

2 Degree based topological indices of graph

3 General entropy of graph

3.1 The first Zagreb entropy

3.2 The second Zagreb entropy

3.3 The hyper Zagreb entropy

3.4 The forgotten entropy

3.5 The augmented Zagreb entropy

3.6 The Balaban entropy

4 Crystallographic structure of first carbon nanosheet T¹C₄C₈(S)[m, n]

4.1 Results for carbon nanosheet T¹C₄C₈(S)[m, n]

4.1.1 The first Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.2 The second Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.3 The hyper Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.4 The forgotten Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.5 The augmented Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.6 The Balaban entropy of T¹C₄C₈(S)[m, n]

5 Carbon nanosheet T²C₄C₈(R)[m, n]

5.1 Results for carbon nanosheet T²C₄C₈(R)[m, n]

5.1.1 The first Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.2 The second Zagreb entropy of T²C₄C₈(S)[m, n]

5.1.3 The hyper Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.4 The forgotten Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.5 The augmented Zagreb entropy of T²C₄C₈(S)[m, n]

5.1.6 The Balaban entropy of T²C₄C₈(R)[m, n]

6 Comparisons and discussion for T^₁C₄C₈(S)[m, n]

7 Comparisons and discussion for T²C₄C₈(R)[m, n]

References

Journal and Issue

Articles in the same Issue

On topological aspects of degree based entropy for two carbon nanosheets

Abstract

1 Introduction

2 Degree based topological indices of graph

3 General entropy of graph

3.1 The first Zagreb entropy

3.2 The second Zagreb entropy

3.3 The hyper Zagreb entropy

3.4 The forgotten entropy

3.5 The augmented Zagreb entropy

3.6 The Balaban entropy

4 Crystallographic structure of first carbon nanosheet T1C4C8(S)[m, n]

4.1 Results for carbon nanosheet T1C4C8(S)[m, n]

4.1.1 The first Zagreb entropy of T1C4C8(S)[m, n]

4.1.2 The second Zagreb entropy of T1C4C8(S)[m, n]

4.1.3 The hyper Zagreb entropy of T1C4C8(S)[m, n]

4.1.4 The forgotten Zagreb entropy of T1C4C8(S)[m, n]

4.1.5 The augmented Zagreb entropy of T1C4C8(S)[m, n]

4.1.6 The Balaban entropy of T1C4C8(S)[m, n]

5 Carbon nanosheet T2C4C8(R)[m, n]

5.1 Results for carbon nanosheet T2C4C8(R)[m, n]

5.1.1 The first Zagreb entropy of T2C4C8(R)[m, n]

5.1.2 The second Zagreb entropy of T2C4C8(S)[m, n]

5.1.3 The hyper Zagreb entropy of T2C4C8(R)[m, n]

5.1.4 The forgotten Zagreb entropy of T2C4C8(R)[m, n]

5.1.5 The augmented Zagreb entropy of T2C4C8(S)[m, n]

5.1.6 The Balaban entropy of T2C4C8(R)[m, n]

6 Comparisons and discussion for T1C4C8(S)[m, n]

7 Comparisons and discussion for T2C4C8(R)[m, n]

References

Journal and Issue

Articles in the same Issue

4 Crystallographic structure of first carbon nanosheet T¹C₄C₈(S)[m, n]

4.1 Results for carbon nanosheet T¹C₄C₈(S)[m, n]

4.1.1 The first Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.2 The second Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.3 The hyper Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.4 The forgotten Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.5 The augmented Zagreb entropy of T¹C₄C₈(S)[m, n]

4.1.6 The Balaban entropy of T¹C₄C₈(S)[m, n]

5 Carbon nanosheet T²C₄C₈(R)[m, n]

5.1 Results for carbon nanosheet T²C₄C₈(R)[m, n]

5.1.1 The first Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.2 The second Zagreb entropy of T²C₄C₈(S)[m, n]

5.1.3 The hyper Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.4 The forgotten Zagreb entropy of T²C₄C₈(R)[m, n]

5.1.5 The augmented Zagreb entropy of T²C₄C₈(S)[m, n]

5.1.6 The Balaban entropy of T²C₄C₈(R)[m, n]

6 Comparisons and discussion for T^₁C₄C₈(S)[m, n]

7 Comparisons and discussion for T²C₄C₈(R)[m, n]