Research Article Study of Vanadium Carbide ... - Hindawi.com

Research ArticleStudyofVanadiumCarbide StructuresBasedonVe andEv-DegreeTopological Indices

AbdulRauf 1 SabaMaqbool1MuhammadNaeem 1AdnanAslam 2HamidehAram3

and Kraidi Anoh Yannick 4

1Department of Mathematics Air University Multan Campus Multan Pakistan2Department of Natural Sciences and Humanities University of Engineering and Technology Lahore Pakistan (RCET) Pakistan3Department of Mathematics Gareziaeddin Center Khoy Branch Islamic Azad University Khoy Iran4UFR of Mathematics and Informatics University Felix Houphouet Boigny of Cocody Abidjan Cote drsquoIvoire

Correspondence should be addressed to Kraidi Anoh Yannick kayanoh2000yahoofr

Received 13 June 2021 Accepted 7 December 2021 Published 27 December 2021

Academic Editor Muhammad Imran

Copyright copy 2021 Abdul Rauf et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Vanadium is a biologically active product with significant industrial and biological applications Vanadium is found in a variety ofminerals and fossil fuels the most common of which are sandstones crude oil and coal Topological descriptors are numericalnumbers assigned to the molecular structures and have the ability to predict certain of their physicalchemical properties In thispaper we have studied topological descriptors of vanadium carbide structure based on ev and ve degrees In particular we havecomputed the closed forms of Zagreb Randic geometric-arithmetic and atom-bond connectivity (ABC) indices of vanadiumcarbide structure based on ev and ve degrees is kind of study may be useful for understanding the biological and chemicalbehavior of the structure

1 Introduction

Vertex degree concept has devised many topological in-dices that are applicable in QSPRQSAR studies Topo-logical indices are widely used in theoretical andmathematical chemistry as they are associated with thetopology of a chemical structure along with its otheridentical properties such as boiling points strain energyand stability [1] In chemical graph theory a chemicalgraph is referred as a molecular structure with atoms as itsvertices and chemical bonds as its edges A topologicalindex is a numerical parameter that creates a link betweenthe physical and chemical properties of a molecule [2]Many topological descriptors based on degree have beenintroduced ese topological indices have provided as-sistance in calculating different parametric calculationsrelated to molecular structures to make them under-standable and beneficial A lot of topological descriptorshave been defined and studied so far but Zagreb indices[3] Weiner index [4] and Randic index [5] are the most

studied among all of them To read more about thechemical applicability of topological descriptors see[5ndash12]

Researchers have attempted to study the varying be-havior of transition metal carbides due to their complexstructures Such mineral metals are available in commercialplaces and their salts are broadly utilized in our enterprisesrelated to electrochemistry and material science Amongthese vanadium carbide complexes have shown crystalmorphologies and stoichiometrics and display a greatvariety of superstructures Very recently many attemptshave been done to purify high quality vanadium carbide bypresenting different binary model system such asVC V2C V4C3 V6C5 and V8C7 For more details see[13ndash16]

Let G be a simple connected graph with its edge set andvertex set denoted by E and V respectively e neighbor setN(v) of a vertex v contains those vertices v1 such thatvv1 isin E e degree of vertex v is denoted by dv and is thecardinality of the set N(v) Let N[v] v1 isin V vv1 isin E1113864 1113865

HindawiJournal of ChemistryVolume 2021 Article ID 7189918 12 pageshttpsdoiorg10115520217189918

cup v be the closed neighborhood of v To read more aboutthe basic concepts related to graph see [17]

M Chellali et al [18] first introduced the concept of evdegree of an edge e and ve degree of a vertex v e ev degreeof an edge e is denoted by dev(e) and is defined as the totalnumber of vertices in the closed neighborhoods of the endvertices of an edge e e ve degree of a vertex v is denoted bydve(v) and is the total number of edges that are adjacent withv and the first neighbor of v Ediz [3] first introduced theconcept of ve degree and ev degree Zagreb and Randicindices e mathematical formulas of these indices arepresented in Table 1 ese newly defined indices werecompared with Zagreb Weiner and Randic indices bymodeling some of the physicalchemical properties of octaneisomers ese indices have been observed to provide bettercorrelation than the Randic Weiner and Zagreb indices forpredicting some specific physical and chemical properties ofoctane isomers Recently a lot of work is done in the di-rection of computing newly defined ve degree and ev degree-based indices [19ndash23]

2 Vanadium Carbide

Vanadium carbide belongs to the family of group IV to VItransition metal carbides and shows homogeneity to metalnitrides monocarbides and carbonitrides ey possessunique associations between physio-chemical propertiessuch as high melting points high temperature resistivitystrength and hardness which are associated with goodelectrical and thermal conductivityese rare combinationsof properties make such compounds very interesting for theresearchers ese materials can be used as wear-resistant

hard alloys and as hard coatings for protection purposes dueto their nanochemical properties [24 25]

