# Topological Indices of Pent-Heptagonal Nanosheets via M-Polynomials

Authors: Hafiza Bushra Mumtaz; Muhammad Javaid; Hafiz Muhammad Awais; Ebenezer Bonyah
Journal: Journal of Mathematics
 (2021)

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## Abstract

The combination of mathematical sciences, physical chemistry, and information sciences leads to a modern field known as cheminformatics. It shows a mathematical relationship between a property and structural attributes of different types of chemicals called quantitative-structures’ activity and qualitative-structures’ property relationships that are utilized to forecast the chemical sciences and biological properties, in the field of engineering and technology. Graph theory has originated a significant usage in the field of physical chemistry and mathematics that is famous as chemical graph theory. The computing of topological indices (TIs) is a new topic of chemical graphs that associates many physiochemical characteristics of the fundamental organic compounds. In this paper, we used the M-polynomial-based TIs such as 1st Zagreb, 2nd Zagreb, modified 2nd Zagreb, symmetric division deg, general Randic´, inverse sum, harmonic, and augmented indices to study the chemical structures of pent-heptagonal nanosheets of VC5C7 and HC5C7. An estimation among the computed TIs with the help of numerical results is also presented.


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## Body

## 1. Introduction

Nanostructures [1, 2] have been studied as new materials with the size of elementals structures that has been engineered at the nanometers’ scale. Most of the materials in this size range usually show novel behavior. Therefore, intervention in the characteristics of structures at the nanoscale allows the formation of devices and nanomaterials with completely or enhanced novel functionalities and properties. Understanding the science of nanostructures is curiosity and important driven not only for the interesting nature of the topic but also for novel and overwhelming usage of nanoscale systems in various fields of science and technology. Nanotechnology can be recognized as a technology of design, application, and fabrication of nanomaterials, and nanostructures [3].The branch of nanotechnology and nanoscience is being perused by chemists, physicists, materials scientists, engineers, biologists, computer scientists, and mathematicians [4]. So, it is also interdisciplinary. Nanostructures may be divided based on modulation and dimensionality. Most of the distinct nanotubes, zeolites, aerogel, core-shell structure, and nanoporous materials have unique properties. Numerous techniques have been utilized for the synthesis of nanomaterials with no. of degrees of success, and several direct as well as indirect methods are used for their properties [5]. The motivation to develop the nanomaterials is that the characteristics become size based in the nanometer range due to quantum confinement effect and surface effect. The chemical bonds, magnetic properties, geometric structure, electronic properties, ionization potential, mechanical strength, optical properties, and thermal properties are affected due to particle size in nanometers range. Nanostructures show characteristics mostly higher than the conventional coarse-grained material. These contain hardness/increased strength, toughness/improved ductility, enhanced diffusivity, reduced density, higher electrical resistance, reduced elastic modulus, lower thermal conductivity, increase specific heat, higher thermal expansion coefficient, increased oscillator and strength luminescence, blue shift absorption, and superior soft magnetic characteristics in comparison to the conventional bulk material. Furthermore, these characteristics are being briefly examined to discover new tools. The interesting branch of nanotechnology has a vast range of different types of applications. The use of nanomaterials has manufactured transistors having low speed and laser having low threshold current. These are utilized in satellite receivers having low noise amplification as a source for fiber optics communications and compact disk player systems. Constructive tools of nanostructures contain UV-resistant wood coating and self-cleaning glass. On the other hand, nanoscale tools are being utilized in the field of medicine for the prevention and treatment of diseases, diagnosis, and in magnetic resonance imaging, drug delivery system, radioactive tracers, etc. [6]. The importance of nanomaterials is rising nowadays. Many other types of tools may be possible with the peculiar and novel characteristics of nanomaterials [7, 8].Therefore, TIs are useful to define molecular nanomaterials. Nanostructures, that have a scale of less than 100 nm, contain nanosheets, nanotubes, and nanoparticles. Nanosheets (two-dimensional nanomaterials) have a sharp edge and large surface area that cause them to play a vital role in various types of tools such as catalysis, energy storage bioelectronics, and optoelectronics [9, 10]. Silicone, borophene, and graphene are specific nanosheets. Due to the rare optical, electrical, mechanical, and structural characteristics, graphene nanosheets received great recognition from industrial and academic researchers [11]. The different properties of the C5C7 nanosheet have become the most advanced field in research. A C5C7 structure is developed by alternating C5andC7 [7]. In 2009, Graovac et al. studied the GA index of TUC4C8 (S) nanotubes. In 2011, Graovac et al. [12] studied the fifth geometric arithmetic index for nanostar dendrimers, and Asadpour et al. calculated, Zagreb, Randi c´, and ABC indices of TUC4C8 (R) and TUC4C8 (S) V-Phenylenic nanotorus and nanotubes. In 2014, Al-Fozan et al. solved Szeged index of H-naphthalene nanosheets (2n, 2m) and C4C8 (S). Loghman and Ashrafi studied the Padmakar–Ivan (PI) index of TUC4C8 (S) nanotubes. For further discussion, see [13–15].However, the combination of three fields such as mathematics, physical chemistry, and information sciences lead to a modern field known as cheminformatics [16–18]. It develops a mathematical relationship between a property and structural attributes of different types of chemicals called by quantitative-structures’ activity and qualitative-structures’ property relationship that are utilized to forecast the organic sciences and biological properties in the field of engineering and technology [19, 20]. Graph theory has originated a significant usage in the field of mathematical chemistry that is famous as chemical graph theory.Polya gave the idea for counting polynomials in the field of chemistry [21], and Wiener introduced the concept of TI related to the paraffin’s boiling point [22]. Computing the TIs is a new field of chemical graphs that associates many physiochemical characteristics of the fundamental chemical compounds [23–27].


