noveldegree-basedtopologicaldescriptorsof carbonnanotubes

Research ArticleNovel Degree-Based Topological Descriptors ofCarbon Nanotubes

M C Shanmukha1 A Usha2 M K Siddiqui3 K C Shilpa4 and A Asare-Tuah 5

1Department of Mathematics Jain Institute of Technology Davanagere-577003 Karnataka India2Department of Mathematics Alliance School of Applied Mathematics Alliance University Bangalore-562106 Karnataka India3Department of Mathematics Comsats University Islamabad Lahore Campus Lahore Pakistan4Department of Computer Science and Engineering Bapuji Institute of Engineering and Technology Davanagere-577004Karnataka India5Department of Mathematics University of Ghana Legon Ghana

Correspondence should be addressed to A Asare-Tuah aasare-tuahugedugh

Received 27 July 2021 Accepted 27 August 2021 Published 8 September 2021

Academic Editor Ajaya Kumar Singh

Copyright copy 2021 M C Shanmukha 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 most significant tool of mathematical chemistry is the numerical descriptor called topological index Topological indices areextensively used in modelling of chemical compounds to analyse the studies on quantitative structure activitypropertytoxicityrelationships and combinatorial library virtual screening In this work an attempt is made in defining three novel descriptorsnamely neighborhood geometric-harmonic harmonic-geometric and neighborhood harmonic-geometric indices Also theaforementioned three indices along with the geometric-harmonic index are tested for physicochemical properties of octaneisomers using linear regression models and computed for some carbon nanotubes

1 Introduction

)e applications of graph theory are diversified in every fieldbut chemistry is the major area of the implementation ofgraph theory In chemical graph theory topological indexplays a vital role which facilitates the chemists with a treasureof data that correlate with the structure of the chemicalcompound )e topological index is a numerical descriptordefines the graph topology of the molecule and predicts anextensive range of molecular properties 5[1ndash6]

From the last two decades topological indices (TIs) areidentified and used in pharmacological medicine bio-inorganic chemistry toxicity and theoretical chemistry andare also used for correlation analysis [7ndash11]

Topological descriptors are frequently used in the dis-covery of drugs as they have rich datasets that give highpredictive values )ese descriptors give the informationdepending on the arrangement of atoms and their bonds of achemical compound )ey are studied for chemical

compounds where generally the hydrogen atoms aresuppressed )e originality of QSARQSPR models dependson physicochemical properties for chemical compoundswith high degree of precision )ese models depend onvarious factors such as selecting the suitable compoundssuitable descriptors and suitable algorithms or tools used inmodel development [12] )e QSARQSPR analysis is basedon the data obtained by the numerical descriptors )esedata are used to verify whether the compound under thestudy is suitable for drug making as the TIs provide com-putational data about the compound Considering the in-formation of the compound QSARQSPRQSTR analysesare carried out

)e TIs have increasing popularity in the field of re-search as they involve only computation without performingany physical experiment Recent years have proved con-siderable attention in TIs as the effects of an atomic type andgroup efforts are considered in QSARQSPR modelling[13ndash15] Distance-based TIs are used in QSAR analysis

HindawiJournal of ChemistryVolume 2021 Article ID 3734185 15 pageshttpsdoiorg10115520213734185

while chirality descriptors are introduced based on molec-ular graphs [16]

Alkanes are acyclic saturated hydrocarbons in whichcarbons and hydrogens are arranged in a tree-like structure)e main use of alkanes is found in crude oil such as pe-troleum cooking gas pesticides and drug synthesis )ecompounds that contain absolutely the same number ofatoms but their arrangement differs are termed as isomers Astudy is carried out for eighteen octane isomers (referFigure 1)

A structure whose size is between the microscopic andmolecular structure is referred to as a nanostructure )ereare different types of nanostructures namely nanocagesnanocomposites nanoparticles nanofabrics etc In the re-cent years nanostructures have attracted a lot of researchersin the areas of biology chemistry and medicines Topo-logical indices of nanostructures can be studied from[17ndash24] )e nanostructures made of carbons with cylin-drical shape are carbon nanotubes (CNTs) )ey have asimilar structure to that of a fullerene and graphene excepttheir cylindrical shape )e shape of fullerene is as that of afootball or basketball design where hexagons are connected

In 1991 Iijima [25] used carbon nanotubes that haveattracted many researchers in nanoscience and nanotech-nology worldwide As they have exotic properties they arewidely used in both research and applications Nanotubeshave a distinctive structure with remarkable mechanical andelectrical properties In case of carbon nanotubes thehexagons are surrounded by squares and each of thesepatterns is linearly arranged Carbon nanotubes reveal ex-ceptional electrical conductivity and possess wonderfultensile strength and thermal conductivity as they havenanostructures in which the carbon atoms are stronglyconnected

