TopologicalIndicesofPent-Heptagonal NanosheetsviaM-Polynomials

Research ArticleTopological Indices of Pent-HeptagonalNanosheets via M-Polynomials

Hafiza Bushra Mumtaz1 Muhammad Javaid 1 Hafiz Muhammad Awais1

and Ebenezer Bonyah 2

1Department of Mathematics School of Science University of Management and Technology (UMT) Lahore Pakistan2Department ofMathematics Education Akenten Appiah-Menka University of Skills Training and Entrepreneurial DevelopmentKumasi 00233 Ghana

Correspondence should be addressed to Ebenezer Bonyah ebbonyagmailcom

Received 16 September 2021 Accepted 20 October 2021 Published 12 November 2021

Academic Editor Ali Ahmad

Copyright copy 2021Hafiza BushraMumtaz et al(is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

(e combination of mathematical sciences physical chemistry and information sciences leads to a modern field known ascheminformatics It shows a mathematical relationship between a property and structural attributes of different types of chemicalscalled quantitative-structuresrsquo activity and qualitative-structuresrsquo property relationships that are utilized to forecast the chemicalsciences and biological properties in the field of engineering and technology Graph theory has originated a significant usage inthe field of physical chemistry and mathematics that is famous as chemical graph theory (e computing of topological indices(TIs) is a new topic of chemical graphs that associates many physiochemical characteristics of the fundamental organiccompounds In this paper we used the M-polynomial-based TIs such as 1st Zagreb 2nd Zagreb modified 2nd Zagreb symmetricdivision deg general Randi c

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

1 Introduction

Nanostructures [1 2] have been studied as new materialswith the size of elementals structures that has been engi-neered at the nanometersrsquo scale Most of the materials in thissize range usually show novel behavior (erefore inter-vention in the characteristics of structures at the nanoscaleallows the formation of devices and nanomaterials withcompletely or enhanced novel functionalities and propertiesUnderstanding the science of nanostructures is curiosity andimportant driven not only for the interesting nature of thetopic but also for novel and overwhelming usage of nano-scale systems in various fields of science and technologyNanotechnology can be recognized as a technology of de-sign application and fabrication of nanomaterials andnanostructures [3]

(e branch of nanotechnology and nanoscience is beingperused by chemists physicists materials scientists

engineers biologists computer scientists and mathemati-cians [4] So it is also interdisciplinary Nanostructures maybe divided based onmodulation and dimensionality Most ofthe distinct nanotubes zeolites aerogel core-shell structureand nanoporous materials have unique properties Nu-merous techniques have been utilized for the synthesis ofnanomaterials with no of degrees of success and severaldirect as well as indirect methods are used for their prop-erties [5] (e motivation to develop the nanomaterials isthat the characteristics become size based in the nanometerrange due to quantum confinement effect and surface effect(e chemical bonds magnetic properties geometricstructure electronic properties ionization potential me-chanical strength optical properties and thermal propertiesare affected due to particle size in nanometers rangeNanostructures show characteristics mostly higher than theconventional coarse-grained material (ese contain hard-nessincreased strength toughnessimproved ductility

HindawiJournal of MathematicsVolume 2021 Article ID 4863993 13 pageshttpsdoiorg10115520214863993

enhanced diffusivity reduced density higher electrical re-sistance reduced elastic modulus lower thermal conduc-tivity increase specific heat higher thermal expansioncoefficient increased oscillator and strength luminescenceblue shift absorption and superior soft magnetic charac-teristics in comparison to the conventional bulk materialFurthermore these characteristics are being briefly exam-ined to discover new tools (e interesting branch ofnanotechnology has a vast range of different types of ap-plications (e use of nanomaterials has manufacturedtransistors having low speed and laser having low thresholdcurrent (ese are utilized in satellite receivers having lownoise amplification as a source for fiber optics communi-cations and compact disk player systems Constructive toolsof nanostructures contain UV-resistant wood coating andself-cleaning glass On the other hand nanoscale tools arebeing utilized in the field of medicine for the prevention andtreatment of diseases diagnosis and in magnetic resonanceimaging drug delivery system radioactive tracers etc [6](e importance of nanomaterials is rising nowadays Manyother types of tools may be possible with the peculiar andnovel characteristics of nanomaterials [7 8]

(erefore TIs are useful to define molecular nano-materials Nanostructures that have a scale of less than100 nm contain nanosheets nanotubes and nanoparticlesNanosheets (two-dimensional nanomaterials) have a sharpedge and large surface area that cause them to play a vitalrole in various types of tools such as catalysis energy storagebioelectronics and optoelectronics [9 10] Silicone bor-ophene and graphene are specific nanosheets Due to therare optical electrical mechanical and structural charac-teristics graphene nanosheets received great recognitionfrom industrial and academic researchers [11] (e differentproperties of the C5C7 nanosheet have become the mostadvanced field in research A C5C7 structure is developed byalternating C5 andC7 [7] In 2009 Graovac et al studied theGA index of TUC4C8 (S) nanotubes In 2011 Graovac et al[12] studied the fifth geometric arithmetic index for nanostardendrimers and Asadpour et al calculated Zagreb Randi c

and ABC indices of TUC4C8 (R) and TUC4C8 (S) V-Phe-nylenic nanotorus and nanotubes In 2014 Al-Fozan et alsolved Szeged index of H-naphthalene nanosheets (2n 2m)and C4C8 (S) Loghman and Ashrafi studied the Padma-karndashIvan (PI) index of TUC4C8 (S) nanotubes For furtherdiscussion see [13ndash15]

However the combination of three fields such asmathematics physical chemistry and information scienceslead to a modern field known as cheminformatics [16ndash18] Itdevelops a mathematical relationship between a propertyand structural attributes of different types of chemicals calledby quantitative-structuresrsquo activity and qualitative-struc-turesrsquo property relationship that are utilized to forecast theorganic sciences and biological properties in the field ofengineering and technology [19 20] Graph theory hasoriginated a significant usage in the field of mathematicalchemistry that is famous as chemical graph theory

Polya gave the idea for counting polynomials in the fieldof chemistry [21] and Wiener introduced the concept of TIrelated to the paraffinrsquos boiling point [22] Computing the

TIs is a new field of chemical graphs that associates manyphysiochemical characteristics of the fundamental chemicalcompounds [23ndash27]

2 Preliminaries

A molecular structure Γ (V(Γ) E(Γ)) V(Γ) s1 s21113864

s3 sn andE(Γ) are nodes (vertices) and edge set of Γ|V(Γ)| v and |E(Γ)| e is the order and size of Γ In aconnected and simple molecular graph a path is representedwithin two vertices and the distance between the two verticess and t is mentioned as φ(s t) in a graph Γ see [28ndash30] Inthis paper a graph is connected and simple having nomultiple edges or loops

1st and 2nd Zagreb indices let Γ be a molecularstructure then its 1st and 2nd Zagreb indices [31] are

M1(Γ) 1113944sisinV(Γ)

[φ(s)]2

1113944stisinE(Γ)

[φ(s) + φ(t)]

M2(Γ) 1113944stisinE(Γ)

[φ(s) times φ(t)](1)

General Randi cindex if R is the real number α isin R

and Γ is a molecular structure the general Randi cindex

[32] is

Rα(Γ) 1113944stisinE(Γ)

[φ(s)φ(t)]α (2)

Symmetric division deg index for a molecular structureΓ the symmetric division deg index [33] is

SDD(Γ) 1113944stisinE(Γ)

min(φ(s)φ(t))

max(φ(s)φ(t))+max(φ(s)φ(t))

min(φ(s) φ(t))1113890 1113891

(3)