Vanadium carbide is the hardest inorganic metal-car-bide with the formula VC VC is an incredibly hard re-fractory ceramic with exceptional wear resistance highmodulus of elasticity (400GPa) and good strength retentioneven at high temperatures [26ndash28] VC coatings are used incorrosion prevention cutting tool application machiningdrilling and dyeing Some industrial uses of VC are given in[27 29ndash31] We denote the crystallographic structure ofvanadium carbide by VC[m n] e molecular structure ofvanadium carbide for m 5 n is depicted in Figure 1 estructure of VC[m n] has total number of 3mn + m + n

vertices and 6mn minus m minus n edges Let Vi denote the vertex setcontaining the vertices of VC[m n] of degree i en thevertex set V(VC[m n]) can be partitioned into six sets with|V1| m + n + 2 |V2| 2m + 2n minus 4 |V3| 4m minus 2n + 2|V4| 1 |V5| m + n minus 2 and |V6| mn minus m minus n + 1 LetE(ij) denote the edge set containing the edges of VC[m n]

with end vertices of degree i and degree j e edge set ofVC[m n] can be partitioned based on the degree of endvertices as follows E(14) with 1 edge E(15) with 2 edgesE(16) with m + n minus 1 edges E(24) with 2 edges E(25) with2m + 2n minus 4 edges E(26) with 2m + 2n minus 6 edges E(34) with1 edge E(35) with 3m + 3n minus 8 edges and E(3 6) having6mn minus 9m minus 9n + 13 edges Ineorem 1 we compute the evdegree Randic and ev degree Zagreb index of VC[m n]

3 Main Results

Theorem 1 Let m nge 2 then

Mev

(VC[m n]) 486mn minus 262m minus 262n + 130

Rev

(VC[m n]) 17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m

+17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n +15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889

(1)

Proof To compute the ev degree Zagreb and ev degreeRandic index of VC[m n] we need to compute the ev degreeof the edges in each partition set E(ij) is calculation is

presented in Table 2 Now using the information presentedin Table 2 and the definition of ev degree Zagreb and evdegree Randic index we get

Mev

(VC[m n]) 1113944eisinE(Z)

dev(e)2

(5)2

E(14)

11138681113868111386811138681113868111386811138681113868 +(6)

2E(15)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(16)

11138681113868111386811138681113868111386811138681113868 +(6)

2E(24)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(25)

11138681113868111386811138681113868111386811138681113868

+(8)2

E(26)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(34)

11138681113868111386811138681113868111386811138681113868 +(8)

2E(35)

11138681113868111386811138681113868111386811138681113868 +(9)

2E(36)

11138681113868111386811138681113868111386811138681113868

(5)2(1) +(6)

2(2) +(7)

2(m + n minus l) +(6)

2(2) +(7)

2(2m + 2n minus 4)

2 Journal of Chemistry

+(8)2(2m + 2n minus 6) +(9)

2(1) +(7)

2(3m + 3n minus 8)

+(8)2(6mn minus 9m minus 9n + 13)

486mn minus 262m minus 262n + 130

Rev


dev(e)minus 12

(5)minus 12

E(14)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(15)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(16)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(24)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(25)

11138681113868111386811138681113868111386811138681113868

+(8)minus 12

E(26)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(34)

11138681113868111386811138681113868111386811138681113868 +(8)

minus 12E(35)

11138681113868111386811138681113868111386811138681113868 +(9)

minus 12E(36)

11138681113868111386811138681113868111386811138681113868

Table 1 Mathematical formula of topological indices

Topological indices Notation Mathematical formulaEv degree Randic index Rev(G) 1113936eisinE(G)dev(e)minus (12)

First Zagreb β-index Mβve1 (G) 1113936uvisinE(G)(dve(u) + dve(v))

Ev degree Zagreb index Mev(G) 1113936eisinE(G)dev(e)2

Second Zagreb β-index Mβve2 (G) 1113936uvisinE(G)(dve(u) times dve(v))

Ve degree harmonic index Hve(G) 1113936uvisinE(G)((2)(dve(u) + dve(v)))

Ve degree sum connectivity index χve(G) 1113936uvisinE(G)(dve(u) + dve(v))minus 12

Ve degree geometric arithmetic index GAve(G) 1113936uvisinE(G)(2dve(u) times dve(v)

1113968)(dve(u) + dve(v))

Ve degree atom bond connectivity index ABCve(G) 1113936uvisinE(G)

(dve(u) + dve(v) minus 2)(dve(u) times dve(v))

1113968

n = 5

n = 4

n = 3

n = 2

n = 1

m = 1m = 2m = 3m = 4m = 5

Figure 1 Molecular structure of vanadium carbide for m 5 and n 5

Table 2 Ev-degrees of edges of vanadium carbide

(d(u) d(v)) dev(e) Frequency

E(14) 5 1E(15) 6 2E(16) 7 m + n minus 1E(24) 6 2E(25) 7 2m + 2n minus 4E(26) 8 2m + 2n minus 6E(34) 7 l

E(35) 8 3m + 3n minus 8E(36) 9 6mn minus 9m minus 9n + 13

Journal of Chemistry 3

(5)minus 12

(1) +(6)minus 12

(2) +(7)minus 12

(m + n minus l) +(6)minus 12

(2) +(7)minus 12

(2m + 2n minus 4)