Carbon nanotubes have applications in orthopaedicimplants especially in total hip replacement and othertreatments pertaining to bone-related ailments )ey areused as a grouting agent placed between the prosthesis andthe bone as a part of their therapeutic use)e CNTs are usedin biomedical fields because of their structural stiffness andeffective optical absorption from UV to IR Also they can bealtered chemically which are expected to be useful in manyfields of technology such as electronics composited mate-rials and carbon fibres )ey have incredible applications inthe field of materials science [26] When the hexagonallattice is rolled in different directions it looks like single-wallcarbon nanotubes have spiral shape and translationalsymmetry along the tube axis It has rotational symmetryalong its own axis Even though nanotubes have favourableapplications in a variety of fields their large-scale productionhas been restricted )e main constraint that obstructs theiruse lies in difficulty in controlling their structure impuritiesand poor process ability To enhance their usage they havegrabbed the attention especially in the formation of com-posites with polymers

)ere are two types of configurations in the arrangementof nanotubes namely zigzag and armchair In the zigzagconfiguration the hexagons are placed one below the otherlinearly whereas in the armchair configuration they are

placed next to each other )is gives two different types ofconfigurations with different terminologies discussed nowTo explain the structure of a nanotube that is infinitely longwe imagine it to be cut open by a parallel axis and placed on aplane )en the atoms and bonds coincide with an imag-inary graphene sheet )e length of the two atoms on op-posite edges of the strip corresponds to circumference of thecylindrical graphene sheet [27ndash29]

)e main objectives of this work are as follows

To define novel indicesTo discuss the physical and chemical applicability ofoctane isomers using regression modelsTo compute defined indices for carbon nanotubes suchas C4C8(S) C4C8(R) and H-naphthalenic nanosheets

Let G (V E) be a graph with a vertex set V(G) and anedge set E(G) such that |V(G)| n and |E(G)| m Forstandard graph terminologies and notations refer to[30 31]where (u v) is an element of E(G) du represents thedegree of the vertex u and Su represents the neighborhooddegree of the vertex u

Definition 1 Recently Usha et al [32] defined the geo-metric-harmonic (GH) index inspired by Vukicevic andFurtula [33] in designing the GA index

GH(G) 1113944du + dv( 1113857

du middot dv

1113968

2 (1)

Motivated by the above work in this paper an attempt ismade to define three novel indices based on degree andneighborhood degree namely harmonic-geometric (HG)neighborhood geometric-harmonic (NGH) and neighbor-hood harmonic-geometric (NHG) indices )ey are definedas follows

HG(G) 11139442

du + dv( 1113857du middot dv

1113968

NGH(G) 1113944Su + Sv( 1113857

Su middot Sv

1113968

2

NHG(G) 11139442

Su + Sv( 1113857Su middot Sv

1113968

(2)

11 Chemical Applicability of GH NGH HG and NHGIndices In this section a linear regression model of fourphysical properties is presented for GH NGH HG andNHG indices )e physical properties such as entropy(S)acentric factor (AF) enthalpy of vaporization (HVAP) andstandard enthalpy of vaporization (DHVAP) of octaneisomers have shown good correlation with the indicesconsidered in the study )e GH HG NGH and NHGindices are tested for the octane isomersrsquo database availableat httpswwwmoleculardescriptorseudatasethtm )eGH HG NGH and NHG indices are computed and tab-ulated in columns 6 7 8 and 9 of Table 1

2 Journal of Chemistry

(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)

Figure 1 (a) n-Octane (b) 2-methylheptane (c) 3-methylheptane (d) 4-methylheptane (e) 3-ethylhexane (f ) 22-dimethylhexane (g) 23-dimethylhexane (h) 24-dimethylhexane (i) 25-dimethylhexane (j) 33-dimethylhexane (k) 34-dimethylhexane (l) 3-ethyl-2-methyl-pentane (m) 3-ethyl-3-methylpentane (n) 223-trimethylpentane (o) 224-trimethylpentane (p) 233-trimethylpentane (q) 234-tri-methylpentane and (r) 2233-trimethylbutane

Table 1 Experimental values of S AF HVAP and DHVAP and the corresponding values of the GH index HG index NGH index andNHG index of octane isomers

Alkane S AF HVAP DHVAP GH NGH HG NHGn-Octane 111700 0398 73190 9915 24423 84496 2193 06792-Methylheptane 109800 0378 70300 9484 27173 98746 1962 05733-Methylheptane 111300 0371 71300 9521 27954 107475 2058 05684-Methylheptane 109300 0372 70910 9483 27954 113356 2058 06003-Ethylhexane 109400 0362 71700 9476 28735 131941 2154 047622-Dimethylhexane 103400 0339 67700 8915 33607 122970 1689 047123-Dimethylhexane 108000 0348 70200 9272 31637 110622 1862 049824-Dimethylhexane 107000 0344 68500 9029 30885 123236 1827 047425-Dimethylhexane 105700 0357 68600 9051 32248 133242 1731 046933-Dimethylhexane 104700 0323 68500 8973 35213 151398 1829 044934-Dimethylhexane 106600 0340 70200 9316 32418 117701 1958 05782-Methyl-3-ethylpentane 106100 0332 69700 9209 32418 176111 1958 03423-Methyl-3-ethylpentane 101500 0307 69300 9081 36820 149222 1968 0377223-Trimethylpentane 101300 0301 67300 8826 38834 164366 1606 0338224-Trimethylpentane 104100 0305 64870 8402 36537 155874 1459 0358233-Trimethylpentane 102100 0293 68100 8897 39659 141657 1649 0462234-Trimethylpentane 102400 0317 68370 9014 35321 168797 1666 03902233-Trimethylbutane 93060 0255 66200 8410 46000 223620 1263 0227

