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Computer Modeling in Engineering & Sciences
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DOI: 10.32604/cmes.2023.027254

ARTICLE

Degree-Based Entropy Descriptors of Graphenylene Using Topological Indices

M. C. Shanmukha1, Sokjoon Lee2,*, A. Usha3, K. C. Shilpa4 and Muhammad Azeem5

1Department of Mathematics, Bapuji Institute of Engineering & Technology, Davanagere, 577004, India
2Department of Computer Engineering (Smart Security), Gachon University, Seongnam, 13120, South Korea
3Department of Mathematics, Alliance School of Applied Mathematics, Alliance University, Bangalore, 562106, India
4Department of Computer Science and Engineering, Bapuji Institute of Engineering & Technology, Davanagere, 577004, India
5Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, 54660, Pakistan
*Corresponding Author: Sokjoon Lee. Email: junny@gachon.ac.kr
Received: 21 October 2022; Accepted: 19 December 2022

Abstract: Graph theory plays a significant role in the applications of chemistry, pharmacy, communication, maps, and aeronautical fields. The molecules of chemical compounds are modelled as a graph to study the properties of the compounds. The geometric structure of the compound relates to a few physical properties such as boiling point, enthalpy, π-electron energy, and molecular weight. The article aims to determine the practical application of graph theory by solving one of the interdisciplinary problems describing the structures of benzenoid hydrocarbons and graphenylene. The topological index is an invariant of a molecular graph associated with the chemical structure, which shows the correlation of chemical structures using many physical, chemical properties and biological activities. This study aims to introduce some novel degree-based entropy descriptors such as ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS using the respective topological indices. Also, the above-mentioned entropy measures and physico-chemical properties of benzenoid hydrocarbons are fitted using linear regression models and calculated for graphenylene structure.

Keywords: Topological descriptors; weighted entropy; graphenylene

1  Introduction

Mathematics and computer science are often used in the applications of research development. One example of such applications is cheminformatics which is a relatively recent area of research in mathematics. The study includes the problem of analysing the structure of a molecule which can be retrieved from the cheminformatics dataset. There are many compounds available in organic and inorganic chemistry whose properties and structures seem promising to search and evaluate the uses of a substance. The study of the structure of a molecule gives information about its chemical properties [1,2].

Graph theory has been a very useful branch of science, especially in the applications of chemistry. It has a very powerful tool known as the topological index which provides a lot of information about a chemical compound. These topological indices are classified based on degree, distance, and eccentricity [3–6].

Most of these indices have a good correlation with the properties of isomers and benzenoid hydrocarbons, such that the compounds are used for various purposes based on their correlation coefficient [7–9].

Graph theory facilitates the mathematical model of a compound to draw information about the chemical compound. In modelling a compound into a graph, the hydrogen atoms are neglected without losing information about the molecule, as carbon atoms have four chemical bonds whereas hydrogen atom has one chemical bond [10–13].

The recent trends in research have attracted a lot of studies involving information science. It provides a good correlation between the biological and structural properties of compounds. Many scientists have done remarkable findings which led to wide applications in graph theory [14–16].

Graph theory being an ideal tool in the hands of the chemist involves representation, synthesis of compounds and numerous chemical activities. Also, chemists are always interested in breaking and making chemical bonds, resulting in different types of structures [17,18].

The degree of amount of energy dispersed and the measure of unavailability of heat energy for work is termed as the entropy. Originally, Shannon introduced the concept of entropy as a part of the communication theory [19]. According to him, data is communicated as a system consisting of three elements: source, channel and receiver. During his learning, Shannon used various methods to encode, transmit and compress the messages which proved that the entropy denotes an absolute limit on how well data can be compressed from the source to reach the receiver in his famous coding theorem.

The measure of uncertainty refers to the entropy of a probability distribution. Indeed, the result of an analysis conducted can be assumed by taking the numerical value equal to the amount of uncertainty of the outcome of the analysis. Furthermore, studies on graphs and networks were studied by various researchers during the late 1950s. More work on entropy measures was carried out using graph invariants which proved advantageous to study important properties of graphs [20–23].

Several theoretic measures and tools have been developed to study the complexity of the structure of chemical compounds and complex networks. The word entropy is dealt with various ways by researchers involving a variety of problems in different fields like discrete mathematics, biology, chemistry, statistics, etc., in investigating entropies of relational structures. In mathematical chemistry, graph entropy is used to characterize the structure of a graph [24–26].

A class of chemical compounds having at least one benzene ring is termed as a benzenoid. They have a high chemical stability because of its bonding with certain molecules. Benzenoids are aromatic hydrocarbons having significant applications in gasoline additives, dry cleaning, manufacture of synthetic fibres, plastics and products in rubber-like materials [27]. Their applications are growing rapidly in the fields of industrial chemistry particularly, in the products of polymers. The work is carried out for twenty-two benzenoid hydrocarbons refer Fig. 1.