## 2. Preliminaries

A molecular structureΓ=VΓ,EΓ; VΓ=s1,s2,s3,…,snandEΓ are nodes (vertices) and edge set of Γ. VΓ=vandEΓ=e is the order and size of Γ. In a connected and simple molecular graph, a path is represented within two vertices and the distance between the two vertices sandt is mentioned as φs,t, in a graph Γ, see [28–30]. In this paper, a graph is connected and simple, having no multiple edges or loops.1st and 2nd Zagreb indices: letΓ be a molecular structure; then, its 1st and 2nd Zagreb indices [31] are(1)M1Γ=∑s∈VΓφs2=∑st∈EΓφs+φt,M2Γ=∑st∈EΓφs×φt.General Randic´ index: if R is the real number, α∈R, and Γ is a molecular structure, the general Randi c´ index [32] is(2)RαΓ=∑st∈EΓφsφtα.Symmetric division deg index: for a molecular structureΓ, the symmetric division deg index [33] is(3)SDDΓ=∑st∈EΓminφs,φtmaxφs,φt+maxφs,φtminφs,φt.Harmonic index: for a molecular structureΓ, the harmonic index [34] is(4)HΓ=∑st∈EΓ2φs+φt.Inverse sum index: for a molecular structureΓ, the inverse sum index [35] is(5)ISΓ=∑st∈EΓφsφtφs+φt.Augmented Zagreb index: for a molecular structureΓ, the augmented Zagreb index [13] is(6)AZIΓ=∑st∈EΓφs×φtφs+φt−23.A graph polynomial is a graph invariant whose values are polynomials. So, all these invariants are discussed in algebraic graph theory [36]. Among such types of algebraic polynomials, the M-polynomial, defined in 2015, shows the same role in finding the much closed form of various degree-based TIs that correlate different types of chemical properties of the various materials under investigation. In 2019, Yang et al. [37] find out the M-polynomial and topological indices of benzene ring embedded in P-type surface network. In 2020, Khalaf et al. [38] computed the M-polynomial and topological indices of book graph and Raza and Sakaiti [2] solved the M-polynomial and degree-based topological indices of some nanostructures. In 2021, Mondal et al. [39] find out the neighborhood M-polynomial of titanium compounds and Irfan et al. [1] computed the M-polynomials and topological indices for line graphs of chain silicate network and H-naphtalenic nanotubes.M-Polynomial: letΓ be a molecular structure and mi,jΓ,i,j≥1, be the number of edges e=st of Γ in such a way that φsφt=i,j. The M-polynomial of Γ is(7)MΓ,μ,ν=∑i≤jΓmi,jΓμiνj.Now, we discussed the relationship between the M-polynomial and some important TIs in the form of Tables1 and 2.Table 1
Derivation of TIs from M-polynomial.
Indicesfμ,νDerivation fromMΓ,μ,νM1μ+νDμ+DνMΓ,μ,ν|μ=1=νM2μνDμDνMΓ,μ,ν|μ=1=νMM21/μνSμSνMΓ,μ,ν|μ=1=νRαμνα,α∈NDμαDναMΓ,μ,ν|μ=1=νRαRα1/μνα,α∈NSμαSναMΓ,μ,ν|μ=1=νSDDμ2+ν2/μνDμSν+DνSμMΓ,μ,ν|μ=1=νTable 2
Other TIs from M-polynomial.
Indicesfμ,νDerivation fromMΓ,μ,νH2/μ+ν2SμJMΓ,μ,ν|μ=1=νISμν/μ+νSμQ2JDμDνMΓ,μ,ν|μ=1=νAZIμν/μ+ν−23Sμ3JDμ3Dν3MΓ,μ,ν|μ=1=ν