Journal of Chemistry 3

Using the method of least squares the linear regressionmodels for S AF HVAP and DHVAP are fitted using thedata of Table 1

)e fitted models for the GH index are

S 133078(plusmn 182) minus 0833(plusmn 0054)GH (3)

acentric factor 0557(plusmn 0009) minus 0007(plusmn 0000)GH

(4)

HVAP 79613(plusmn 1878) minus 0315(plusmn 0056)GH (5)

DHVAP 11273(plusmn 0285) minus 0065(plusmn 0008)GH (6)

)e fitted models for the HG index are

S 76608(plusmn 4486) + 15766(plusmn 2435)HG (7)

acentric factor 0114(plusmn 0037) + 0121(plusmn 0020)HG

(8)

HVAP 54960(plusmn 1356) + 7773(plusmn 0736)HG (9)

DHVAP 6428(plusmn 0249) + 1477(plusmn 0135)HG (10)

)e fitted models for the NGH index are

S 12177(plusmn 235) minus 0119(plusmn 0017)NGH (11)

acentric factor 0465(plusmn 0018) minus 0001(plusmn 0000)NGH

(12)

HVAP 75007(plusmn 1552) minus 0043(plusmn 0011)NGH (13)

DHVAP 10363(plusmn 0257) minus 0009(plusmn 0002)NGH (14)

)e fitted models for the NHG index are

S 89524(plusmn 2505) + 3435(plusmn 5271)NHG (15)

acentric factor 0207(plusmn 0018) + 0277(plusmn 0038)NHG

(16)

HVAP 62667(plusmn 1352) + 14044(plusmn 2846)NHG (17)

DHVAP 7793(plusmn 0219) + 2882(plusmn 0460)NHG (18)

Note in equations (3)ndash(18) the errors of the regressioncoefficients are represented within brackets

Tables 2ndash5 and Figures 2ndash5 show the correlation coef-ficient and residual standard error for the regression modelsof four physical properties with GH HG NGH and NHGindices

From Table 2 and Figure 2 it is obvious that the GHindex highly correlates with the acentric factor and thecorrelation coefficient |r| 0987 Also the GH index hasgood correlation coefficient |r| 0968 with entropy |r|

0815 with HVAP and |r| 0885 with DHVAPFrom Table 3 and Figure 3 it is noticed that the HG

index highly correlates with DHVAP and the correlationcoefficient r 0939 Also the HG index has good corre-lation coefficient r 085 with entropy r 0833 with theacentric factor and r 0935 with HVAP

From Table 4 and Figure 4 it is clear that the NGH indexhighly correlates with the acentric factor and the correlationcoefficient |r| 0877 Also the NGH index has good cor-relation coefficient |r| 0873 with entropy |r| 0695 withHVAP and |r| 0778 with DHVAP

From Table 5 and Figure 5 it is clear that the NHG indexhighly correlates with the acentric factor and the correlationcoefficient r 0877 Also the NHG index has good cor-relation coefficient r 0852 with entropy r 0777 withHVAP and r 0843 with DHVAP

2 GH NGH HG and NHG Indices of C4C8(S)C4C8(R) and H-Naphthalenic Nanosheets

21 Results for the C4C8(S) Nanosheet )e alternatingpattern of 4 carbon atoms forming squares and 8 carbonatoms forming octagons constitutes the TUC4C8(S)[a b]

nanosheetIn this section GH HG NGH and NHG indices of the

C4C8(S) nanosheet are computed )e pattern of carbonatoms gives rise to two types of nanosheets namelyT1[a b] and T2[a b] )e 2-dimensional nanosheet isrepresented by T1[a b] where a and b are parameters(Figure 6) In T1[a b] C4 acts as a square while C8 is anoctagon in which a and b represent the column and rowrespectively Figure 7 depicts the type 1 minus C4C8(S)

nanosheet )e number of vertices of the C4C8(S)

nanosheet is 8ab and the number of edges is12ab minus 2a minus 2b

)e edge partition of the T1[a b] nanosheet based on thedegree of vertices is detailed in Table 6

Theorem 1 Let T1[a b] be an (a b)-dimensional nano-sheet then GH and HG indices are equal to

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(19)

Proof Using Table 6 the definitions of GH and HG indicesare as follows


Table 2 Parameters of regression models for the GH index

Physical properties Value of the correlation coefficient Residual standard errorEntropy minus0968 117Acentric factor minus0987 00059HVAP minus0815 121DHVAP minus0885 0184

Table 3 Parameters of regression models for the HG index

Physical properties Value of the correlation coefficient Residual standard errorEntropy 085 2448Acentric factor 0833 0020HVAP 0935 074DHVAP 0939 0136

Table 4 Parameters of regression models for the NGH index


Table 5 Parameters of regression models for the NHG index


90

95

100

105

110

115

S

30 35 40 45 5025GH index

(a)