Harmonic index for a molecular structure Γ the har-monic index [34] is

H(Γ) 1113944stisinE(Γ)

2φ(s) + φ(t)

(4)

Inverse sum index for a molecular structure Γ the in-verse sum index [35] is

IS(Γ) 1113944stisinE(Γ)

φ(s)φ(t)

φ(s) + φ(t) (5)

Augmented Zagreb index for a molecular structure Γthe augmented Zagreb index [13] is

AZI(Γ) 1113944stisinE(Γ)

φ(s) times φ(t)

φ(s) + φ(t) minus 21113890 1113891

3

(6)

A graph polynomial is a graph invariant whose valuesare polynomials So all these invariants are discussed inalgebraic graph theory [36] Among such types of alge-braic polynomials the M-polynomial defined in 2015shows the same role in finding the much closed form ofvarious degree-based TIs that correlate different types ofchemical properties of the various materials under

2 Journal of Mathematics

investigation In 2019 Yang et al [37] find out theM-polynomial and topological indices of benzene ringembedded in P-type surface network In 2020 Khalafet al [38] computed the M-polynomial and topologicalindices of book graph and Raza and Sakaiti [2] solved theM-polynomial and degree-based topological indices ofsome nanostructures In 2021 Mondal et al [39] find outthe neighborhood M-polynomial of titanium compoundsand Irfan et al [1] computed the M-polynomials andtopological indices for line graphs of chain silicate net-work and H-naphtalenic nanotubes

M-Polynomial let Γ be a molecular structure andmijΓ i jge 1 be the number of edges e st of Γ in such a waythat φ(s)φ(t)1113864 1113865 i j1113864 1113865 (e M-polynomial of Γ is

M(Γ μ ]) 1113944ile j(Γ)

mijΓμi]j

1113872 1113873(7)

Now we discussed the relationship between theM-polynomial and some important TIs in the form ofTables 1 and 2

3 Pent-Heptagonal Nanosheet

Firstly we discuss the structure of pent-heptagonal nano-sheet VC5C7 For nanosheet of VC5C7(a b) we representthe number of pentagons in the first row by b and the firstfour rows of nodes as well as edges are repeated (ereforewe represent the number of repetitions as a (e nanosheetVC5C7(2 4) has 16ab + 2a + 5b nodes or vertices and24ab + 4b edges Additionally it has 6a + 7b nodes havingdegree 2 and 16ab minus 4a minus 2b nodes having degree 3 (edegree-based edge partition of nanosheet a 2 and b 4 isshown in Table 3

From Figure 1 we note that 2 distinct types of vertices inVC5C7 are 2 and 3 So

V1 s isin V Γ1( 1113857|φ(s) 21113864 1113865

V2 s isin V Γ1( 1113857|φ(s) 31113864 1113865(8)

We have 3 different types of edges that is based on thedegree of end nodes in (Γ1) that are

E22 st isin Γ1( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ1( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ1( 1113857|φ(s) 3φ(t) 31113864 1113865

(9)

where |E1| (2a + 2b + 4) |E2| (8a + 10b minus 8)|E3| (24ab minus 10a minus 8b + 4) and a 2 and b 4 (en

E Γ1( 11138571113868111386811138681113868

1113868111386811138681113868 E11113868111386811138681113868

1113868111386811138681113868 + E21113868111386811138681113868

1113868111386811138681113868 + E31113868111386811138681113868

1113868111386811138681113868 16 + 48 + 144 208 (10)

Now we discuss the structure of pent-heptagonalnanosheet HC5C7 For the nanosheet HC5C7(a b) werepresent the number of pentagons in the first row by b andthe 1st four rows of nodes and edges are repeated So werepresent the number of repetitions as a (e nanosheetsHC5C7(2 4) have 16ab + 2a + 4b vertices and 24ab + 3b

edges Moreover it has 6a + 6b vertices with degree 2 and16ab minus 4a minus 2b vertices with degree 3(e degree-based edge

partition of nanosheets for a 2 and b 4 is shown inTable 4

From Figure 2 we note that 2 distinct types of vertices inHC5C7 are 2 and 3 So

V1 s isin V Γ2( 1113857|φ(u) 21113864 1113865

V2 s isin V Γ2( 1113857|φ(u) 31113864 1113865(11)

We have 3 different types of edges that is based on thedegree of end nodes in (Γ1)

E22 st isin Γ2( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ2( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ2( 1113857|φ(s) 3φ(t) 31113864 1113865

(12)

4 Main Results

(is section deals with the main results consisting ofpolynomials and TIs of the nanosheets

Table 1 Derivation of TIs from M-polynomial

Indices f(μ ]) Derivation from M(Γ μ ])

M1 μ + ] (Dμ + D])(M(Γ μ ]))|μ1]M2 μ] (DμD])(M(Γ μ ]))|μ1]MM2 1μ] (SμS])(M(Γ μ ]))|μ1]Rα (μ])α α isin N (Dα

μDα] )(M(Γ μ ]))|μ1]

RαRα 1(μ])α α isin N (SαμSα] )(M(Γ μ ]))|μ1]SDD μ2 + ]2μ] (DμS] + D]Sμ)(M(Γ μ ]))|μ1]

Table 2 Other TIs from M-polynomial


H 2μ + ] 2SμJ(M(Γ μ ]))|μ1]IS μ]μ + ] SμQ2JDμD](M(Γ μ ]))|μ1]

AZI (μ]μ + ] minus 2)3 S3μJD3μD3

](M(Γ μ ]))|μ1]

Table 3 Partition of edge set VC5C7

Edges partitions E1 E22 E2 E23 E3 E33

Cardinality 2a + 2b + 4 8a + 10b minus 8 24ab minus 10a minus 8b + 4

Figure 1 Pent-heptagonal nanosheet VC5C7

Journal of Mathematics 3

Theorem 1 Let Γ1 VC5C7 be the pent-heptagonal nano-sheet en the M-polynomial of Γ is

M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3

+(24ab minus 10a minus 8b + 4)μ3]3(13)

Proof Now by using definition of M-polynomial of (Γ1)we obtain

M Γ1 μ ]( 1113857 1113944sle t

Est Γ1( 1113857μs]t1113960 1113961

11139442le 2

E22 Γ1( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ1( 1113857μ2]31113960 1113961

+ 11139443le 3

E331113960 1113872Γ11113857μ3]31113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


(e M-polynomial of (Γ1) is

M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


Theorem 2 Let Γ VC5C7 be the pent-heptagonal nano-sheet en the M-polynomial of Γ is

M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


So the 1st Zagreb index (M1(Γ1)) 2nd Zagreb index(M2(Γ1)) 2nd modified Zagreb (MM2(Γ1)) general Randic(Rc(Γ1)) reciprocal general Randic RRc(Γ1) where c isin αand the symmetric division deg index (SDD(Γ1)) obtainedfrom M-polynomial are as follows

(a) M1(Γ1) 144ab minus 12a + 10b

(b) M2(Γ1) 216ab minus 34a minus 4b + 4(c) MM2(Γ1) 83ab + 1318a + 2318b + 19(d) Rc(Γ1) (4)c(2a + 2b + 4) + (6)c(8a + 10b minus

8) + (9)c(24ab minus 10a minus 8b + 4)