Figure 1: Molecular structure of benzenoid hydrocarbons

Graphenylene is a cyclic hydrocarbon in which each hexagon has a square adjacent to it. Two such hexagons separated by a square are termed as biphenylenes. Chemically, it is a cyclobutadiene ring in between two benzene rings. Biphenylene is a building block of graphenylene which is pale yellowish powder having melting temperature of 110∘C. Biphenylene is a hydrocarbon whose chemical formula is C12H8. A 2D graphene is a prospective compound that has significant applications in the next-generation electronic and optical devices. Biphenylene becomes an interesting forerunner of a 2D porous graphene-like molecular network called Graphenylene. This new material has a good dispersion and gap separation in the characterization of delocalized band [28,29].

Numerous studies on graphene have grabbed researchers across the globe due to its magnificent properties and promising potential applications because of its unique 2D structure. Graphene can be wrapped up into fullerenes, carbon nanotubes and even along a specific direction that forms graphene nanoribbon. These have extremely enriched the family of carbon nanomaterials. Also, these studies have created an interest in exploiting new 2D carbon allotropes through both experimental techniques and theoretical calculations [30]. Biphenylene carbon is a product of cyclotrimerization of graphene whose structure is 2D network of hydrogen free carbon atoms.

The main aims/objectives of this work are

•   To introduce novel entropy measures.

•   To understand their physical/chemical applicability of benzenoid hydrocarbons by regression models.

•   To calculate defined entropies of graphenylene structure.

In this article, all standard graph terminologies and notations are referred from [31–33]. In the literature, various studies are carried out using topological indices of which the below mentioned indices are considered in this work.

Definition 1.1.Gutman defined Sombor index [34] as

SO(G)=∑ϑω∈E(G)(d(ϑ))2+(d(ω))2

Definition 1.2.Usha et al. [35] defined Geometric-Harmonic index as

GH(G)=∑ϑω∈E(G)((d(ϑ)+d(ω))(d(ϑ)×d(ω))2

Definition 1.3.Shanmukha et al. [36] defined Harmonic-Geometric index as

HG(G)=∑ϑω∈E(G)2(d(ϑ)+d(ω))(d(ϑ)×d(ω))

Definition 1.4.Zhao et al. [37] defined SS index as

SS(G)=∑ϑω∈E(G)d(ϑ)×d(ω)d(ϑ)+d(ω)

In continuation to SS index, the neighborhood version of SS index is defined as

NSS(G)=∑ϑω∈E(G)S(ϑ)×S(ω)S(ϑ)+S(ω)

Definition 1.5.Kulli [38] defined neighborhood Sombor index as

NSO(G)=∑ϑω∈E(G)(S(ϑ))2+(S(ω))2

Definition 1.6.Shanmukha et al. [39] defined neighborhood redefined first and second Zagreb indices as

NReZ1(G)=∑ϑω∈E(G)S(ϑ)+S(ω)S(ϑ)×S(ω)NReZ2(G)=∑ϑω∈E(G)S(ϑ)×S(ω)S(ϑ)+S(ω)

1.1 Graph Entropy Based on Degree and Edge Weight

The theory of edge weighted graph based on entropy was first introduced by Chen et al. [40] in the year 2014. Let G be an edge weighted graph, denoted by (V(G),E(G),ψ(ϑω)). Here, V(G) and E(G) are the usual notations of the set of vertices and edges respectively, where ψ(ϑω) denotes the edge weight of the graph G. The edge weight of ψ(ϑω) is computed by adding the degrees of the vertices ϑ and ω of the edge ϑω. Then, the graph entropy based on edge weight is defined in Eq. (1).

ENTψ(G)=−∑ϑ′ω′∈E(G)ψ(ϑ′ω′)∑ϑω∈E(G)ψ(ϑω)log[ψ(ϑ′ω′)∑ϑω∈E(G)ψ(ϑω)](1)

•   Sombor Entropy

If ψ(ϑω)=d(ϑ)2+d(ω)2, then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)d(ϑ)2+d(ω)2=SO(G)

Using the definition of Eq. (1) for Sombor index results in Sombor entropy given by

ENTSO(G)=log(SO(G))−1SO(G)log[∏ϑω∈E(G)[d(ϑ)2+d(ω)2][d(ϑ)2+d(ω)2]](2)

•   Geometric-Harmonic Entropy

If ψ(ϑω)=(d(ϑ)+d(ω))(d(ϑ)×d(ω))2, then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)(d(ϑ)+d(ω))(d(ϑ)×d(ω))2=GH(G)

Using the definition of Eq. (1) for Geometric-Harmonic index results in Geometric-Harmonic entropy given by

ENTGH(G)=log(GH(G))−1GH(G)log[∏ϑω∈E(G)[(d(ϑ)+d(ω))(d(ϑ)×d(ω))2][(d(ϑ)+d(ω))(d(ϑ)×d(ω))2]](3)