024026028030032034036038040

AF

30 35 40 45 5025GH index

(b)

64

66

68

70

72

74

HV

AP

30 35 40 45 5025GH index

(c)

828486889092949698

100102

DH

VA

P

30 35 40 45 5025GH index

(d)

Figure 2 Scatter diagram of physical properties S AF HVAP and DHVAP with the GH index


90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)

Figure 3 Scatter diagram of physical properties S AF HVAP and DHVAP with the HG index

90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)

Figure 4 Scatter diagram of physical properties S AF HVAP and DHVAP with the NGH index


90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)

Figure 5 Scatter diagram of physical properties S AF HVAP and DHVAP with the NHG index

Figure 6 A TUC4C8(S)[a b] nanotube

1 2 3 a

1

2

b

3

Figure 7 Type I-C4C8(S) nanosheet T1[a b]


GH T1[a b]1113872 1113873 1113944

uvisinE(G)

du + dv( 1113857du times dv

1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897

+(12ab minus 8a minus 8b + 4)2

(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)

)e edge partition of the T1[a b] nanosheet based on theneighborhood degree of vertices is detailed in Table 7

Theorem 2 Let T1[a b] be an (a b)-dimensional nano-sheet then NGH and NHG indices are equal to

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)

Proof Using Table 7 the definitions of NGH and NHGindices are as follows

Table 6 )e edge partition of T1[a b]

(du dv) with uv isin E(G) Number of edges

(2 2) 2(a + b + 2)

(2 3) 4a + 4b minus 8(3 3) 12ab minus 8a minus 8b + 4


NGH T1[a b]1113872 1113873 1113944

uvisinE(G)

Su + Sv( 1113857Su times Sv

1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)

22 Results for the C4C8(R) Nanosheet )is structure isformed by 4 carbon atoms forming a rhombus that arelinearly bridged by edges whose sequence looks like 4rhombuses connected by 4 edges row and column wiseresulting in an alternating pattern of rhombuses and octa-gons and is represented as T2[a b] )e 2-dimensionallattice of the TUC4C8(R)[a b] nanosheet where a and b areparameters is shown in Figure 8 Figure 9 shows the type2 minus C4C8(R) nanosheet In the following theorem GH HGNGH andNHG indices of this nanosheet are computed)enumber of vertices of the type-2 structure is 4ab + 4(a + b) + 4 and the number of edges is 6ab + 5a + 5b + 4

)e edge partition of the T2[a b] nanosheet for degree-based vertices is detailed in Table 8


GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)

Table 7 Edge partition of T1[a b] for neighborhood degree-based vertices

(Su Sv) with uv isin E(G) Number of edges

(4 4) 4(4 5) 8(5 5) 2a + 2b minus 8(5 8) 4a + 4b minus 8(8 8) 2a + 2b minus 4(8 9) 4a + 4b minus 8(9 9) 12ab minus 14a minus 14b + 16



GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)

Figure 8 A TUC4C8(R)[a b] nanotube

1 2 3 a

1

2

b

3

Figure 9 Type II-C4C8(R) nanosheet T2[a b]

Table 8 Edge partition of T2[a b]


(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(25) Proof Using Table 9 the definitions of NGH and NHGindices are as follows

NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)

23Results for theH-NaphthalenicNanosheet Carbon atomsbonded in the form of a hexagonal structure constitutecarbon nanotubes )ey are peri-condensed benzenoidswhich mean three or more rings share the same atomsH-Naphthalenic nanosheet is constituted by the alternatingsequence of squares C4 hexagons C6 and octagons C8 and isrepresented as T3[a b] where a and b are the parameters)e number of vertices of the H-naphthalenic nanosheet is10ab and the edges are 15ab minus 2a minus 2b )e GH HG NGHand NHG indices of this nanosheet are computed seeFigure 10



GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)



(5 5) 4(5 8) 8(6 8) 4a + 4b minus 8(8 8) 2a + 2b + 4(8 9) 4a + 4b minus 8(9 9) 6ab minus 5a minus 5b + 4



GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3

Figure 10 An H-naphthalenic nanosheet T3[a b]

Table 10 )e details of edges and types of the T3[a b] nanosheet based on the degree of vertices


(2 2) 2a + 4(2 3) 8a + 4b minus 8(3 3) 15ab minus 10a minus 8b + 4




NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)



(4 5) 8(5 5) 2b minus 4(5 7) 4(5 8) 4b minus 4(6 7) 4a minus 4(6 8) 4a minus 4(7 9) 2a

(8 8) 2a + 2b minus 4(8 9) 4a + 4b minus 8(9 9) 15ab minus 18a minus 14b + 16


3 Conclusion

)is paper is devoted to defining NGH HG and NHGindices and the chemical applicability is studied for somephysical and chemical properties of octane isomers usingregression models including the recently introduced GHindex )e GH index has a high negative correlation withacentric factor having r 0987 with a residual standarderror of 00059)eHG index has a high positive correlationwith DHVAP having r 0939 with a residual standarderror of 0136 )e NGH index has a high negative corre-lation with acentric factor having r 0877 with a residualstandard error of 00176 )e NHG index has a high positivecorrelation with acentric factor having r 0877 with aresidual standard error of 0018 )e applications of carbonnanotubes have considerably increased because of theirexcellent mechanical thermal and electrical properties )enovel indices introduced in this paper would be of great helpto understand the physicochemical and biological propertiesof various compounds in addition to the existing degree-based indices