## 3. Pent-Heptagonal Nanosheet

Firstly, we discuss the structure of pent-heptagonal nanosheetVC5C7. For nanosheet of VC5C7a,b, we represent the number of pentagons in the first row by b, and the first four rows of nodes as well as edges are repeated. Therefore, we represent the number of repetitions as a. The nanosheet VC5C72,4 has 16ab+2a+5b nodes or vertices and 24ab+4b edges. Additionally, it has 6a+7b nodes having degree 2 and 16ab−4a−2b nodes having degree 3. The degree-based edge partition of nanosheet a=2 and b=4 is shown in Table 3.Table 3
Partition of edge set,VC5C7.
Edges partitionsE1=E2,2E2=E2,3E3=E3,3Cardinality2a+2b+48a+10b−824ab−10a−8b+4From Figure1, we note that 2 distinct types of vertices in VC5C7 are 2 and 3. So,(8)V1=s∈VΓ1|φs=2V2=s∈VΓ1|φs=3.Figure 1
Pent-heptagonal nanosheetVC5C7.We have 3 different types of edges that is based on the degree of end nodes inΓ1 that are(9)E2,2=st∈Γ1|φs=2,φt=2E2,3=st∈Γ1|φs=2,φt=3E3,3=st∈Γ1|φs=3,φt=3,where E1=2a+2b+4, E2=8a+10b−8, E3=24ab−10a−8b+4, and a=2 and b=4. Then,(10)EΓ1=E1+E2+E3=16+48+144=208.Now, we discuss the structure of pent-heptagonal nanosheetHC5C7. For the nanosheet HC5C7a,b, we represent the number of pentagons in the first row by b, and the 1st four rows of nodes and edges are repeated. So, we represent the number of repetitions as a. The nanosheets HC5C72,4 have 16ab+2a+4b vertices and 24ab+3b edges. Moreover, it has 6a+6b vertices with degree 2 and 16ab−4a−2b vertices with degree 3. The degree-based edge partition of nanosheets for a=2 and b=4 is shown in Table 4.Table 4
Partition of edge set ofHC5C7.
Edges’ partitionsE1=E2,2E2=E2,3E3=E3,3Cardinality2a+3b+28a+6b−424ab−10a−6b+10From Figure2, we note that 2 distinct types of vertices in HC5C7 are 2 and 3. So,(11)V1=s∈VΓ2|φu=2,V2=s∈VΓ2|φu=3.Figure 2
Pent-heptagonal nanosheetHC5C7.We have 3 different types of edges that is based on the degree of end nodes inΓ1:(12)E2,2=st∈Γ2|φs=2,φt=2,E2,3=st∈Γ2|φs=2,φt=3,E3,3=st∈Γ2|φs=3,φt=3.