•   Harmonic-Geometric Entropy

If ψ(ϑω)=2(d(ϑ)+d(ω))(d(ϑ)×d(ω)), then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)2(d(ϑ)+d(ω))(d(ϑ)×d(ω))=HG(G)

Using the definition of Eq. (1) for Harmonic-Geometric index results in Harmonic-Geometric entropy given by

ENTHG(G)=log(HG(G))−1HG(G)log[∏ϑω∈E(G)[2(d(ϑ)+d(ω))(d(ϑ)×d(ω))][2(d(ϑ)+d(ω))(d(ϑ)×d(ω))]](4)

•   The SS Entropy

If ψ(ϑω)=d(ϑ)×d(ω)d(ϑ)+d(ω), then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)d(ϑ)×d(ω)d(ϑ)+d(ω)=SS(G)

Using the definition of Eq. (1) for SS index results in SS entropy given by

ENTSS(G)=log(SS(G))−1SS(G)log[∏ϑω∈E(G)[d(ϑ)×d(ω)d(ϑ)+d(ω)][d(ϑ)×d(ω)d(ϑ)+d(ω)]](5)

•   The Neighborhood Sombor Entropy

If ψ(ϑω)=S(ϑ)2+S(ω)2, then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)S(ϑ)2+S(ω)2=NSO(G)

Using the definition of Eq. (1) for neighborhood Sombor index results in neighborhood Sombor entropy given by

ENTNSO(G)=log(NSO(G))−1NSO(G)log[∏ϑω∈E(G)[S(ϑ)2+S(ω)2][S(ϑ)2+S(ω)2]](6)

•   The NReZ1 Entropy

If ψ(ϑω)=S(ϑ)+S(ω)S(ϑ)×S(ω), then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)S(ϑ)+S(ω)S(ϑ)×S(ω)=NReZ1(G)

Using the definition of Eq. (1) for NReZ1 index results in NReZ1 entropy given by

ENTNReZ1(G)=log(NReZ1(G))−1NReZ1(G)log[∏ϑω∈E(G)[S(ϑ)+S(ω)S(ϑ)×S(ω)][S(ϑ)+S(ω)S(ϑ)×S(ω)]](7)

•   The NReZ2 Entropy

If ψ(ϑω)=S(ϑ)×S(ω)S(ϑ)+S(ω), then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)S(ϑ)×S(ω)S(ϑ)+S(ω)=NReZ2(G)

Using the definition of Eq. (1) for NReZ2 index results in NReZ2 entropy given by

ENTNReZ2(G)=log(NReZ2(G))−1NReZ2(G)log[∏ϑω∈E(G)[S(ϑ)×S(ω)S(ϑ)+S(ω)][S(ϑ)×S(ω)S(ϑ)+S(ω)]](8)

•   The Neighborhood SS Entropy

If ψ(ϑω)=S(ϑ)×S(ω)S(ϑ)+S(ω), then

∑ϑω∈E(G)ψ(ϑω)=∑ϑω∈E(G)S(ϑ)×S(ω)S(ϑ)+S(ω)=SS(G)

Using the definition of Eq. (1) for neighborhood SS index results in neighborhood SS entropy given by

ENTNSS(G)=log(NSS(G))−1NSS(G)log[∏ϑω∈E(G)[S(ϑ)×S(ω)S(ϑ)+S(ω)][S(ϑ)×S(ω)S(ϑ)+S(ω)]](9)

1.2 Chemical Applicability of Defined Degree-Based Entropies

This section concentrates on framing the linear regression model for the properties viz., boiling point (BP), enthalpy (E), π-electron energy (π−ele) and molecular weight (MW) of benzene derivatives. They are presented for entropies such as ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS. The properties considered [41–44] in the study have shown a good correlation with the defined entropies.

The ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS are calculated and shown in Table 1.

The linear regression models for BP, E, π-ele and MW are fitted using least squares method for the data presented in Table 1.

The models fitted for ENTSO are

BP=190.6 (±6.276)ENTSO−655.096 (±37.014)(10)

E=108.097 (±8.357)ENTSO−345.434 (±49.282)(11)

π−ele=10.080 (±0.449)ENTSO−32.197 (±2.646)(12)

MW=86.748 (±4.116)ENTSO−266.352 (±24.272)(13)

The models fitted for ENTGH are

BP=160.894 (±6.565)ENTGH−593.023 (±43.342)(14)

E=90.611 (±7.801)ENTGH−306.043 (±51.508)(15)

π−ele=8.521 (±0.419)ENTGH−28.997 (±2.766)(16)

MW=73.130 (±3.996)ENTGH−237.460 (±26.384)(17)

The models fitted for ENTHG are

BP=331.917 (±16.403)ENTHG+401.306 (±6.781)(18)

E=197.177(±9.610)ENTHG+252.060(±3.973)(19)

π−ele=17.788(±0.788)ENTHG+23.625(±0.326)(20)

MW=154.499 (±5.532)ENTHG+213.819 (±2.287)(21)

The models fitted for ENTSS are

BP=204.078 (±6.950)ENTSS−392.055 (±29.394)(22)