Data Availability

)e data used to support the findings of this study are citedat relevant places within the text as references

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally to this work

References

[1] D Bonchev Information 7eoretic Indices for Characteriza-tion of Chemical Structures Research Studies Press 1983

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

[3] I Gutman and O E Polansky Mathematical Concepts inOrganic Chemistry Springer Berlin Germany 2012

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

[5] R Todeschini and V Consonni Handbook of MolecularDescriptors Wiley VCH Weinheim Germany 2000

[6] N Trinajstic Chemical Graph7eory CRC Press Boca RatonFL USA 1992

[7] S Hayat M Imran and J B Liu ldquoCorrelation between theEsrada index and π-electron energies for benzenoid hydro-carbons with applications to boron nanotubesrdquo InternationalJournal of Quantum Chemistry vol 119 pp 1ndash13 2019

[8] M Randic ldquoQuantitative structure-property relationshipBoiling points of planar benzenoidsrdquo New Journal ofChemistry vol 20 pp 1001ndash1009 1996

[9] M C Shanmukha A Usha N S Basavarajappa andK C Shilpa ldquoNovel Neighbourhood redefined First andSecond Zagreb indices on Carborundum Structuresrdquo Journal

of Applied Mathematics and Computing vol 66 pp 1ndash142020

[10] M C Shanmukha N S Basavarajappa K C Shilpa andA Usha ldquoDegree-based topological indices on anticancerdrugs with QSPR analysisrdquo Heliyon vol 6 Article ID e042352020

[11] S A K Kirmani P Ali F Azam and P A Alvi ldquoTopologicalindices and QSPRQSAR analysis of some antiviral drugsbeing investigated for the treatment of COVID-19 patientsrdquoInternational Journal of Quantum Chemistry vol 121 pp 1ndash22 2021

[12] K Roy ldquoTopological descriptors in drug design and modelingstudiesrdquo Molecular Diversity vol 8 no 4 pp 321ndash323 2004

[13] A T Balaban and J Devillers Topological Indices and RelatedDescriptors in QSAR and QSPR CRC Press London UK2014

[14] J C Dearden Advances in QSAR Modeling Springer ChamSwitzerland pp 57ndash88 2017

[15] E Estrada L Torres L Roriguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo Indian Journal of Chemistry vol 37 pp 849ndash8551998

[16] H Wiener ldquoStructural determination of parafin boilingpointsrdquo Journal of the American Chemical Society vol 69no 1 pp 17ndash20 1947

[17] B Basavangouda and S Policepatil ldquoChemical applicability ofGourava and hyper-Gourava indicesrdquo Nanosystems PhysicsChemistry Mathematics vol 12 pp 142ndash150 2021

[18] P S R Harisha and V Lokesha ldquoM-Polynomial of valency-based topological indices of operators on titania nanotubesTiO2[m n] networksrdquo Advances and Applications in Math-ematical Sciences vol 20 pp 1493ndash1508 2021

[19] P S Hemavathi V Lokesha M Manjunath P S KReddyand R Shruti ldquoTopological aspects boron triangular nanotubeand boron-α nanotuberdquo Vladikavkaz Mathematical Journalvol 22 pp 66ndash77 2020

[20] V Lokesha T Deepika P S Ranjini and I N CangulldquoOperations of nanostructures via SDD ABC 4 and GA 5indicesrdquo Applied Mathematics and Nonlinear Sciences vol 2no 1 pp 173ndash180 2017

[21] V Lokesha R Shruti P S Ranjini and S Cevik ldquoOn certaintopological indices of nanostructures using Q(G) and R(G)

peratorsrdquo Communications vol 67 pp 178ndash187 2018[22] S Mondal A Bhosale N De and A Pal ldquoTopological

properties of some nanostructuresrdquo Nanosystems PhysicsChemistry Mathematics vol 11 no 1 pp 14ndash24 2020

[23] M F Nadeem S Zafar and Z Zahid ldquoOn topologicalproperties of the line graphs of subdivision graphs of certainnanostructuresrdquo Applied Mathematics and Computationvol 273 pp 125ndash130 2016

[24] Y Shanthakumari P Siva Kota Reddy V Lokesha andP S Hemavathi ldquoTopological aspects boron triangularnanotube and boron-α nanotube-II south east asianrdquo Journalof Mathematics and Mathematical Sciences (SEAJMMS)vol 16 pp 145ndash156 2020

[25] S Iijima ldquoSynthesis of Carbon Nanotubesrdquo Nature vol 354pp 56ndash58 1991

[26] S Hayat M K Shafiq A Khan et al ldquoOn topologicalproperties of 2-dimensional lattices of carbon nanotubesrdquoJournal of Computational and7eoretical Nanoscience vol 13no 10 pp 6606ndash6615 2016