## 4. Main Results

This section deals with the main results consisting of polynomials and TIs of the nanosheets.Theorem 1.
LetΓ1=VC5C7 be the pent-heptagonal nanosheet. Then, the M-polynomial of Γ is(13)MΓ1,μ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.Proof.
Now, by using definition of M-polynomial ofΓ1, we obtain(14)MΓ1,μ,ν=∑s≤tEs,tΓ1μsνt=∑2≤2E2,2Γ1μ2ν2+∑2≤3E2,3Γ1μ2ν3+∑3≤3E3,3Γ1μ3ν3=E1μ2ν2+E2μ2ν3+E3μ3ν3=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.
The M-polynomial ofΓ1 is(15)MΓ1,μ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.Theorem 2.
LetΓ=VC5C7 be the pent-heptagonal nanosheet. Then, the M-polynomial of Γ is(16)MΓ1,μ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.
So, the 1st Zagreb indexM1Γ1, 2nd Zagreb index M2Γ1, 2nd modified Zagreb MM2Γ1, general Randic RγΓ1, reciprocal general Randic RRγΓ1, where γ∈α, and the symmetric division deg index SDDΓ1 obtained from M-polynomial are as follows:aM1Γ1=144ab−12a+10bbM2Γ1=216ab−34a−4b+4cMM2Γ1=8/3ab+13/18a+23/18b+1/9dRγΓ1=4γ2a+2b+4+6γ8a+10b−8+9γ24ab−10a−8b+4eRRγΓ1=2a+2b+4/4γ+8a+10b−8/6γ+24ab−10a−8b+4/9γfSSDΓ1=48ab+4/3a+29/3b−4/3Proof.
Letfμ,ν=MΓ1,μ,ν be the M-polynomial of the pent-heptagonal nanosheet VC5C7; then,(17)fμ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.
Firstly, we find out the required partial derivatives and integrals asDμfμ,ν=22a+2b+4μν2+28a+10b−8μν3+324ab−10a−8b+4μ2ν3Dνfμ,ν=22a+2b+4μ2ν+38a+10b−8μ2ν2+324ab−10a−8b+4μ3ν2DμDνfμ,ν=42a+2b+4μν+68a+10b−8μν2+924ab−10a−8b+4μ2ν2Tμfμ,ν=a+b+cμ2ν2+4a+5b−4μ2ν3+8ab−10/3a−8/3b+4/3μ3ν3Tνfμ,ν=a+b+cμ2ν2+8a+10b−8/3μ2ν3+8ab−10/3a−8/3b+4/3μ3ν3TμTνfμ,ν=TμTνfμ,ν=a+b+2/2μ2ν2+8a+10b−8/6μ2ν3+4ab−5/3a−4/3b+2/3μ3ν3DνTμfμ,ν=DνTμfμ,ν=2a+b+cμ2ν+34a+5b−4μ2ν2+38ab−10/3a−8/3b+4/3μ3ν2DμTνfμ,ν=DμTνfμ,ν=2a+b+cμν2+2/38a+10b−8μν3+24ab−10a−8b+4μ2ν3DμγDνγ=4γ2a+2b+4μν+6γ8a+10b−8μν2+9γ24ab−10a−8b+4μ2ν2TμγTνγ=2a+2b+4/4γμ2ν2+8a+10b−8/6μ2ν3+24ab−10a−8b+4/9γμ3ν3
Now, we obtainμ=ν=1:Dμfμ,ν|μ=ν=1=72ab−10a+4Dνfμ,ν|μ=ν=1=72ab−2a+10b−4DμDνfμ,ν|μ=ν=1=216ab−34a−4b+4Tμfμ,ν|μ=ν=1=8ab+5/3a+10/3b−8/3Tνfμ,ν|μ=ν=1=8ab+1/3a+5/3b−4/3TμTνfμ,ν|μ=ν=1=8/3ab+13/18a+23/18b−8/9DνTμfμ,ν|μ=ν=1=24ab+4a+9b−4DμTνfμ,ν|μ=ν=1=24ab−8/3a+2/3b+8/3DμγDνγfμ,ν|μ=ν=1=4γ2a+2b+4+68a+10b−8+9γ24ab−10a−8b+4TμγTνγfμ,ν|μ=ν=1=2a+2b+4/4γ+8a+10b−8/6γ+24ab−10a−8b+4/9γ