(e) RRc(Γ1) 2a + 2b + 4(4)c + 8a + 10b minus 8(6)c +

24ab minus 10a minus 8b + 4(9)c

(f) SSD(Γ1) 48ab + 43a + 293b minus 43

Proof Let f(μ ]) M(Γ1 μ ]) be theM-polynomial of thepent-heptagonal nanosheet VC5C7 then

f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


Firstly we find out the required partial derivatives andintegrals as

Dμf(μ ]) 2(2a + 2b + 4)μ]2 + 2(8a + 10b minus 8)μ]3 +

3(24ab minus 10a minus 8b + 4)μ2]3

D]f(μ ]) 2(2a + 2b + 4)μ2] + 3(8a + 10b minus 8)μ2]2 +


Dμ(D]f(μ ])) 4(2a + 2b + 4)μ] + 6(8a + 10b minus 8)

μ]2 + 9(24ab minus 10a minus 8b + 4)μ2]2

Tμ(f(μ ])) (a + b + c)μ2]2 + (4a + 5b minus 4)μ2]3 +

(8ab minus 103a minus 83b + 43μ3]3)T](f(μ ])) (a + b + c)μ2]2 + (8a + 10b minus 8)3μ2]3 + (8ab minus 103a minus 83b + 43)μ3]3TμT](f(μ ])) Tμ(T](f(μ ]))) (a + b + 2)2μ2]2 + (8a + 10b minus 8)6μ2]3 + (4ab minus 53a minus 43b +

23)μ3]3D]Tμ(f(μ ])) D](Tμ(f(μ ]))) 2(a + b + c)

μ2] + 3(4a + 5b minus 4)μ2]2 + 3(8ab minus 103a minus 83b +

43μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) 2(a + b + c)

μ]2 + 23(8a + 10b minus 8)μ]3 + (24ab minus 10a minus 8b +

4)μ2]3

DcμD

c] (4)c(2a + 2b + 4)μ] +(6)c(8a + 10b minus 8)μ]2 +

(9)c(24ab minus 10a minus 8b + 4)μ2]2

TcμT

c] (2a + 2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)

μ2]3 + (24ab minus 10a minus 8b + 4)(9)cμ3]3

Now we obtain μ ] 1

Dμf(μ ])|μ]1 72ab minus 10a + 4D]f(μ ])|μ]1 72ab minus 2a + 10b minus 4Dμ(D]f(μ ]))|μ]1 216ab minus 34a minus 4b + 4Tμ(f(μ ]))|μ]1 8ab + 53a + 103b minus 83T](f(μ ]))|μ]1 8ab + 13a + 53b minus 43

Figure 2 Pent-heptagonal nanosheet HC5C7

Table 4 Partition of edge set of HC5C7

Edgesrsquo partitions E1 E22 E2 E23 E3 E33



Tμ(T](f(μ ])))|μ]1 83ab + 1318a + 2318b minus 89D](Tμ(f(μ ])))|μ]1 24ab + 4a + 9b minus 4Dμ(T](f(μ ])))|μ]1 24ab minus 83a + 23b + 83D

cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 2b + 4) + (6)(8a +

10b minus 8) + (9)c(24ab minus 10a minus 8b + 4)

Tcμ(T

c](f(μ ])))|μ]1 (2a + 2b + 4)(4)c + (8a +

10b minus 8)(6)c + (24ab minus 10a minus 8b + 4)(9)c

Consequently

(i) First Zagreb index M1(Γ1) (Dμ + D]) (f(μ ]))

|μ]1 Dμ(f(μ ]))|μ]1 + D](f(μ ]))|μ]1 (72ab minus 10a + 4) + (72ab minus 2a + 10b minus 4) 144ab minus

12a + 10b

(ii) Second Zagreb index M2(Γ1) (DμD])

(f(μ ]))|μ]1 Dμ(D](f(μ ])))|μ]1 216ab minus

34a minus 4b + 4(iii) Second modified Zagreb index MM2(Γ1)

(TμT])(f(μ ]))|μ]1 Tμ(T](f(μ ])))|μ]1 83ab + 1318a + 2318b + 19

(iv) General Randic index Rc(Γ1) (DcμD

c])(f(μ ]))

|μ]1 (4)c (2a + 2b + 4) + (6)c(8a + 10b minus 8) +

(9)c(24ab minus 10a minus 8b + 4)

(v) Reciprocal general Randic index RRc(Γ1) (2a +

2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)cμ2]3 + (24ab

minus 10a minus 8b + 4)(9)c

(vi) Symmetric division deg index SDD(Γ1) (DμT]+

D]Tμ) (f(μ]))|μ]1 DμT](f(μ]))|μ]1 + D]Tμ) (f(μ]))|μ]1 (24ab minus 83+23b +83) + (24ab +4a +9b minus 4) 48ab+ 43a +293bminus 43

Theorem 3 Let Γ1 VC5C7 be the pent-heptagonal nano-sheets en the M-polynomial of Γ1 is

M(Γ μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


en harmonic index (H(Γ1)) inverse index (IS(Γ1))and augmented Zagreb index (AZI(Γ1)) obtained fromM-polynomial are as follows

(a) H(Γ1) 1315a + 73b + 8ab + 215(b) IS(Γ1) 36ab minus 175a minus 44b + 25(c) AZI(Γ1) 273375ab minus 3390625a + 4875b +

135625

Proof Let f(μ ]) M(Γ1 μ ]) be theM-polynomial of thepent-heptagonal nanosheets VC5C7 then

f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


Firstly we find out the required partial derivatives andintegrals are as follows

J(f(μ ])) (2a + 2b + 4)μ4 + (8a + 10b minus 8)μ5 +

(24ab minus 10a minus 8b + 4)μ6

Tμ(J(f(μ ]))) (a2 + b2 + 1)μ4 + (85a + 2b minus 85)

μ5 + (4ab minus 53a minus 43b + 23)μ6

J(Dμ(D](f(μ ])))) (8a + 8b + 16)μ2 + (48a +

60b minus 48)μ3 + (216ab minus 90a minus 72b + 36)μ4

Q2(J(Dμ(D](f(μ ]))))) (8a + 8b + 16)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 2b + 4)μ4 + 15(48a + 60b minus 48)μ5 + (36ab minus 15a minus 12b + 6)μ6

D3μ(D3

](f(μ ]))) (4)3(2a + 2b + 4)μ] + (6)3(8a +

10b minus 8)μ]2 + (9)3(24ab minus 10a minus 8b + 4)μ2]2

J(D3μD3

](f(μ ]))) (4)3(2a + 2b + 4)μ2 + (6)3(8a +

10b minus 8)μ3 + (9)3(24ab minus 10a minus 8b + 4)μ4

T3μ(J(D3

μD3](f(μ])))) |(4)3(2a +2b +4)2μ2 + (6)3

(8a +10b minus 8)3μ3 + (9)3(24ab minus 10a minus 8b +4)4μ4


Tμ(J(f(μ ])))|μ]1 1330

a +76

b + 4ab +115

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 36ab minus175

a + 2b +25

T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 2b + 4) + 8(8a + 10b minus 8)

+72964

1113874 1113875(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(20)

Consequently (i) Harmonic index


H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)

+(36ab minus 15a minus 12b + 6)

36ab minus175

a + 2b +25

(22)

(iii) Augmented Zagreb index

AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)

Theorem 4 Let Γ2 HC5C7 be the second pent-heptagonalnanosheets the M-polynomial of (Γ2) is

M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)

μ2]3 +(24ab minus 10a minus 6b + 10)μ3]3(24)

Proof Now by using definition of M-polynomial for (Γ2)

M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3


Theorem 5 Let Γ2 HC5C7 be the pent-heptagonalnanosheets en the M-polynomial of Γ2 is