[27] A Ali W Nazeer MMunir andM K Shin ldquoM-polynomialsand topological indices of zigzag and rhombic benzenoidsystemsrdquo Open Chemistry vol 16 pp 73ndash78 2017


[28] A Q Baig M Imran and H Ali ldquoComputing omega sad-hana and PI polynomials of benzenoid carbon nanotubesrdquoOptoelectronics and advanced materials-rapid communica-tions vol 9 pp 248ndash255 2015

[29] M Nadeem A Yousaf and A Razaq ldquoCertain polynomialsand related topological indices for the series of benzenoidgraphsrdquo Scientific Reports vol 9 pp 1ndash6 2019

[30] F Harary Graph 7eory Addison-Wesely Reading MassBoston MA USA 1969

[31] V R Kulli College Graph 7eory Vishwa InternationalPublications Gulbarga India 2012

[32] A Usha M C Shanmukha K N Anil Kumar andK C Shilpa ldquoComparision of novel index with geometric-arithmetic and sum-connectivity indicesrdquo Journal of Math-ematical and Computational Science vol 11 pp 5344ndash53602021

[33] D Vukicevic and B Furtula ldquoTopological index based on theratios of geometric and arithmetical means of end-vertexdegrees of edgesrdquo Journal of Mathematical Chemistry vol 46pp 1369ndash1376 2009


while chirality descriptors are introduced based on molec-ular graphs [16]

Alkanes are acyclic saturated hydrocarbons in whichcarbons and hydrogens are arranged in a tree-like structure)e main use of alkanes is found in crude oil such as pe-troleum cooking gas pesticides and drug synthesis )ecompounds that contain absolutely the same number ofatoms but their arrangement differs are termed as isomers Astudy is carried out for eighteen octane isomers (referFigure 1)

A structure whose size is between the microscopic andmolecular structure is referred to as a nanostructure )ereare different types of nanostructures namely nanocagesnanocomposites nanoparticles nanofabrics etc In the re-cent years nanostructures have attracted a lot of researchersin the areas of biology chemistry and medicines Topo-logical indices of nanostructures can be studied from[17ndash24] )e nanostructures made of carbons with cylin-drical shape are carbon nanotubes (CNTs) )ey have asimilar structure to that of a fullerene and graphene excepttheir cylindrical shape )e shape of fullerene is as that of afootball or basketball design where hexagons are connected

In 1991 Iijima [25] used carbon nanotubes that haveattracted many researchers in nanoscience and nanotech-nology worldwide As they have exotic properties they arewidely used in both research and applications Nanotubeshave a distinctive structure with remarkable mechanical andelectrical properties In case of carbon nanotubes thehexagons are surrounded by squares and each of thesepatterns is linearly arranged Carbon nanotubes reveal ex-ceptional electrical conductivity and possess wonderfultensile strength and thermal conductivity as they havenanostructures in which the carbon atoms are stronglyconnected

Carbon nanotubes have applications in orthopaedicimplants especially in total hip replacement and othertreatments pertaining to bone-related ailments )ey areused as a grouting agent placed between the prosthesis andthe bone as a part of their therapeutic use)e CNTs are usedin biomedical fields because of their structural stiffness andeffective optical absorption from UV to IR Also they can bealtered chemically which are expected to be useful in manyfields of technology such as electronics composited mate-rials and carbon fibres )ey have incredible applications inthe field of materials science [26] When the hexagonallattice is rolled in different directions it looks like single-wallcarbon nanotubes have spiral shape and translationalsymmetry along the tube axis It has rotational symmetryalong its own axis Even though nanotubes have favourableapplications in a variety of fields their large-scale productionhas been restricted )e main constraint that obstructs theiruse lies in difficulty in controlling their structure impuritiesand poor process ability To enhance their usage they havegrabbed the attention especially in the formation of com-posites with polymers

)ere are two types of configurations in the arrangementof nanotubes namely zigzag and armchair In the zigzagconfiguration the hexagons are placed one below the otherlinearly whereas in the armchair configuration they are

placed next to each other )is gives two different types ofconfigurations with different terminologies discussed nowTo explain the structure of a nanotube that is infinitely longwe imagine it to be cut open by a parallel axis and placed on aplane )en the atoms and bonds coincide with an imag-inary graphene sheet )e length of the two atoms on op-posite edges of the strip corresponds to circumference of thecylindrical graphene sheet [27ndash29]

)e main objectives of this work are as follows

To define novel indicesTo discuss the physical and chemical applicability ofoctane isomers using regression modelsTo compute defined indices for carbon nanotubes suchas C4C8(S) C4C8(R) and H-naphthalenic nanosheets

Let G (V E) be a graph with a vertex set V(G) and anedge set E(G) such that |V(G)| n and |E(G)| m Forstandard graph terminologies and notations refer to[30 31]where (u v) is an element of E(G) du represents thedegree of the vertex u and Su represents the neighborhooddegree of the vertex u

Definition 1 Recently Usha et al [32] defined the geo-metric-harmonic (GH) index inspired by Vukicevic andFurtula [33] in designing the GA index

GH(G) 1113944du + dv( 1113857

du middot dv

1113968

2 (1)