E=115.902 (±8.905)ENTSS−196.928 (±37.661)(23)

π−ele=10.777 (±0.506)ENTSS−18.225 (±2.139)(24)

MW=92.772 (±4.594)ENTSS−146.175 (±19.430)(25)

The models fitted for ENTNSO are

BP=164.761 (±23.838)ENTNSO−604.233 (±155.021)(26)

E=89.087 (±16.788)ENTNSO−288.401 (±109.175)(27)

π−ele=8.820 (±1.261)ENTNSO−30.199 (±8.200)(28)

MW=74.984 (±11.262)ENTNSO−243.183 (±73.241)(29)

The models fitted for ENTNReZ1 are

BP=161.292 (±20.581)ENTNReZ1+136.973 (±43.764)(30)

E=92.088 (±13.555)ENTNReZ1+102.544 (±28.822)(31)

π−ele=8.263 (±1.222)ENTNReZ1+10.226 (±2.598)(32)

MW=71.638 (±10.428)ENTNReZ1+97.710 (±22.174)(33)

The models fitted for ENTNReZ2 are

BP=194.930 (±17.009)ENTNReZ2−576.413 (±91.188)(34)

E=106.499 (±14.300)ENTNReZ2−279.217 (±76.661)(35)

π−ele=10.303 (±0.960)ENTNReZ2−28.006 (±5.149)(36)

MW=88.044 (±8.732)ENTNReZ2−226.952 (±46.811)(37)

The models fitted for ENTNSS are

BP=95.807 (±13.311)ENTNSS+24.289(±62.604)(38)

E=53.271(±9.070)ENTNSS+44.738(±42.659)(39)

π−ele=4.916(±0.775)ENTNSS+4.417(±3.647)(40)

MW=42.370(±6.707)ENTNSS+48.494(±31.544)(41)

Note: The errors associated with the regression coefficients are enclosed with in brackets of Eqs. (10)–(41).

Tables 2 to 9 and Fig. 2 show the r (coefficient of correlation) and RSE (residual standard error) of the above properties using regression models with ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS entropies.

Figure 2: Scatter diagram of physical property with entropy descriptor

By inspection of Table 2, it shows that ENTSO and BP are highly correlated with r=0.989. Also, ENTSO and E have r=0.945, ENTSO and π−ele have r=0.981, ENTSO and MW have r=0.978.

From Table 3, it is observed that ENTGH and BP are are highly correlated with r=0.984. Also, ENTGH and E have r=0.933, ENTGH and π−ele have r=0.977, ENTGH and MW have r=0.971.

From Table 4, it is observed that ENTHG and MW highly correlated with r=0.987. Also, ENTHG and BP have r=0.976, ENTHG and E have r=0.977, ENTHG and π−ele have r=0.981.

From Table 5, it is observed that ENTSS and BP highly correlated with r=0.989. Also, ENTSS and E have r=0.946, ENTSS and π−ele have r=0.979, ENTSS and MW have r=0.976.

From Table 6, it is observed that ENTNSO and π−ele highly correlated with r=0.843. Also, ENTNSO and BP have r=0.840, ENTNSO and E have r=0.765, ENTNSO and MW have r=0.830.

From Table 7, it is observed that ENTNReZ1 and BP highly correlated with r=0.869. Also, ENTNReZ1 and E have r=0.835, ENTNReZ1 and π−ele have r=0.834, ENTNReZ1 and MW have r=0.838.

From Table 8, it is observed that ENTNReZ2 and BP highly correlated with r=0.932. Also, ENTNReZ2 and E have r=0.857, ENTNReZ2 and π−ele have r=0.923, ENTNReZ2 and MW have r=0.914.

From Table 9, it is observed that ENTNSS and BP highly correlated with r=0.849. Also, ENTNSS and E have r=0.796, ENTNSS and π−ele have r=0.817, ENTNSS and MW have r=0.816.

2  ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS of Graphenylene Molecular Graph

Graphenylene is a cyclic hydrocarbon, where each hexagon has a square adjacent to it. Two such hexagons separated by a square are termed as biphenylene. Chemically, it is a cyclobutadiene ring in between two benzene rings. Graphenylene is modelled as a molecular graph for which the vertices and edges are computed and details are tabulated in Tables 10 and 11 based on degrees and neighborhood degrees of end vertices respectively. From the Fig. 3, the total number of vertices and edges of Graphenylene are 12mn and 18mn−2m−2n, respectively.

Figure 3: Planar view of 4×4 supercells of biphenylene (graphenylene)

In this section, ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS of graphenylene are computed.

2.1 Results for Graphenylene for Degree Based Vertices

Theorem 2.1. Consider graphenylene structure as a molecular graph G, then the Sombor entropy is:

ENTSO(G)=log⁡(76.368mn−13.862m−13.862n−0.28)−log⁡[(2m+2n+2)×(8)8](76.368mn−13.862m−13.862n−0.28)−log⁡[(4m+4n−4)×(13)13](76.368mn−13.862m−13.862n−0.28)−log⁡[(18mn−8m−8n+2)×(18)18](76.368mn−13.862m−13.862n−0.28).