M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3


So the 1st Zagreb index (M1(Γ2)) 2nd modified Zagreb(MM2(Γ2)) general Randic (Rc(Γ2)) where c isin α recip-rocal general Randic (RRc(Γ2)) where c isin α and thesymmetric division deg index (SDD(Γ2)) obtained fromM-polynomial are as follows

(a) M1(Γ2) 144ab minus 12a + 6b + 48(b) M2(Γ2) 216ab minus 34a minus 6b + 74(c) MM2(Γ2) 83ab + 1318a + 1312b + 1718(d) Rc(Γ2) (4)c(2a + 3b + 2) + (6)c(8a + 6b minus 4) +




(f) SSD(Γ2) 48ab + 43a + 7b + 463

Proof Let f(μ ]) M(Γ2 μ ]) be theM-polynomial of thepent-heptagonal nanosheets HC5C7 then

f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3


Firstly we find out the required partial derivatives andintegrals as follows



D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)

μ2]3 + (8ab minus (103)a minus 2b + (103)μ3]3)T](f(μ ])) (a + (32)b + 1)μ2]2 + ((83)a + 2b minus

(43))μ2]3 + (8ab minus (103)a minus 2b + (103))μ3]3

TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)

2μ2]2 + 16(8a + 6b minus 4)μ2]3 + 19(24ab minus 10a minus 6b +

10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)

μ]2 + 2(83a + 2b minus 43)μ]3 + (24ab minus 10a minus 6b + 10)

μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c



Dμf(μ ])|μ]1 72ab minus 10a + 26D]f(μ ])|μ]1 72ab minus 2a + 6b + 22Dμ(D]f(μ ]))|μ]1 216ab minus 34a minus 6b + 74Tμ(f(μ ]))|μ]1 8ab + 53a + 52b + 73T](f(μ ]))|μ]1 8ab + 13a + 32b + 3Tμ(T](f(μ ])))|μ]1 83ab + 1318a + 1312b +

1718D](Tμ(f(μ ])))|μ]1 24ab + 4a + 6b + 6Dμ(T](f(μ ])))|μ]1 24ab minus 83a + b + 283D

cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+

(6) c((8a + 6b minus 4) + (9)c(24ab minus 10a minus 6b + 10)

Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently

(i) First Zagreb index M1(Γ2) (Dμ + D])(f(μ ]))

|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1

144ab minus 12a + 6b + 48(ii) Second Zagreb index M2(Γ2) (DμD])(f(μ ]))

|μ]1 Dμ(D](f(μ ])))|μ]1 216ab minus 34a minus 6b

+ 74(iii) Secondmodified Zagreb index MM2(Γ2) (TμT])

(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +

1318a + 1312b + 1718(iv) General Randic index Rc(Γ2) (D

cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +


(v) Reciprocal general Randic index RRc(Γ2) (TcμT

c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus

4)(6)c + (24ab minus 10a minus 6b + 10)(9)c

(vi) Symmetric division deg index SDD(Γ2) (DμT] +

D]Tμ)(f(μ ])) |μ]1 DμT](f(μ ]))|μ]1 + D]Tμ)(f(μ ]))|μ]1 (24ab minus 83a + b + 283) + (24ab + 4a + 6b + 6) 48ab+ 43a + 7b + 463


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3


(en harmonic index (H(Γ2)) inverse index (IS(Γ2))and augmented Zagreb index (AZI(Γ2)) obtained fromM-polynomial are as follows

(a) H(Γ2) 1315a + 1910b + 8ab minus 113(b) IS(Γ2) 36ab minus 175a minus 95b + 615(c) AZI((Γ2) 2733744ab minus 3390625a + 36564b +

97906

Proof Let f(μ ]) M(Γ2 μ ]) be theM-polynomial of thepent-heptagonal nanosheet HC5C7 then

f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3


First we find out the required partial derivatives andintegrals as

J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +


Tμ(J(f(μ ]))) 2a + 3b + 24μ4 + 8a + 6b minus 45μ5 +

24ab minus 10a minus 6b + 106μ6

J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +

15(48a + 36b minus 24)μ5 + (36ab minus 15a minus 9b + 15)μ6

D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)

1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)

8(2a + 3b + 2) + 8(8a + 6b minus 4) +(113906)(24ab minus 10a minus 6b + 10)

2733744ab minus 3390625a + 36564b + 97906

(31)


Table 5 Comparison between M1(Γ1) M2(Γ1) MM1(Γ1) and SDD(Γ1) of VC5C7

a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)

a 2 b 4 1168 1648 28016 42391a 4 b 6 3468 5028 7468 1214a 6 b 8 6920 10134 14269 239068a 8 b 10 11524 16972 23202 394599a 10 b 12 17280 25536 343086 588799a 12 b 14 24188 35828 475248 8214a 14 b 16 32248 47848 62877 1092414a 16 b 18 41460 61596 803652 1401816a 18 b 20 51824 77072 999894 1749618a 20 b 22 63340 94276 1217496 213582

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD

Figure 3 Graphical comparison between M1(Γ1) M2(Γ1) MM1(Γ1) and SDD(Γ1) of VC5C7 and comparison between M1(Γ2) M2(Γ2)MM1(Γ2) and SDD(Γ2) of HC5C7

Table 6 Comparison between M1(Γ2) M2(Γ2) MM1(Γ2) and SDD(Γ2) of HC5C7

a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1

Figure 4 Graphical comparison between M1(Γ2) M2(Γ2) MM1(Γ2) and SDD(Γ2) of HC5C7 and comparison between H(Γ1) IS(Γ1)and AZI(Γ1) of VC5C7

Table 7 Comparison between H(Γ1) IS(Γ1) and AZI(Γ1) of VC5C7

a b H(Γ1) IS(Γ1) AZI(Γ1)

a 2 b 4 75208 1056 215225a 4 b 6 20968 586 646818a 6 b 8 408 1354 1297112a 8 b 10 670454 24132 216611125a 10 b 12 996864 37584 325380625a 12 b 14 1387274 53916 456020125a 14 b 16 1841684 73128 608529625a 16 b 18 2360094 9522 782909125a 18 b 20 2942504 120192 979158625a 20 b 22 3588914 148044 1197278125


IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH

Figure 5 Graphical comparison between H(Γ1) IS(Γ1) and AZI(Γ1) of VC5C7 and comparison between H(Γ2) IS(Γ2) and AZI(Γ2) ofHC5C7

Table 8 Comparison between H(Γ2) IS(Γ2) and AZI(Γ2) of HC5C7




AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion

In this section we used the various degree-based TIs andshow the comparison in the form of tables and figuresComparison between M1(Γ1) M2(Γ1) MM1(Γ1) andSDD(Γ1) of VC5C7

(e comparison of 1st Zagreb 2nd Zagreb 2ndmodified Zagreb and symmetric division deg indices ofpent-heptagonal nanosheets (Γ1) is computationallycomputed by using these M-polynomials We calculatedthese indices for different values of a and b in Table 5 andwe noted that when we increase the values of a and b thenall of the TIs of VC5C7 are increasing with the same orderas shown in Figure 3

(e comparison of 1st Zagreb 2nd Zagreb 2ndmodifiedZagreb and symmetric division deg indices of pent-hep-tagonal nanosheets (Γ2) is computationally computed byusing these M-polynomials We calculated these indices fordifferent values of a and b in Table 6 and we noted that whenwe increase the values of a and b then all of the TIs ofHC5C7 are increasing with the same order as shown inFigure 4