Motivated by the above work in this paper an attempt ismade to define three novel indices based on degree andneighborhood degree namely harmonic-geometric (HG)neighborhood geometric-harmonic (NGH) and neighbor-hood harmonic-geometric (NHG) indices )ey are definedas follows

HG(G) 11139442

du + dv( 1113857du middot dv

1113968

NGH(G) 1113944Su + Sv( 1113857

Su middot Sv

1113968

2

NHG(G) 11139442

Su + Sv( 1113857Su middot Sv

1113968

(2)

11 Chemical Applicability of GH NGH HG and NHGIndices In this section a linear regression model of fourphysical properties is presented for GH NGH HG andNHG indices )e physical properties such as entropy(S)acentric factor (AF) enthalpy of vaporization (HVAP) andstandard enthalpy of vaporization (DHVAP) of octaneisomers have shown good correlation with the indicesconsidered in the study )e GH HG NGH and NHGindices are tested for the octane isomersrsquo database availableat httpswwwmoleculardescriptorseudatasethtm )eGH HG NGH and NHG indices are computed and tab-ulated in columns 6 7 8 and 9 of Table 1


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)









(4)






(8)






(12)






(16)


















GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(19)











90

95

100

105

110

115

S

30 35 40 45 5025GH index

(a)

024026028030032034036038040

AF

30 35 40 45 5025GH index

(b)

64

66

68

70

72

74

HV

AP

30 35 40 45 5025GH index

(c)

828486889092949698

100102

DH

VA

P

30 35 40 45 5025GH index

(d)



90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)


90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)



90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

(p) (q) (r)









(4)






(8)






(12)






(16)


















GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(19)











90

95

100

105

110

115

S

30 35 40 45 5025GH index

(a)

024026028030032034036038040

AF

30 35 40 45 5025GH index

(b)

64

66

68

70

72

74

HV

AP

30 35 40 45 5025GH index

(c)

828486889092949698

100102

DH

VA

P

30 35 40 45 5025GH index

(d)



90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)


90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)



90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References










































(4)






(8)






(12)






(16)


















GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(19)











90

95

100

105

110

115

S

30 35 40 45 5025GH index

(a)

024026028030032034036038040

AF

30 35 40 45 5025GH index

(b)

64

66

68

70

72

74

HV

AP

30 35 40 45 5025GH index

(c)

828486889092949698

100102

DH

VA

P

30 35 40 45 5025GH index

(d)



90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)


90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)



90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References














































90

95

100

105

110

115

S

30 35 40 45 5025GH index

(a)

024026028030032034036038040

AF

30 35 40 45 5025GH index

(b)

64

66

68

70

72

74

HV

AP

30 35 40 45 5025GH index

(c)

828486889092949698

100102

DH

VA

P

30 35 40 45 5025GH index

(d)



90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)


90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)



90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































90

95

100

105

110

115S

14 16 18 20 2212HG index

(a)

024026028030032034036038040

AF

14 16 18 20 2212HG index

(b)

64

66

68

70

72

74

HV

AP

14 16 18 20 2212HG index

(c)

828486889092949698

100102

DH

VA

P14 16 18 20 2212

HG index

(d)


90

95

100

105

110

115

S

100 120 140 160 180 200 220 24080NGH index

(a)

024026028030032034036038040

AF

100 120 140 160 180 200 220 24080NGH index

(b)

64

66

68

70

72

74

HV

AP

100 120 140 160 180 200 220 24080NGH index

(c)

828486889092949698

100102

DH

VA

P

100 120 140 160 180 200 220 24080NGH index

(d)



90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































90

95

100

105

110

115S

03 04 05 06 0702NHG index

(a)

024026028030032034036038040

AF

03 04 05 06 0702NHG index

(b)

64

66

68

70

72

74

HV

AP

03 04 05 06 0702NHG index

(c)

828486889092949698

100102

DH

VA

P03 04 05 06 0702

NHG index

(d)



1 2 3 a

1

2

b

3



GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































GH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2a + 2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(12ab minus 8a minus 8b + 4)(2 + 3)(

2 times 3

radic)

21113896 1113897

GH T1[a b]1113872 1113873 108ab minus

4938125

a minus4938125

b +376125

HG T1[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2a + 2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(2 + 3)(2 times 3

radic)

1113896 1113897

HG T1[a b]1113872 1113873

43

ab +33125

a +33125

b +69500

(20)



NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(21)




(2 2) 2(a + b + 2)



NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































NGH T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(4 + 4)(

4 times 4

radic)

21113896 1113897 + 8

(4 + 5)(4 times 5

radic)

21113896 1113897 +(2a + 2b minus 8)

(5 + 5)(5 times 5

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897

+(12ab minus 14a minus 14b + 16)(9 + 9)(

9 times 9

radic)

21113896 1113897

NGH T1[a b]1113872 1113873 972ab minus

251531500

a minus251531500

b +1591211000

NHG T1[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

42

(4 + 4)(4 times 4

radic)

1113896 1113897 + 82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2a + 2b minus 8)2

(5 + 5)(5 times 5

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897


(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T1[a b]1113872 1113873

37250

ab +911000

a +911000

b +1571000

(22)




GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(23)






GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References







































GH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(2 + 2)(

2 times 2

radic)

21113896 1113897 +(4a + 4b)

(2 + 3)(2 times 3

radic)

21113896 1113897 +(6ab + a + b)

(3 + 3)(3 times 3

radic)

21113896 1113897

GH T2[a b]1113872 1113873 54ab +

66920

a +66920

b + 16

HG T2[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

22

(2 + 2)(2 times 2

radic)

1113896 1113897 +(4a + 4b)2

(2 + 3)(2 times 3

radic)

1113896 1113897 +(6ab + a + b)2

(3 + 3)(3 times 3

radic)

1113896 1113897

HG T2[a b]1113872 1113873

23

ab +191250

a +191250

b + 1

(24)


1 2 3 a

1

2

b

3




(2 2) 4(2 3) 4(a + b)

(3 3) 6ab + a + b




NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References








































NGH T2[a b]1113872 1113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100


NGH T2[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

4(5 + 5)(

5 times 5

radic)

21113896 1113897 + 8

(5 + 8)(5 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(6 + 8)(6 times 8

radic)

21113896 1113897 +(2a + 2b + 4)

middot(8 + 8)(

8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

(8 + 9)(8 times 9

radic)

21113896 1113897 +(6ab minus 5a minus 5b + 4)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T2[a b]1113872 11138731113873 486ab +

20549100

a +20549100

b +21549500

NHG T2[a b]1113872 1113873 1113944

uvisinE(G)

2Su + Sv( 1113857

Su times Sv

1113968

42

(5 + 5)(5 times 5

radic)

1113896 1113897 + 82

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(6 + 8)(6 times 8

radic)

1113896 1113897 +(2a + 2b + 4)

middot2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(6ab minus 5a minus 5b + 4)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T2[a b]1113872 1113873

37500

ab +1071000

a +1071000

b +19100

(26)




GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(27)






GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References







































GH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

(2b + 4)(2 + 2)(

2 times 2

radic)

21113896 1113897 +(8a + 8b minus 8)

(2 + 3)(2 times 3

radic)

21113896 1113897

+(15ab minus 10a minus 8b + 4)(3 + 3)(

3 times 3

radic)

21113896 1113897

GH T3[a b]1113872 1113873 135ab minus

4101100

a minus7901200

b +301100

HG T3[a b]1113872 1113873 1113944

uvisinE(G)

2du + dv( 1113857

du times dv

1113968

(2b + 4)2

(2 + 2)(2 times 2

radic)

1113896 1113897 +(8a + 8b minus 8)2

(2 + 3)(2 times 3

radic)

1113896 1113897


(3 + 3)(3 times 3

radic)

1113896 1113897

HG T3[a b]1113872 1113873

16671000

ab +39200

a +33125

b +691500

(28)

1 2 3 a

1

2

b

3








NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References








































NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(29)


NGH T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

8(4 + 5)(

4 times 5

radic)

21113896 1113897 +(2b minus 4)

(5 + 5)(5 times 5

radic)

21113896 1113897 + 4

(5 + 7)(7 times 7

radic)

21113896 1113897 +(4b minus 4)

middot(5 + 8)(

5 times 8

radic)

21113896 1113897 +(4a minus 4)

(6 + 7)(6 times 7

radic)

21113896 1113897 +(4a minus 4)

(6 + 8)(6 times 8

radic)

21113896 1113897

+ 2a(7 + 9)(

7 times 9

radic)

21113896 1113897 +(2a + 2b minus 4)

(8 + 8)(8 times 8

radic)

21113896 1113897 +(4a + 4b minus 8)

middot(8 + 9)(

8 times 9

radic)

21113896 1113897 +(15ab minus 18a minus 14b + 16)

(9 + 9)(9 times 9

radic)

21113896 1113897

NGH T3[a b]1113872 1113873 1215ab minus

1119920

a minus251531500

b +17382125

NHG T3[a b]1113872 1113873 1113944

uvisinE(G)


1113968

2

82

(4 + 5)(4 times 5

radic)

1113896 1113897 +(2b minus 4)2

(5 + 5)(5 times 5

radic)

1113896 1113897 + 42

(5 + 7)(7 times 7

radic)

1113896 1113897 +(4b minus 4)

middot2

(5 + 8)(5 times 8

radic)

1113896 1113897 +(4a minus 4)2

(6 + 7)(6 times 7

radic)

1113896 1113897 +(4a minus 4)2

(6 + 8)(6 times 8

radic)

1113896 1113897

+ 2a2

(7 + 9)(7 times 9

radic)

1113896 1113897 +(2a + 2b minus 4)2

(8 + 8)(8 times 8

radic)

1113896 1113897 +(4a + 4b minus 8)

middot2

(8 + 9)(8 times 9

radic)

1113896 1113897 +(15ab minus 18a minus 14b + 16)2

(9 + 9)(9 times 9

radic)

1113896 1113897

NHG T3[a b]1113872 1113873

37200

ab +15200

a +911000

b +2492500

(30)






3 Conclusion


Data Availability






References






































3 Conclusion


Data Availability






References






































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