Proof. The Sombor entropy and Sombor index are computed as per the above definitions and Table 10.

Then the Sombor index is given by

SO(G)=76.368mn−13.862m−13.862n−0.28.

The Sombor entropy is computed for graphenylene structure using Eq. (2) and Table 10, which results in:

ENTSO(G)=log(SO(G))−1SO(G)log[∏(2,2)∈E1(G)[1(2)2+(2)2]1(2)2+(2)2×∏(2,3)∈E2(G)[1(2)2+(3)2]1(2)2+(3)2+∏(3,3)∈E3(G)[1(3)2+(3)2]1(3)2+(3)2]ENTSO(G)=log(SO(G))−1SO(G)log⁡[[(2m+2n+2)×(8)8]×[(4m+4n−4)×(13)13]×[(18mn−8m−8n+2)×(18)18]]ENTSO(G)=log⁡(76.368mn−13.862m−13.862n−0.28)−log⁡[(2m+2n+2)×(8)8](76.368mn−13.862m−13.862n−0.28)−log⁡[(4m+4n−4)×(13)13](76.368mn−13.862m−13.862n−0.28)−log⁡[(18mn−8m−8n+2)×(18)18](76.368mn−13.862m−13.862n−0.28).

Theorem 2.2. Consider graphenylene structure as a molecular graph G, then the GH entropy is:

ENTGH(G)=log⁡(162mn−39.504m−39.504n+1.504)−log⁡[(2m+2n+2)×(4)4](162mn−39.504m−39.504n+1.504)−log⁡[(4m+4n−4)×(6.124)6.124](162mn−39.504m−39.504n+1.504)−log⁡[(18mn−8m−8n+2)×(9)9](162mn−39.504m−39.504n+1.504).

Proof. The GH entropy and GH index are computed as per the above definitions and Table 10.

Then, the GH index is given by

GH(G)=162mn−39.504m−39.504n+1.504.

The GH entropy is calculated for graphenylene structure using Eq. (3) and Table 10, results in:

ENTGH(G)=log(GH(G))−1GH(G)log[∏(2,2)∈E1(G)[(2+2)(2×2)2](2+2)(2×2)2×∏(2,3)∈E2(G)[(2+3)(2×3)2](2+3)(2×3)2×∏(3,3)∈E3(G)[(3+3)(3×3)2](3+3)(3×3)2]ENTGH(G)=log(GH(G))−1GH(G)log⁡[[(2m+2n+2)×(4)4]×[(4m+4n−4)×(6.124)6.124]×[(18mn−8m−8n+2)×(9)9]]ENTGH(G)=log⁡(162mn−39.504m−39.504n+1.504)−log⁡[(2m+2n+2)×(4)4](162mn−39.504m−39.504n+1.504)−log⁡[(4m+4n−4)×(6.124)6.124](162mn−39.504m−39.504n+1.504)−log⁡[(18mn−8m−8n+2)×(9)9](162mn−39.504m−39.504n+1.504).

Theorem 2.3. Consider graphenylene structure as a molecular graph G, then the HG entropy is:

ENTHG(G)=log⁡(1.998mn+0.264m+0.264n+0.07)−log⁡[(2m+2n+2)×(0.25)0.25](1.998mn+0.264m+0.264n+0.07)−log⁡[(4m+4n−4)×(0.163)0.163](1.998mn+0.264m+0.264n+0.07)−log⁡[(18mn−8m−8n+2)×(0.111)0.111](1.998mn+0.264m+0.264n+0.07).

Proof. The HG entropy and HG index are computed as per the above definitions and Table 10.

Then, the HG index is given by

HG(G)=1.998mn+0.264m+0.264n+0.07.

The HG entropy is calculated for graphenylene structure using Eq. (4) and Table 10, results in:

ENTHG(G)=log(HG(G))−1HG(G)log[∏(2,2)∈E1(G)[2(2+2)(2×2)]2(2+2)(2×2)×∏(2,3)∈E2(G)[2(2+3)(2×3)]2(2+3)(2×3)×∏(3,3)∈E3(G)[2(3+3)(3×3)]2(3+3)(3×3)]ENTHG(G)=log(HG(G))−1HG(G)log⁡[[(2m+2n+2)×(0.25)0.25]×[(4m+4n−4)×(0.163)0.163]×[(18mn−8m−8n+2)×(0.111)0.111]]ENTHG(G)=log⁡(1.998mn+0.264m+0.264n+0.07)−log⁡[(2m+2n+2)×(0.25)0.25](1.998mn+0.264m+0.264n+0.07)−log⁡[(4m+4n−4)×(0.163)0.163](1.998mn+0.264m+0.264n+0.07)−log⁡[(18mn−8m−8n+2)×(0.111)0.111](1.998mn+0.264m+0.264n+0.07).