(e comparison of the harmonic index the inverse sumindex and the augmented Zagreb index of pent-heptagonalnanosheets (Γ1) is computationally computed by these

M-polynomials We calculated these indices for differentvalues of a and b in Table 7 and we noted that when weincrease the values of a and b then all of the TIs of VC5C7are increasing with the same order as shown in Figure 5

(e comparison of the harmonic index the inverse sumindex and the augmented Zagreb index of pent-heptagonalnanosheets (Γ2) is computationally computed by theseM-polynomials We calculated these indices for differentvalues of a and b in Table 8 and we noted that when weincrease the values of a and b then all of the TIs of HC5C7are increasing with the same order as shown in Figure 6

In this paper the calculated M-polynomials and enu-merated TIs assist us to recognize the physical characteristicchemical sensitivity and biological animation of the pent-heptagonal nanosheets (Γ1) and (Γ2) (ese consequencesgive us remarkable ascertainment in the field of pharma-ceutical production

However the problem is still open to compute thedifferent TIs (degree and distance based) for variousnanosheets

(i) To compute the nanosheet for other topologicalindices

(ii) To compute the various nanosheets for differenttopological indices

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH

Figure 6 Graphical comparison between H(Γ2) IS(Γ2) and AZI(Γ2) of HC5C7


Data Availability

No data were used to support this study

Conflicts of Interest

(e authors declare that there are no conflicts of interestregarding this publication

References

[1] M Irfan H U Rehman H Almusawa S Rasheed andI A Baloch ldquoM-polynomials and topological indices for linegraphs of chain silicate network and h-naphtalenic nano-tubesrdquo Journal of Mathematics vol 2021 Article ID 555182511 pages 2021

[2] Z Raza and E K Sukaiti ldquoM-polynomial and degree basedtopological indices of some nanostructuresrdquo Symmetryvol 12 no 5 p 831 2020

[3] K Culik Applications of Graph eory to Mathematical Logicand Linguistics eory of Graphs and it Applications CzechAcademy of Sciences Prague Czechia 1963

[4] M Baca J Horvthov M Mokriov A Semanicov-Fenovckovand A Suhnyiov ldquoOn topological indices of a carbonnanotube networkrdquo Canadian Journal of Chemistry vol 93no 10 pp 1157ndash1160 2015

[5] O M Yaghi M OrsquoKeeffe N W Ockwig H K ChaeM Eddaoudi and J Kim ldquoA route to high surface areaporosity and inclusion of large molecules in crystalsrdquo Naturevol 423 no 6941 pp 705ndash714 2003

[6] H Gonzalez-Diaz S Vilar L Santana and E UriarteldquoMedicinal chemistry and bioinformatics-current trends indrugs discovery with networks topological indicesrdquo CurrentTopics in Medicinal Chemistry vol 7 no 10 pp 1015ndash10292007

[7] M R Farahani ldquoConnectivity indices of pent-heptagonalnanotubesrdquo Advance in Materials and Corrosion vol 2pp 33ndash35 2013

[8] S Hayat and M Imran ldquoComputation of certain topologicalindices of nanotubes covered by C 5 and C 7rdquo Journal ofComputational and eoretical Nanoscience vol 12 no 4pp 533ndash541 2015

[9] S Klavar and I Gutman ldquoA comparison of the schultzmolecular topological index with the wiener indexrdquo Journal ofChemical Information and Computer Sciences vol 36 no 5pp 1001ndash1003 1996

[10] X Li and J Zheng ldquoExtremal chemical trees withminimum ormaximum general randic indexrdquo MATCH Communicationsin Mathematical and in Computer Chemistry vol 55 no 2pp 381ndash390 2006

[11] A W Bharati Rajan C Grigorious and S Stephen ldquoOncertain topological indices of silicate honeycomb and hex-agonal networksrdquo Journal of Computer and MathematicalSciences vol 3 no 5 pp 498ndash556 2012

[12] A Graovac M Ghorbani and M A Hosseinzadeh ldquoCom-puting fifth geometric-arithmetic index for nanostar den-drimersrdquo Journal of Mathematical Nanoscience vol 1no 1ndash2 pp 33ndash42 2011

[13] B Furtula A Graovac and D Vukicevic ldquoAugmented zagrebindexrdquo Journal of Mathematical Chemistry vol 48 no 2pp 370ndash380 2010

[14] K C Das and N Trinajstic ldquoRelationship between the ec-centric connectivity index and zagreb indicesrdquo Computers amp

Mathematics with Applications vol 62 no 4 pp 1758ndash17642011

[15] C K Gupta V Lokesha S B Shwetha and P S Ranjini ldquoOnthe symmetric division deg index of graphrdquo Southeast AsianBulletin of Mathematics vol 40 no 1 2016

[16] A R Matamala and E Estrada ldquoGeneralised topologicalindices optimisation methodology and physico-chemicalinterpretationrdquo Chemical Physics Letters vol 410 no 4ndash6pp 343ndash347 2005

[17] A Rani and U Ali ldquoDegree-Based topological indices ofpolysaccharides amylose and blue starch-iodine complexrdquoJournal of Chemistry vol 2021 Article ID 6652014 10 pages2021

[18] G Abbas A Rani M Salman T Noreen and U Ali ldquoHosoyaproperties of the commuting graph associated with the groupof symmetriesrdquo Main Group Metal Chemistry vol 44 no 1pp 173ndash184 2021

[19] J Devillers D Domine C Guillon S Bintein andW Karcher ldquoPrediction of partition coefficients (log p oct)using autocorrelation descriptorsrdquo SAR and QSAR in Envi-ronmental Research vol 7 no 1ndash4 pp 151ndash172 1997

[20] I Gutman and O E Polansky Mathematical Concepts inOrganic Chemistry Springer Science amp Business MediaBerlin Germany 2012

[21] G Plya ldquoAlgebraische berechnung der anzahl der isomereneiniger organischer verbindungenrdquo Zeitschrift fr Kristallog-raphie-Crystalline Materials vol 93 no 1 pp 415ndash443 1936

[22] H Wiener ldquoStructural determination of paraffin boilingpointsrdquo Journal of the American Chemical Society vol 69pp 17ndash20 1947

[23] S Akhter andM Imran ldquoOnmolecular topological propertiesof benzenoid structuresrdquo Canadian Journal of Chemistryvol 94 no 8 pp 687ndash698 2016

[24] M Baca J Horvthov M Mokriov and A Suhnyiov ldquoOntopological indices of fullerenesrdquo Applied Mathematics andComputation vol 251 pp 154ndash161 2015

[25] FM Brckler T Dolic A Graovac and I Gutman ldquoOn a classof distance-based molecular structure descriptorsrdquo ChemicalPhysics Letters vol 503 no 4ndash6 pp 336ndash338 2011

[26] B Furtula and I Gutman ldquoA forgotten topological indexrdquoJournal of Mathematical Chemistry vol 53 no 4 pp 1184ndash1190 2015

[27] M Javaid M U Rehman and J Cao ldquoTopological indices ofrhombus type silicate and oxide networksrdquo Canadian Journalof Chemistry vol 95 no 2 pp 134ndash143 2017

[28] A Vasilyev ldquoUpper and lower bounds of symmetric divisiondeg indexrdquo Iranian Journal of Mathematical Chemistry vol 5no 2 pp 91ndash98 2014

[29] M Javaid J B Liu M A Rehman and S Wang ldquoOn thecertain topological indices of titania nanotube TiO2 [m n]rdquoZeitschrift fur Naturforschung A vol 72 no 7 pp 647ndash6542017