Consequently,(i)
First Zagreb index:M1Γ1=Dμ+Dνfμ,ν|μ=ν=1=Dμfμ,ν|μ=ν=1+Dνfμ,ν|μ=ν=1=72ab−10a+4+72ab−2a+10b−4=144ab−12a+10b(ii)
Second Zagreb index:M2Γ1=DμDνfμ,ν|μ=ν=1=DμDνfμ,ν|μ=ν=1=216ab−34a−4b+4(iii)
Second modified Zagreb index:MM2Γ1=TμTνfμ,ν|μ=ν=1=TμTνfμ,ν|μ=ν=1=8/3ab+13/18a+23/18b+1/9(iv)
General Randic index:RγΓ1=DμγDνγfμ,ν|μ=ν=1=4γ2a+2b+4+6γ8a+10b−8+9γ24ab−10a−8b+4(v)
Reciprocal general Randic index:RRγΓ1=2a+2b+4/4γμ2ν2+8a+10b−8/6γμ2ν3+24ab−10a−8b+4/9γ(vi)
Symmetric division deg index:SDDΓ1=DμTν+DνTμfμ,ν|μ=ν=1=DμTνfμ,ν|μ=ν=1+DνTμfμ,ν|μ=ν=1=24ab−8/3+2/3b+8/3+24ab+4a+9b−4=48ab+4/3a+29/3b−4/3Theorem 3.
LetΓ1=VC5C7 be the pent-heptagonal nanosheets. Then, the M-polynomial of Γ1 is(18)MΓ,μ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.
Then, harmonic indexHΓ1, inverse index ISΓ1 and augmented Zagreb index AZIΓ1 obtained from M-polynomial are as follows:aHΓ1=13/15a+7/3b+8ab+2/15bISΓ1=36ab−17/5a−44b+2/5cAZIΓ1=273.375ab−33.90625a+4.875b+13.5625Proof.
Letfμ,ν=MΓ1,μ,ν be the M-polynomial of the pent-heptagonal nanosheets VC5C7; then,(19)fμ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.
Firstly, we find out the required partial derivatives and integrals are as follows:Jfμ,ν=2a+2b+4μ4+8a+10b−8μ5+24ab−10a−8b+4μ6TμJfμ,ν=a/2+b/2+1μ4+8/5a+2b−8/5μ5+4ab−5/3a−4/3b+2/3μ6JDμDνfμ,ν=8a+8b+16μ2+48a+60b−48μ3+216ab−90a−72b+36μ4Q2JDμDνfμ,ν=8a+8b+16μ4+48a+60b−48μ5+216ab−90a−72b+36μ6TμQ2JDμDνfμ,ν=2a+2b+4μ4+1/548a+60b−48μ5+36ab−15a−12b+6μ6Dμ3Dν3fμ,ν=432a+2b+4μν+638a+10b−8μν2+9324ab−10a−8b+4μ2ν2JDμ3Dν3fμ,ν=432a+2b+4μ2+638a+10b−8μ3+9324ab−10a−8b+4μ4Tμ3JDμ3Dν3fμ,ν=|432a+2b+4/2μ2+638a+10b−8/3μ3+9324ab−10a−8b+4/4μ4
Now, we obtainμ=ν=1:(20)TμJfμ,ν|μ=ν=1=1330a+76b+4ab+115TμQ2JDμDνfμ,ν|μ=ν=1=36ab−175a+2b+25Tμ3JDμ3Dν3fμ,ν|μ=ν=1=82a+2b+4+88a+10b−8+7296424ab−10a−8b+4=273.375ab−33.90625a+4.875b+13.5625.
Consequently,(i)
Harmonic index:(21)HΓ1=2TμJfμ,ν|μ=ν=1=21330a+76b+4ab+115=1315a+73b+8ab+215.(ii)
Inverse index:(22)ISΓ1=TμQ2JDμDνfμ,ν|μ=ν=1=2a+2b+4+1548a+60b−48+36ab−15a−12b+6=36ab−175a+2b+25.(iii)
Augmented Zagreb index:(23)AZIΓ1=Tμ3JDμ3Dν3fμ,ν|μ=ν=1=4232a+2b+4+6338a+10b−8+94324ab−10a−8b+4=273.375ab−33.90625a+4.875b+13.5625.Theorem 4.