Theorem 2.4. Consider graphenylene structure as a molecular graph G, then the SS entropy is:

ENTSS(G)=log⁡(22.05mn−3.42m−3.42n+0.07)−log⁡[(2m+2n+2)×(1)1](22.05mn−3.42m−3.42n+0.07)−log⁡[(4m+4n−4)×(1.095)1.095](22.05mn−3.42m−3.42n+0.07)−log⁡[(18mn−8m−8n+2)×(1.225)1.225](22.05mn−3.42m−3.42n+0.07).

Proof. The SS entropy and SS index are computed as per the above definitions and Table 10.

Then, the SS index is given by

SS(G)=22.05mn−3.42m−3.42n+0.07.

The SS entropy is calculated for graphenylene structure using Eq. (5) and Table 10, which results in:

ENTSS(G)=log(SS(G))−1SS(G)log[∏(2,2)∈E1(G)[2×22+2]2×22+2×∏(2,3)∈E2(G)[2×32+3]2×32+3×∏(3,3)∈E3(G)[3×33+3]3×33+3]ENTSS(G)=log(SS(G))−1SS(G)log⁡[[(2m+2n+2)×(1)1]×[(4m+4n−4)×(1.095)1.095]×[(18mn−8m−8n+2)×(1.225)1.225]]ENTSS(G)=log⁡(22.05mn−3.42m−3.42n+0.07)−log⁡[(2m+2n+2)×(1)1](22.05mn−3.42m−3.42n+0.07)−log⁡[(4m+4n−4)×(1.095)1.095](22.05mn−3.42m−3.42n+0.07)−log⁡[(18mn−8m−8n+2)×(1.225)1.225](22.05mn−3.42m−3.42n+0.07).

2.2 Results for Graphenylene Using Neighbourhood Degree of End Vertices

Theorem 2.5. Consider graphenylene structure as a molecular graph G, then the neighborhood Sombor entropy is:

ENTNSO(G)=log⁡(229.102mn−55.436m−55.436n+1.686)−log⁡[(2)×(32)32](229.102mn−55.436m−55.436n+1.686)−log⁡[(4)×(41)41](229.102mn−55.436m−55.436n+1.686)−log⁡[(2m+2n−4)×(50)50](229.102mn−55.436m−55.436n+1.686)−log⁡[(4m+4n−4)×(89)89](229.102mn−55.436m−55.436n+1.686)−log⁡[(4)×(28)28](229.102mn−55.436m−55.436n+1.686)−log⁡[(8m+8n−16)×(145)145](229.102mn−55.436m−55.436n+1.686)−log⁡[(18mn−16m−16n+14)×(162)162](229.102mn−55.436m−55.436n+1.686).

Proof. The neighborhood Sombor entropy and neighborhood Sombor index are computed as per the above definitions and Table 11.

Then, the neighborhood Sombor index is given by

NSO(G)=229.102mn−55.436m−55.436n+1.686.

The neighborhood Sombor entropy is calculated for graphenylene structure using Eq. (6) and Table 11, which results in:

ENTNSO(G)=log(NSO(G))−1NSO(G)log[∏(4,4)∈E1(G)[(4)2+(4)2](4)2+(4)2×∏(4,5)∈E2(G)[(4)2+(5)2](4)2+(5)2×∏(5,5)∈E3(G)[(5)2+(5)2](5)2+(5)2×∏(5,8)∈E2(G)[(5)2+(8)2](5)2+(8)2×∏(8,8)∈E3(G)[(8)2+(8)2](8)2+(8)2×∏(8,9)∈E4(G)[(8)2+(9)2](8)2+(9)2×∏(9,9)∈E4(G)[(9)2+(9)2](9)2+(9)2]ENTNSO(G)=log(NSO(G))−1NSO(G)log⁡[[(2)×(32)32]×[(4)×(41)41]×[(2m+2n−4)×(50)50]×[(4m+4n−4)×(89)89]×[(4)×(128)128]×[(8m+8n−16)×(145)145]×[(18mn−16m−16n+14)×(162)162]]ENTNSO(G)=log⁡(229.102mn−55.436m−55.436n+1.686)−log⁡[(2)×(32)32](229.102mn−55.436m−55.436n+1.686)−log⁡[(4)×(41)41](229.102mn−55.436m−55.436n+1.686)−log⁡[(2m+2n−4)×(50)50](229.102mn−55.436m−55.436n+1.686)−log⁡[(4m+4n−4)×(89)89](229.102mn−55.436m−55.436n+1.686)−log⁡[(4)×(28)28](229.102mn−55.436m−55.436n+1.686)−log⁡[(8m+8n−16)×(145)145](229.102mn−55.436m−55.436n+1.686)−log⁡[(18mn−16m−16n+14)×(162)162](229.102mn−55.436m−55.436n+1.686).