[30] H M Awais M Jamal andM Javaid ldquoTopological propertiesof metal-organic frameworksrdquo Main Group Metal Chemistryvol 43 no 1 pp 67ndash76 2020

[31] I Gutman and N Trinajstic ldquoGraph theory and molecularorbitals total f-electron energy of alternant hydrocarbonsrdquoChemical Physics Letters vol 17 no 4 pp 535ndash538 1972

[32] D Amic D Belo B Lucic S Nikolic and N Trinajstic ldquo(evertex-connectivity index revisitedrdquo Journal of ChemicalInformation and Computer Sciences vol 38 no 5 pp 819ndash822 1998


[33] V Lokesha and T Deepika ldquoSymmetric division deg index oftricyclic and tetracyclic graphsrdquo International journal ofScience and Engineering Research vol 7 pp 53ndash55 2016

[34] L Zhong ldquo(e harmonic index for graphsrdquo Applied Math-ematics Letters vol 25 no 3 pp 561ndash566 2012

[35] K Pattabiraman ldquoInverse sum indeg index of graphsrdquo AKCEInternational Journal of Graphs and Combinatorics vol 15no 2 pp 155ndash167 2018

[36] Y Shi M Dehmer X Li and I Gutman Graph PolynomialsCRC Press Boca Raton FL USA 2016

[37] H Yang A Q Baig W Khalid M R Farahani and X ZhangldquoM-polynomial and topological indices of benzene ringembedded in P-type surface networkrdquo Journal of Chemistryvol 2019 Article ID 7297253 9 pages 2019

[38] A J M Khalaf S Hussain D Afzal F Afzal and AMaqboolldquoM-polynomial and topological indices of book graphrdquoJournal of Discrete Mathematical Sciences and Cryptographyvol 23 no 6 pp 1217ndash1237 2020

[39] S Mondal M Imran N De and A Pal ldquoNeighborhoodM-polynomial of titanium compoundsrdquo Arabian Journal ofChemistry vol 14 no 8 Article ID 103244 2021


enhanced diffusivity reduced density higher electrical re-sistance reduced elastic modulus lower thermal conduc-tivity increase specific heat higher thermal expansioncoefficient increased oscillator and strength luminescenceblue shift absorption and superior soft magnetic charac-teristics in comparison to the conventional bulk materialFurthermore these characteristics are being briefly exam-ined to discover new tools (e interesting branch ofnanotechnology has a vast range of different types of ap-plications (e use of nanomaterials has manufacturedtransistors having low speed and laser having low thresholdcurrent (ese are utilized in satellite receivers having lownoise amplification as a source for fiber optics communi-cations and compact disk player systems Constructive toolsof nanostructures contain UV-resistant wood coating andself-cleaning glass On the other hand nanoscale tools arebeing utilized in the field of medicine for the prevention andtreatment of diseases diagnosis and in magnetic resonanceimaging drug delivery system radioactive tracers etc [6](e importance of nanomaterials is rising nowadays Manyother types of tools may be possible with the peculiar andnovel characteristics of nanomaterials [7 8]

(erefore TIs are useful to define molecular nano-materials Nanostructures that have a scale of less than100 nm contain nanosheets nanotubes and nanoparticlesNanosheets (two-dimensional nanomaterials) have a sharpedge and large surface area that cause them to play a vitalrole in various types of tools such as catalysis energy storagebioelectronics and optoelectronics [9 10] Silicone bor-ophene and graphene are specific nanosheets Due to therare optical electrical mechanical and structural charac-teristics graphene nanosheets received great recognitionfrom industrial and academic researchers [11] (e differentproperties of the C5C7 nanosheet have become the mostadvanced field in research A C5C7 structure is developed byalternating C5 andC7 [7] In 2009 Graovac et al studied theGA index of TUC4C8 (S) nanotubes In 2011 Graovac et al[12] studied the fifth geometric arithmetic index for nanostardendrimers and Asadpour et al calculated Zagreb Randi c

and ABC indices of TUC4C8 (R) and TUC4C8 (S) V-Phe-nylenic nanotorus and nanotubes In 2014 Al-Fozan et alsolved Szeged index of H-naphthalene nanosheets (2n 2m)and C4C8 (S) Loghman and Ashrafi studied the Padma-karndashIvan (PI) index of TUC4C8 (S) nanotubes For furtherdiscussion see [13ndash15]

However the combination of three fields such asmathematics physical chemistry and information scienceslead to a modern field known as cheminformatics [16ndash18] Itdevelops a mathematical relationship between a propertyand structural attributes of different types of chemicals calledby quantitative-structuresrsquo activity and qualitative-struc-turesrsquo property relationship that are utilized to forecast theorganic sciences and biological properties in the field ofengineering and technology [19 20] Graph theory hasoriginated a significant usage in the field of mathematicalchemistry that is famous as chemical graph theory

Polya gave the idea for counting polynomials in the fieldof chemistry [21] and Wiener introduced the concept of TIrelated to the paraffinrsquos boiling point [22] Computing the

TIs is a new field of chemical graphs that associates manyphysiochemical characteristics of the fundamental chemicalcompounds [23ndash27]

2 Preliminaries

A molecular structure Γ (V(Γ) E(Γ)) V(Γ) s1 s21113864

s3 sn andE(Γ) are nodes (vertices) and edge set of Γ|V(Γ)| v and |E(Γ)| e is the order and size of Γ In aconnected and simple molecular graph a path is representedwithin two vertices and the distance between the two verticess and t is mentioned as φ(s t) in a graph Γ see [28ndash30] Inthis paper a graph is connected and simple having nomultiple edges or loops

1st and 2nd Zagreb indices let Γ be a molecularstructure then its 1st and 2nd Zagreb indices [31] are

M1(Γ) 1113944sisinV(Γ)

[φ(s)]2

1113944stisinE(Γ)

[φ(s) + φ(t)]

M2(Γ) 1113944stisinE(Γ)

[φ(s) times φ(t)](1)

General Randi cindex if R is the real number α isin R

and Γ is a molecular structure the general Randi cindex

[32] is

Rα(Γ) 1113944stisinE(Γ)

[φ(s)φ(t)]α (2)

Symmetric division deg index for a molecular structureΓ the symmetric division deg index [33] is

SDD(Γ) 1113944stisinE(Γ)

min(φ(s)φ(t))

max(φ(s)φ(t))+max(φ(s)φ(t))

min(φ(s) φ(t))1113890 1113891

(3)

Harmonic index for a molecular structure Γ the har-monic index [34] is

H(Γ) 1113944stisinE(Γ)

2φ(s) + φ(t)

(4)

Inverse sum index for a molecular structure Γ the in-verse sum index [35] is

IS(Γ) 1113944stisinE(Γ)

φ(s)φ(t)

φ(s) + φ(t) (5)

Augmented Zagreb index for a molecular structure Γthe augmented Zagreb index [13] is

AZI(Γ) 1113944stisinE(Γ)

φ(s) times φ(t)

φ(s) + φ(t) minus 21113890 1113891

3

(6)

A graph polynomial is a graph invariant whose valuesare polynomials So all these invariants are discussed inalgebraic graph theory [36] Among such types of alge-braic polynomials the M-polynomial defined in 2015shows the same role in finding the much closed form ofvarious degree-based TIs that correlate different types ofchemical properties of the various materials under




M(Γ μ ]) 1113944ile j(Γ)

mijΓμi]j

1113872 1113873(7)





V1 s isin V Γ1( 1113857|φ(s) 21113864 1113865

V2 s isin V Γ1( 1113857|φ(s) 31113864 1113865(8)