LetΓ2=HC5C7 be the second pent-heptagonal nanosheets; the M-polynomial of Γ2 is(24)MΓ2,μ,ν=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.Proof.
Now, by using definition of M-polynomial forΓ2,(25)MΓ2,μ,ν=∑s≤tEs,tΓμsνt=∑2≤2E2,2Γ2μ2ν2+∑2≤3E2,3Γ2μ2ν3+∑3≤3E3,3Γ2μ3ν3=E1μ2ν2+E2μ2ν3+E3μ3ν3=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.
The M-polynomial ofΓ2 is(26)MΓ2,μ,ν=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.Theorem 5.
LetΓ2=HC5C7 be the pent-heptagonal nanosheets. Then, the M-polynomial of Γ2 is(27)MΓ2,μ,ν=2a+2b+4μ2ν2+8a+10b−8μ2ν3+24ab−10a−8b+4μ3ν3.So, the 1st Zagreb indexM1Γ2, 2nd modified Zagreb MM2Γ2, general Randic RγΓ2 where γ∈α, reciprocal general Randic RRγΓ2, where γ∈α, and the symmetric division deg index SDDΓ2 obtained from M-polynomial are as follows:aM1Γ2=144ab−12a+6b+48bM2Γ2=216ab−34a−6b+74cMM2Γ2=8/3ab+13/18a+13/12b+17/18dRγΓ2=4γ2a+3b+2+6γ8a+6b−4+9γ24ab−10a−6b+10eRRγΓ2=2a+3b+2/4γ+8a+6b−4/6γ+24ab−10a−6b+10/9γfSSDΓ2=48ab+4/3a+7b+46/3Proof.
Letfμ,ν=MΓ2,μ,ν be the M-polynomial of the pent-heptagonal nanosheets HC5C7; then,(28)fμ,ν=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.
Firstly, we find out the required partial derivatives and integrals as follows:Dμfμ,ν=22a+3b+2μν2+28a+6b−4μν3+324ab−10a−6b+10μ2ν3Dνfμ,ν=22a+3b+2μ2ν+38a+6b−4μ2ν2+324ab−10a−6b+10μ3ν2DμDνfμ,ν=42a+3b+2μν+68a+6b−4μν2+924ab−10a−6b+10μ2ν2Tμfμ,ν=a+3/2b+1μ2ν2+4a+3b−2μ2ν3+8ab−10/3a−2b+10/3μ3ν3Tνfμ,ν=a+3/2b+1μ2ν2+8/3a+2b−4/3μ2ν3+8ab−10/3a−2b+10/3μ3ν3TμTνfμ,ν=TμTνfμ,ν=1/42a+3b+22μ2ν2+1/68a+6b−4μ2ν3+1/924ab−10a−6b+10μ3ν3DνTμfμ,ν=DνTμfμ,ν=2a+3b+2μ2ν+34a+3b−2μ2ν2+38ab−10/3a−2b+10/3μ3ν2DμTνfμ,ν=DμTνfμ,ν=2a+3b+2μν2+28/3a+2b−4/3μν3+24ab−10a−6b+10μ2ν3DμγDνγ=4γ2a+3b+2μν+6γ8a+6b−4μν2+9γ24ab−10a−6b+10μ2ν2TμγTνγ=2a+3b+2/4γμ2ν2+8a+6b−4/6γμ2ν3+24ab−10a−6b+10/9γμ3ν3
Now, we obtainμ=ν=1:Dμfμ,ν|μ=ν=1=72ab−10a+26Dνfμ,ν|μ=ν=1=72ab−2a+6b+22DμDνfμ,ν|μ=ν=1=216ab−34a−6b+74Tμfμ,ν|μ=ν=1=8ab+5/3a+5/2b+7/3Tνfμ,ν|μ=ν=1=8ab+1/3a+3/2b+3TμTνfμ,ν|μ=ν=1=8/3ab+13/18a+13/12b+17/18DνTμfμ,ν|μ=ν=1=24ab+4a+6b+6DμTνfμ,ν|μ=ν=1=24ab−8/3a+b+28/3DμγDνγfμ,ν|μ=ν=1=4γ2a+3b+2+6γ8a+6b−4+9γ24ab−10a−6b+10TμγTνγfμ,ν|μ=ν=1=2a+3b+2/4γ+8a+6b−4/6γ+24ab−10a−6b+10/9γ
Consequently,(i)
First Zagreb index:M1Γ2=Dμ+Dνfμ,ν|μ=ν=1=Dμfμ,ν|μ=ν=1+Dνfμ,ν|μ=ν=1=144ab−12a+6b+48(ii)