Theorem 2.6. Consider graphenylene structure as a molecular graph G, then NReZ1 entropy is:

ENTNReZ1(G)=log⁡(3.966mn+0.436mn+0.436n+0.232)−log⁡[(2)×(0.5)0.5](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4)×(0.45)0.45](3.966mn+0.436mn+0.436n+0.232)−log⁡[(2m+2n−4)×(0.4)0.4](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4m+4n−4)×(0.325)0.325](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4)×(0.25)0.25](3.966mn+0.436mn+0.436n+0.232)−log⁡[(8m+8n−16)×(0.236)0.236](3.966mn+0.436mn+0.436n+0.232)−log⁡[(18mn−16m−16n+14)×(0.222)0.222](3.966mn+0.436mn+0.436n+0.232).

Proof.

The NReZ1 entropy and NReZ1 index are computed as per the above definitions and Table 11.

Then, the NReZ1 index is given by

NReZ1(G)=3.966mn+0.436mn+0.436n+0.232.

The NReZ1 entropy is calculated for graphenylene using Eq. (7) and Table 11, which results in:

ENTNReZ1(G)=log(NReZ1(G))−1NReZ1(G)log[∏(4,4)∈E1(G)[4+44×4]4+44×4×∏(4,5)∈E2(G)[4+54×5]4+54×5×∏(5,5)∈E3(G)[5+55×5]5+55×5×∏(5,8)∈E2(G)[5+85×8]5+85×8×∏(8,8)∈E3(G)[8+88×8]8+88×8×∏(8,9)∈E4(G)[8+98×9]8+98×9×∏(9,9)∈E4(G)[9+99×9]9+99×9]ENTNReZ1(G)=log(NReZ1(G))−1NReZ1(G)log⁡[[(2)×(0.5)0.5]×[(4)×(0.45)0.45]×[(2m+2n−4)×(0.4)0.4]×[(4m+4n−4)×(0.325)0.325]×[(4)×(0.25)0.25]×[(8m+8n−16)×(0.236)0.236]×[(18mn−16m−16n+14)×(0.222)0.222]]ENTNReZ1(G)=log⁡(3.966mn+0.436mn+0.436n+0.232)−log⁡[(2)×(0.5)0.5](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4)×(0.45)0.45](3.966mn+0.436mn+0.436n+0.232)−log⁡[(2m+2n−4)×(0.4)0.4](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4m+4n−4)×(0.325)0.325](3.966mn+0.436mn+0.436n+0.232)−log⁡[(4)×(0.25)0.25](3.966mn+0.436mn+0.436n+0.232)−log⁡[(8m+8n−16)×(0.236)0.236](3.966mn+0.436mn+0.436n+0.232)−log⁡[(18mn−16m−16n+14)×(0.222)0.222](3.966mn+0.436mn+0.436n+0.232).

Theorem 2.7. Consider graphenylene structure as a molecular graph G, then NReZ2 entropy is:

ENTNReZ2(G)=log⁡(81mn−20.812m−20.812n+1.82)−log⁡[(2)×(2)2](81mn−20.812m−20.812n+1.82)−log⁡[(4)×(2.222)2.222](81mn−20.812m−20.812n+1.82)−log⁡[(2m+2n−4)×(2.5)2.5](81mn−20.812m−20.812n+1.82)−log⁡[(4m+4n−4)×(3.077)3.077](81mn−20.812m−20.812n+1.82)−log⁡[(4)×(4)4](81mn−20.812m−20.812n+1.82)−log⁡[(8m+8n−16)×(4.235)4.235](81mn−20.812m−20.812n+1.82)−log⁡[(18mn−16m−16n+14)×(4.5)4.5](81mn−20.812m−20.812n+1.82).

Proof. The NReZ2 entropy and NReZ2 index are computed as per the above definitions and Table 11.

Then, the NReZ2 index is given by

NReZ2(G)=3.966mn+0.436mn+0.436n+0.232.

The NReZ2 entropy is calculated for graphenylene using Eq. (8) and Table 11, which results in:

ENTNReZ2(G)=log(NReZ2(G))−1NReZ2(G)log[∏(4,4)∈E1(G)[4×44+4]4×44+4×∏(4,5)∈E2(G)[4×54+5]4×54+5×∏(5,5)∈E3(G)[5×55+5]5×55+5×∏(5,8)∈E2(G)[5×85+8]5×85+8×∏(8,8)∈E3(G)[8×88+8]8×88+8×∏(8,9)∈E4(G)[8×98+9]8×98+9×∏(9,9)∈E4(G)[9×99+9]9×99+9]ENTNReZ2(G)=log(NReZ2(G))−1NReZ2(G)log⁡[[(2)×(2)2]×[(4)×(2.222)2.222]×[(2m+2n−4)×(2.5)2.5]×[(4m+4n−4)×(3.077)3.077]×[(4)×(4)4]×[(8m+8n−16)×(4.235)4.235]×[(18mn−16m−16n+14)×(4.5)4.5]]ENTNReZ2(G)=log⁡(81mn−20.812m−20.812n+1.82)−log⁡[(2)×(2)2](81mn−20.812m−20.812n+1.82)−log⁡[(4)×(2.222)2.222](81mn−20.812m−20.812n+1.82)−log⁡[(2m+2n−4)×(2.5)2.5](81mn−20.812m−20.812n+1.82)−log⁡[(4m+4n−4)×(3.077)3.077](81mn−20.812m−20.812n+1.82)−log⁡[(4)×(4)4](81mn−20.812m−20.812n+1.82)−log⁡[(8m+8n−16)×(4.235)4.235](81mn−20.812m−20.812n+1.82)−log⁡[(18mn−16m−16n+14)×(4.5)4.5](81mn−20.812m−20.812n+1.82).