E22 st isin Γ1( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ1( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ1( 1113857|φ(s) 3φ(t) 31113864 1113865

(9)


E Γ1( 11138571113868111386811138681113868

1113868111386811138681113868 E11113868111386811138681113868

1113868111386811138681113868 + E21113868111386811138681113868

1113868111386811138681113868 + E31113868111386811138681113868

1113868111386811138681113868 16 + 48 + 144 208 (10)





V1 s isin V Γ2( 1113857|φ(u) 21113864 1113865

V2 s isin V Γ2( 1113857|φ(u) 31113864 1113865(11)


E22 st isin Γ2( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ2( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ2( 1113857|φ(s) 3φ(t) 31113864 1113865

(12)

4 Main Results





μDα] )(M(Γ μ ]))|μ1]






](M(Γ μ ]))|μ1]







M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 1113944sle t

Est Γ1( 1113857μs]t1113960 1113961

11139442le 2

E22 Γ1( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ1( 1113857μ2]31113960 1113961

+ 11139443le 3

E331113960 1113872Γ11113857μ3]31113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



(a) M1(Γ1) 144ab minus 12a + 10b





(f) SSD(Γ1) 48ab + 43a + 293b minus 43


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3





D]f(μ ]) 2(2a + 2b + 4)μ2] + 3(8a + 10b minus 8)μ2]2 +






23)μ3]3D]Tμ(f(μ ])) D](Tμ(f(μ ]))) 2(a + b + c)


43μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) 2(a + b + c)


4)μ2]3

DcμD

c] (4)c(2a + 2b + 4)μ] +(6)c(8a + 10b minus 8)μ]2 +


TcμT

c] (2a + 2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)










cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 2b + 4) + (6)(8a +


Tcμ(T

c](f(μ ])))|μ]1 (2a + 2b + 4)(4)c + (8a +


Consequently



12a + 10b






c])(f(μ ]))

|μ]1 (4)c (2a + 2b + 4) + (6)c(8a + 10b minus 8) +



2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)cμ2]3 + (24ab





M(Γ μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3




135625


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



J(f(μ ])) (2a + 2b + 4)μ4 + (8a + 10b minus 8)μ5 +




J(Dμ(D](f(μ ])))) (8a + 8b + 16)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 8b + 16)μ4 + (48a +



D3μ(D3

](f(μ ]))) (4)3(2a + 2b + 4)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 2b + 4)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ])))) |(4)3(2a +2b +4)2μ2 + (6)3



Tμ(J(f(μ ])))|μ]1 1330

a +76

b + 4ab +115


a + 2b +25

T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 2b + 4) + 8(8a + 10b minus 8)

+72964

1113874 1113875(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(20)



H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)


36ab minus175

a + 2b +25

(22)


AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)



M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3







(f) SSD(Γ2) 48ab + 43a + 7b + 463


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3





D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)



TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)


10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)


μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References













































M(Γ μ ]) 1113944ile j(Γ)

mijΓμi]j

1113872 1113873(7)





V1 s isin V Γ1( 1113857|φ(s) 21113864 1113865

V2 s isin V Γ1( 1113857|φ(s) 31113864 1113865(8)


E22 st isin Γ1( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ1( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ1( 1113857|φ(s) 3φ(t) 31113864 1113865

(9)


E Γ1( 11138571113868111386811138681113868

1113868111386811138681113868 E11113868111386811138681113868

1113868111386811138681113868 + E21113868111386811138681113868

1113868111386811138681113868 + E31113868111386811138681113868

1113868111386811138681113868 16 + 48 + 144 208 (10)





V1 s isin V Γ2( 1113857|φ(u) 21113864 1113865

V2 s isin V Γ2( 1113857|φ(u) 31113864 1113865(11)


E22 st isin Γ2( 1113857|φ(s) 2φ(t) 21113864 1113865

E23 st isin Γ2( 1113857|φ(s) 2φ(t) 31113864 1113865

E33 st isin Γ2( 1113857|φ(s) 3φ(t) 31113864 1113865

(12)

4 Main Results





μDα] )(M(Γ μ ]))|μ1]






](M(Γ μ ]))|μ1]







M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 1113944sle t

Est Γ1( 1113857μs]t1113960 1113961

11139442le 2

E22 Γ1( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ1( 1113857μ2]31113960 1113961

+ 11139443le 3

E331113960 1113872Γ11113857μ3]31113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



(a) M1(Γ1) 144ab minus 12a + 10b





(f) SSD(Γ1) 48ab + 43a + 293b minus 43


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3





D]f(μ ]) 2(2a + 2b + 4)μ2] + 3(8a + 10b minus 8)μ2]2 +






23)μ3]3D]Tμ(f(μ ])) D](Tμ(f(μ ]))) 2(a + b + c)


43μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) 2(a + b + c)


4)μ2]3

DcμD

c] (4)c(2a + 2b + 4)μ] +(6)c(8a + 10b minus 8)μ]2 +


TcμT

c] (2a + 2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)










cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 2b + 4) + (6)(8a +


Tcμ(T

c](f(μ ])))|μ]1 (2a + 2b + 4)(4)c + (8a +


Consequently



12a + 10b






c])(f(μ ]))

|μ]1 (4)c (2a + 2b + 4) + (6)c(8a + 10b minus 8) +



2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)cμ2]3 + (24ab





M(Γ μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3




135625


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



J(f(μ ])) (2a + 2b + 4)μ4 + (8a + 10b minus 8)μ5 +




J(Dμ(D](f(μ ])))) (8a + 8b + 16)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 8b + 16)μ4 + (48a +



D3μ(D3

](f(μ ]))) (4)3(2a + 2b + 4)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 2b + 4)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ])))) |(4)3(2a +2b +4)2μ2 + (6)3



Tμ(J(f(μ ])))|μ]1 1330

a +76

b + 4ab +115


a + 2b +25

T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 2b + 4) + 8(8a + 10b minus 8)

+72964

1113874 1113875(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(20)



H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)


36ab minus175

a + 2b +25

(22)


AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)



M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3







(f) SSD(Γ2) 48ab + 43a + 7b + 463


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3





D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)



TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)


10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)


μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References












































M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 1113944sle t

Est Γ1( 1113857μs]t1113960 1113961

11139442le 2

E22 Γ1( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ1( 1113857μ2]31113960 1113961

+ 11139443le 3

E331113960 1113872Γ11113857μ3]31113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



M Γ1 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



(a) M1(Γ1) 144ab minus 12a + 10b





(f) SSD(Γ1) 48ab + 43a + 293b minus 43


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3





D]f(μ ]) 2(2a + 2b + 4)μ2] + 3(8a + 10b minus 8)μ2]2 +






23)μ3]3D]Tμ(f(μ ])) D](Tμ(f(μ ]))) 2(a + b + c)


43μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) 2(a + b + c)


4)μ2]3

DcμD

c] (4)c(2a + 2b + 4)μ] +(6)c(8a + 10b minus 8)μ]2 +


TcμT

c] (2a + 2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)










cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 2b + 4) + (6)(8a +


Tcμ(T

c](f(μ ])))|μ]1 (2a + 2b + 4)(4)c + (8a +


Consequently



12a + 10b






c])(f(μ ]))

|μ]1 (4)c (2a + 2b + 4) + (6)c(8a + 10b minus 8) +



2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)cμ2]3 + (24ab





M(Γ μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3




135625


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



J(f(μ ])) (2a + 2b + 4)μ4 + (8a + 10b minus 8)μ5 +




J(Dμ(D](f(μ ])))) (8a + 8b + 16)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 8b + 16)μ4 + (48a +