Second Zagreb index:M2Γ2=DμDνfμ,ν|μ=ν=1=DμDνfμ,ν|μ=ν=1=216ab−34a−6b+74(iii)
Second modified Zagreb index:MM2Γ2=TμTνfμ,ν|μ=ν=1=TμTνfμ,ν|μ=ν=1=8/3ab+13/18a+13/12b+17/18(iv)
General Randic index:RγΓ2=DμγDνγfμ,ν|μ=ν=1=4γ2a+3b+2+6γ8a+6b−4+9γ24ab−10a−6b+10(v)
Reciprocal general Randic index:RRγΓ2=TμγTνγfμ,ν|μ=ν=1=2a+3b+2/4γ+8a+6b−4/6γ+24ab−10a−6b+10/9γ(vi)
Symmetric division deg index:SDDΓ2=DμTν+DνTμfμ,ν|μ=ν=1=DμTνfμ,ν|μ=ν=1+DνTμfμ,ν|μ=ν=1=24ab−8/3a+b+28/3+24ab+4a+6b+6=48ab+4/3a+7b+46/3Theorem 6.
LetΓ2=HC5C7 be the pent-heptagonal nanosheets. Then, the M-polynomial of Γ2 is(29)MΓ2,μ,ν=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.Then, harmonic indexHΓ2, inverse index ISΓ2, and augmented Zagreb index AZIΓ2 obtained from M-polynomial are as follows:aHΓ2=13/15a+19/10b+8ab−11/3bISΓ2=36ab−17/5a−9/5b+61/5cAZIΓ2=273.3744ab−33.90625a+3.6564b+97.906Proof.
Letfμ,ν=MΓ2,μ,ν be the M-polynomial of the pent-heptagonal nanosheet HC5C7; then,(30)fμ,ν=2a+3b+2μ2ν2+8a+6b−4μ2ν3+24ab−10a−6b+10μ3ν3.
First, we find out the required partial derivatives and integrals asJfμ,ν=2a+3b+2μ4+8a+6b−4μ5+24ab−10a−6b+10μ6TμJfμ,ν=2a+3b+2/4μ4+8a+6b−4/5μ5+24ab−10a−6b+10/6μ6JDμDνfμ,ν=8a+12b+8μ2+48a+36b−24μ3+216ab−90a−54b+90μ4Q2JDμDνfμ,ν=8a+12b+8μ4+48a+36b−24μ5+216ab−90a−54b+90μ6TμQ2JDμDνfμ,ν=2a+3b+2μ4+1/548a+36b−24μ5+36ab−15a−9b+15μ6Dμ3Dν3fμ,ν=432a+3b+2μν+638a+6b−4μν2+9324ab−10a−6b+10μ2ν2JDμ3Dν3fμ,ν=432a+3b+2μ2+638a+6b−4μ3+9324ab−10a−6b+10μ4Tμ3JDμ3Dν3fμ,ν=82a+3b+2μ2+88a+6b−4μ3+9/4324ab−10a−6b+10/4μ4
Now, we obtainμ=ν=1:(31)TμJfμ,ν|μ=ν=1=142a+3b+2+158a+6b−4+1624ab−10a−6b+10=1330a+1920+4ab+4130,TμQ2JDμDνfμ,ν|μ=ν=1=148a+12b+8+1548a+36b−24+16216ab−90a−54b+90=15180ab−17a−6b+36,Tμ3JDμ3Dν3fμ,ν|μ=ν=1=82a+3b+2+88a+6b−4+94324ab−10a−6b+10=82a+3b+2+88a+6b−4+11.390624ab−10a−6b+10=273.3744ab−33.90625a+3.6564b+97.906.
Consequently,(i)
Harmonic index:(32)HΓ2=2TμJfμ,ν|μ=ν=1=2142a+3b+2+158a+6b−4+1624ab−10a−6b+10=8ab+1315a+1910b−113.(ii)
Inverse index:(33)ISΓ2=TμQ2JDμDνfμ,ν|μ=ν=1=148a+12b+8+1548a+36b−24+16216ab−90a−54b+90=36ab−175a−95b+615.(iii)
Augmented Zagreb index:(34)AZIΓ2=Tμ3JDμ3Dν3fμ,ν|μ=ν=1=2a+3b+2+88a+6b−4+94324ab−10a−6b+10=82a+3b+2+88a+6b−4+11.390624ab−10a−6b+10=273.3744ab−33.90625a+3.6564b+97.906