Theorem 2.8. Consider graphenylene structure as a molecular graph G, then neighborhood SS entropy is:

ENTNSS(G)=log⁡(38.18mn−7.298m−7.298n+0.221)−log⁡[(2)×(1.414)1.414](38.18mn−7.298m−7.298n+0.221)−log⁡[(4)×(1.491)1.491](38.18mn−7.298m−7.298n+0.221)−log⁡[(2m+2n−4)×(1.581)1.581](38.18mn−7.298m−7.298n+0.221)−log⁡[(4m+4n−4)×(1.754)1.754](38.18mn−7.298m−7.298n+0.221)−log⁡[(4)×(2)2](38.18mn−7.298m−7.298n+0.221)−log⁡[(8m+8n−16)×(2.058)2.058](38.18mn−7.298m−7.298n+0.221)−log⁡[(18mn−16m−16n+14)×(2.121)2.121](38.18mn−7.298m−7.298n+0.221).

Proof. The neighborhood SS entropy and neighborhood SS index are computed as per the above definitions and Table 11.

Then, the neighborhood SS index is given by

NSS(G)=38.18mn−7.298m−7.298n+0.221.

The neighborhood SS entropy is calculated for graphenylene structure using Eq. (9) and Table 11, which results in:

ENTNSS(G)=log(NSS(G))−1NSS(G)log[∏(4,4)∈E1(G)[4×44+4]4×44+4×∏(4,5)∈E2(G)[4×54+5]4×54+5×∏(5,5)∈E3(G)[5×55+5]5×55+5×∏(5,8)∈E2(G)[5×85+8]5×85+8×∏(8,8)∈E3(G)[8×88+8]8×88+8×∏(8,9)∈E4(G)[8×98+9]8×98+9×∏(9,9)∈E4(G)[9×99+9]9×99+9]ENTNSS(G)=log(NSS(G))−1NSS(G)log⁡[[(2)×(1.414)1.414]×[(4)×(1.491)1.491]×[(2m+2n−4)×(1.581)1.581]×[(4m+4n−4)×(1.754)1.754]×[(4)×(2)2]×[(8m+8n−16)×(2.058)2.058]×[(18mn−16m−16n+14)×(2.121)2.121]]ENTNSS(G)=log⁡(38.18mn−7.298m−7.298n+0.221)−log⁡[(2)×(1.414)1.414](38.18mn−7.298m−7.298n+0.221)−log⁡[(4)×(1.491)1.491](38.18mn−7.298m−7.298n+0.221)−log⁡[(2m+2n−4)×(1.581)1.581](38.18mn−7.298m−7.298n+0.221)−log⁡[(4m+4n−4)×(1.754)1.754](38.18mn−7.298m−7.298n+0.221)−log⁡[(4)×(2)2](38.18mn−7.298m−7.298n+0.221)−log⁡[(8m+8n−16)×(2.058)2.058](38.18mn−7.298m−7.298n+0.221)−log⁡[(18mn−16m−16n+14)×(2.121)2.121](38.18mn−7.298m−7.298n+0.221).

3  Conclusion

This work is dedicated to defining and discussing the chemical applicability of ENTSO, ENTGH, ENTHG, ENTSS, ENTNSO, ENTNReZ1, ENTNReZ2 and ENTNSS for a few properties of by-products of benzene molecules by applying regression models and computing entropy descriptors of graphenylene structure. It is observed from the above study that the graph entropies considered have proved to be correlated well with the four physical properties mentioned in this work. The following are the observations made from the study. The ENTSO has a high positive correlation against BP with r=0.989. The ENTGH has a high positive correlation against BP with r=0.984. The ENTHG has a high positive correlation against MW with r=0.987. There is a high positive correlation between ENTSS against BP with r=0.989. The ENTNSO has a high positive correlation against π−ele with r=0.843. The ENTNReZ1 has a high positive correlation with BP having r=0.869. The ENTNReZ2 has a high positive correlation with BP with r=0.932. The ENTNSS has a high positive correlation against BP with r=0.849.

There has been a remarkable increase in the applications of graphenylene as it has distinguished mechanical, electrical, and thermal properties. This study provides a way for researchers to study physico-chemical and biological properties of various compounds associated with degree-based entropies.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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