D3μ(D3

](f(μ ]))) (4)3(2a + 2b + 4)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 2b + 4)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ])))) |(4)3(2a +2b +4)2μ2 + (6)3



Tμ(J(f(μ ])))|μ]1 1330

a +76

b + 4ab +115


a + 2b +25

T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 2b + 4) + 8(8a + 10b minus 8)

+72964

1113874 1113875(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(20)



H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)


36ab minus175

a + 2b +25

(22)


AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)



M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3







(f) SSD(Γ2) 48ab + 43a + 7b + 463


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3





D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)



TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)


10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)


μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References












































cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 2b + 4) + (6)(8a +


Tcμ(T

c](f(μ ])))|μ]1 (2a + 2b + 4)(4)c + (8a +


Consequently



12a + 10b






c])(f(μ ]))

|μ]1 (4)c (2a + 2b + 4) + (6)c(8a + 10b minus 8) +



2b + 4)(4)cμ2]2 + (8a + 10b minus 8)(6)cμ2]3 + (24ab





M(Γ μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3




135625


f(μ ]) (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3



J(f(μ ])) (2a + 2b + 4)μ4 + (8a + 10b minus 8)μ5 +




J(Dμ(D](f(μ ])))) (8a + 8b + 16)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 8b + 16)μ4 + (48a +



D3μ(D3

](f(μ ]))) (4)3(2a + 2b + 4)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 2b + 4)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ])))) |(4)3(2a +2b +4)2μ2 + (6)3



Tμ(J(f(μ ])))|μ]1 1330

a +76

b + 4ab +115


a + 2b +25

T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 2b + 4) + 8(8a + 10b minus 8)

+72964

1113874 1113875(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(20)



H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)


36ab minus175

a + 2b +25

(22)


AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)



M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3







(f) SSD(Γ2) 48ab + 43a + 7b + 463


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3





D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)



TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)


10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)


μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































H Γ1( 1113857 2Tμ(J(f(μ ])))|μ]1

21330

a +76

b + 4ab +115

1113874 1113875

1315

a +73

b + 8ab +215

(21)

(ii) Inverse index

IS Γ1( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

(2a + 2b + 4) +15

(48a + 60b minus 48)


36ab minus175

a + 2b +25

(22)


AZI Γ1( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

42

1113874 11138753(2a + 2b + 4) +

63

1113874 11138753(8a + 10b minus 8)

+94

1113874 11138753(24ab minus 10a minus 8b + 4)

273375ab minus 3390625a + 4875b + 135625

(23)


M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)



M Γ2 μ ]( 1113857 1113944sle t

Est(Γ)μs]t

1113960 1113961

11139442le 2

E22 Γ2( 1113857μ2]21113960 1113961 + 11139442le 3

E23 Γ2( 1113857μ2]31113960 1113961

+ 11139443le 3

E33 Γ2( 1113857μ3]31113960 1113961

E11113868111386811138681113868

1113868111386811138681113868μ2]2 + E21113868111386811138681113868

1113868111386811138681113868μ2]3 + E31113868111386811138681113868

1113868111386811138681113868μ3]3

(2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



M Γ2 μ ]( 1113857 (2a + 2b + 4)μ2]2 +(8a + 10b minus 8)μ2]3







(f) SSD(Γ2) 48ab + 43a + 7b + 463


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3





D]f(μ ]) 2(2a + 3b + 2)μ2] + 3(8a + 6b minus 4)μ2]2 +




Tμ(f(μ ])) (a + (32)b + 1)μ2]2 + (4a + 3b minus 2)



TμT](f(μ ])) Tμ(T](f(μ ]))) 14(2a + 3b + 2)


10)μ3]3

D]Tμ(f(μ ])) D](Tμ(f(μ ]))) (2a + 3b + 2)


103μ3]2)DμT](f(μ ])) Dμ(T](f(μ ]))) (2a + 3b + 2)


μ2]3


DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































DcμD

c] (4)c(2a + 3b + 2)μ] + (6)c(8a + 6b minus 4)μ]2 +


TcμT

c] (2a + 3b + 2)(4)cμ2]2 + (8a + 6b minus 4)(6)c





cμ(D

c](f(μ ])))|μ]1 (4)c(2a + 3b + 2)+


Tcμ(T

c](f(μ ])))|μ]1 (2a + 3b + 2)(4)c + (8a +


Consequently


|μ]1 Dμ(f(μ ]))|μ]1 + D] (f(μ ]))|μ]1




(f(μ ]))|μ]1 Tμ (T](f(μ ])))|μ]1 83ab +


cμD

c])(f(μ ]))

|μ]1 (4)c(2a + 3b + 2) + (6)c (8a + 6b minus 4) +



c])

(f (μ ]))|μ]1 (2a + 3b + 2)(4)c + (8a + 6b minus





M Γ2 μ ]( 1113857 (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3




97906


f(μ ]) (2a + 3b + 2)μ2]2 +(8a + 6b minus 4)μ2]3



J(f(μ ])) (2a + 3b + 2)μ4 + (8a + 6b minus 4)μ5 +




J(Dμ(D](f(μ ])))) (8a + 12b + 8)μ2 + (48a +


Q2(J(Dμ(D](f(μ ]))))) (8a + 12b + 8)μ4 + (48a +


Tμ(Q2(J(Dμ(D](f(μ ])))))) (2a + 3b + 2)μ4 +


D3μ(D3

](f(μ ]))) (4)3(2a + 3b + 2)μ] + (6)3(8a +


J(D3μD3

](f(μ ]))) (4)3(2a + 3b + 2)μ2 + (6)3(8a +


T3μ(J(D3

μD3](f(μ ])))) 8(2a + 3b + 2)μ2 + 8(8a +



Tμ(J(f(μ ])))|μ]1 14

(2a + 3b + 2) +15

(8a + 6b minus 4) +16


1330

a +1920

+ 4ab +4130

Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1 14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


15



T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1 8(2a + 3b + 2) + 8(8a + 6b minus 4) +

94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(31)



a b M1(Γ1) M2(Γ1) MM1(Γ1) SDD(Γ1)


200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

M1M2

MM1SSD



a b M1(Γ2) M2(Γ2) MM1(Γ2) SDD(Γ2)



H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































H Γ2( 1113857 2Tμ(J(f(μ ])))|μ]1

214

(2a + 3b + 2) +15

(8a + 6b minus 4) +16

(24ab minus 10a minus 6b + 10)1113876 1113877

8ab +1315

a +1910

b minus113

(32)

(ii) Inverse index

200000

180000

160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

SSDMM1

M2M1






IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































IS Γ2( 1113857 Tμ Q2 J Dμ D](f(μ ]))( 11138571113872 11138731113872 11138731113872 1113873|μ]1

14

(8a + 12b + 8) +15

(48a + 36b minus 24) +16


36ab minus175

a minus95

b +615

(33)


160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH






AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































AZI Γ2( 1113857 T3μ J D

3μD

3](f(μ ]))1113872 11138731113872 1113873|μ]1

(2a + 3b + 2) + 8(8a + 6b minus 4) +94

1113874 11138753(24ab minus 10a minus 6b + 10)


2733744ab minus 3390625a + 36564b + 97906

(34)

5 Conclusion











160000

140000

120000

100000

80000

60000

40000

20000

01 2 3 4

AZIISH



Data Availability




References











































Data Availability




References



















































Documents

TopologicalIndicesofPent-Heptagonal NanosheetsviaM-Polynomials