## 5. Conclusion

In this section, we used the various degree-based TIs and show the comparison in the form of tables and figures. Comparison betweenM1Γ1, M2Γ1, MM1Γ1, and SDDΓ1 of VC5C7The comparison of 1st Zagreb, 2nd Zagreb, 2nd modified Zagreb, and symmetric division deg indices of pent-heptagonal nanosheetsΓ1 is computationally computed by using these M-polynomials. We calculated these indices for different values of a and b in Table 5, and we noted that when we increase the values of a and b, then all of the TIs of VC5C7 are increasing with the same order, as shown in Figure 3.Table 5
Comparison betweenM1Γ1, M2Γ1, MM1Γ1, and SDDΓ1 of VC5C7.
a,bM1Γ1M2Γ1MM1Γ1SDDΓ1a=2,b=41168164828.016423.91a=4,b=63468502874.681214a=6,b=8692010134142.692390.68a=8,b=101152416972232.023945.99a=10,b=121728025536343.0865887.99a=12,b=142418835828475.2488214a=14,b=163224847848628.7710924.14a=16,b=184146061596803.65214018.16a=18,b=205182477072999.89417496.18a=20,b=2263340942761217.49621358.2Figure 3
Graphical comparison betweenM1Γ1, M2Γ1, MM1Γ1, and SDDΓ1 of VC5C7 and comparison between M1Γ2, M2Γ2, MM1Γ2, and SDDΓ2 of HC5C7.The comparison of 1st Zagreb, 2nd Zagreb, 2nd modified Zagreb, and symmetric division deg indices of pent-heptagonal nanosheetsΓ2 is computationally computed by using these M-polynomials. We calculated these indices for different values of a and b in Table 6, and we noted that when we increase the values of a and b, then all of the TIs of HC5C7 are increasing with the same order, as shown in Figure 4.Table 6
Comparison betweenM1Γ2, M2Γ2, MM1Γ2, and SDDΓ2 of HC5C7.
a,bM1Γ2M2Γ2MM1Γ2SDDΓ2a=2,b=41200171028.07429.97a=4,b=63492508674.331214.66a=6,b=8693610190141.942383.33a=8,b=101153217022231.1053936.06a=10,b=121728025582341.5055872.74a=12,b=142418035870473.2658193.42a=14,b=163223247886626.38510898.1a=16,b=184143661630800.86513986.78a=18,b=205179277102996.70517459.46a=20,b=2263300943021211.74521316.14Figure 4
Graphical comparison betweenM1Γ2, M2Γ2, MM1Γ2, and SDDΓ2 of HC5C7 and comparison between HΓ1, ISΓ1, and AZIΓ1 of VC5C7.The comparison of the harmonic index, the inverse sum index, and the augmented Zagreb index of pent-heptagonal nanosheetsΓ1 is computationally computed by these M-polynomials. We calculated these indices for different values of a and b in Table 7, and we noted that when we increase the values of a and b, then all of the TIs of VC5C7 are increasing with the same order, as shown in Figure 5.Table 7
Comparison betweenHΓ1, ISΓ1, and AZIΓ1 of VC5C7.
a,bHΓ1ISΓ1AZIΓ1a=2,b=475.208105.62152.25a=4,b=6209.685866468.18a=6,b=8408135412971.12a=8,b=10670.4542413.221661.1125a=10,b=12996.8643758.432538.0625a=12,b=141387.2745391.645602.0125a=14,b=161841.6847312.860852.9625a=16,b=182360.094952278290.9125a=18,b=202942.50412019.297915.8625a=20,b=223588.91414804.4119727.8125Figure 5
Graphical comparison betweenHΓ1, ISΓ1, and AZIΓ1 of VC5C7 and comparison between HΓ2, ISΓ2, and AZIΓ2 of HC5C7.The comparison of the harmonic index, the inverse sum index, and the augmented Zagreb index of pent-heptagonal nanosheetsΓ2 is computationally computed by these M-polynomials. We calculated these indices for different values of a and b in Table 8, and we noted that when we increase the values of a and b, then all of the TIs of HC5C7 are increasing with the same order, as shown in Figure 6.Table 8
Comparison betweenHΓ2, ISΓ2, and AZIΓ2 of HC5C7.
a,bHΓ2ISΓ2AZIΓ2a=2,b=469.66236.22231.7143a=4,b=6203.20851.806545.22a=6,b=8400.731705.4013045.69a=8,b=10662.266284721732.8a=10,b=12987.84276.632607.1a=12,b=141377.34599.245668.36a=14,b=161830.86800060916.58a=16,b=182348.40210293.478351.76a=18,b=202929.9361287597974.1a=20,b=223575.4715744.6119782.2Figure 6
Graphical comparison betweenHΓ2, ISΓ2, and AZIΓ2 of HC5C7.In this paper, the calculated M-polynomials and enumerated TIs assist us to recognize the physical characteristic, chemical sensitivity, and biological animation of the pent-heptagonal nanosheetsΓ1 and Γ2. These consequences give us remarkable ascertainment in the field of pharmaceutical production.However, the problem is still open to compute the different TIs (degree and distance based) for various nanosheets:(i)
To compute the nanosheet for other topological indices(ii)
To compute the various nanosheets for different topological indices



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