+(8)minus 12

(2m + 2n minus 6) +(9)minus 12

(1) +(7)minus 12

(3m + 3n minus 8)

+(8)minus 12

(6mn minus 9m minus 9n + 13)

17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n

+15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889 (2)


Mβve1 (VC[m n]) 216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1944mn minus 51m minus 1887n + 1798

ABCve

(VC[m n]) 634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888

+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970 22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889

GAve

(VC[m n]) 12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

301113888

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

36⎞⎠

Hve

(VC[m n]) 13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(3a)

χve(Z) mn +

120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m +120

radic1113888

+223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic 1113889n +112

radic +216

radic +316

radic1113888

minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic minus1026

radic +122

radic minus630

radic +227

radic


minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic minus1035

radic +128

radic +3736

radic 1113889

Rve

(VC[m n]) 6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic +2121

radic1113888

minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic +2182

radic minus8208

radic

+2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic +1180

radic +37324

radic 1113889 (3b)

Proof To compute the ve degree-based indices we need tofind the edge partition of E(VC[m n]) based on the vedegree of end vertices of each edge is partition is

presented in Table 3 Now using the values from Table 3 andthe definition of ve degree indices we get

Mβve1 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857

(12) E11113868111386811138681113868

1113868111386811138681113868 +(16) E21113868111386811138681113868

1113868111386811138681113868 +(16) E31113868111386811138681113868

1113868111386811138681113868 +(20) E41113868111386811138681113868

1113868111386811138681113868 +(17) E51113868111386811138681113868

1113868111386811138681113868 +(21) E61113868111386811138681113868

1113868111386811138681113868

+(22) E71113868111386811138681113868

1113868111386811138681113868 +(23) E81113868111386811138681113868

1113868111386811138681113868 +(22) E91113868111386811138681113868

1113868111386811138681113868 +(25) E101113868111386811138681113868

1113868111386811138681113868 +(22) E111113868111386811138681113868

1113868111386811138681113868 +(26) E121113868111386811138681113868

1113868111386811138681113868 +(22) E131113868111386811138681113868

1113868111386811138681113868

+(30) E141113868111386811138681113868

1113868111386811138681113868 +(27) E151113868111386811138681113868

1113868111386811138681113868 +(29) E161113868111386811138681113868

1113868111386811138681113868 +(27) E171113868111386811138681113868

1113868111386811138681113868 +(28) E181113868111386811138681113868

1113868111386811138681113868

+(31) E191113868111386811138681113868

1113868111386811138681113868 +(32) E201113868111386811138681113868

1113868111386811138681113868 +(34) E211113868111386811138681113868

1113868111386811138681113868 +(35) E221113868111386811138681113868

1113868111386811138681113868

+(28) E231113868111386811138681113868

1113868111386811138681113868 +(36) E241113868111386811138681113868

1113868111386811138681113868

(12)(1) +(16)(2) +(16)(3) +(20)(m + n minus 4) +(17)(2) +(21)(2) +(22)(2)

+(23)(2m + 2n minus 10) +(22)(2) +(25)(2) +(22)(2) +(26)(2m + 2n minus 10)

+(22)(1) +(30)(m + n minus 6) +(27)(2) +(29)(2m + 2n minus 8) +(27)(2) +(28)(2)

+(31)(2) +(32)(3m + 3n minus 14) +(34)(m + n minus 3) +(35)(8m + 2n minus 10)

+(28)(1) +(36)(6mn minus 15m minus 15n + 37)

216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) times dve(v)( 1113857

(32) E11113868111386811138681113868

1113868111386811138681113868 +(55) E21113868111386811138681113868

1113868111386811138681113868 +(60) E31113868111386811138681113868

1113868111386811138681113868 +(84) E41113868111386811138681113868

1113868111386811138681113868 +(72) E51113868111386811138681113868

1113868111386811138681113868 +(110) E61113868111386811138681113868

1113868111386811138681113868

+(168) E121113868111386811138681113868

1113868111386811138681113868 +(112) E131113868111386811138681113868

1113868111386811138681113868 +(121) E71113868111386811138681113868

1113868111386811138681113868 +(130) E81113868111386811138681113868

1113868111386811138681113868 +(117) E91113868111386811138681113868

1113868111386811138681113868

+(154) E101113868111386811138681113868

1113868111386811138681113868 +(120) E111113868111386811138681113868

1113868111386811138681113868 +(121) E141113868111386811138681113868

1113868111386811138681113868 +(182) E151113868111386811138681113868

1113868111386811138681113868 +(208) E161113868111386811138681113868

1113868111386811138681113868

+(176) E171113868111386811138681113868

1113868111386811138681113868 +(187) E181113868111386811138681113868

1113868111386811138681113868 +(238) E191113868111386811138681113868

1113868111386811138681113868 +(252) E201113868111386811138681113868

1113868111386811138681113868 +(288) E211113868111386811138681113868

1113868111386811138681113868


+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)

+(306)(8m + 2n minus 10) +(180)(1) +(324)(6mn minus 15m minus 15n + 37)

1944mn minus 51m minus 1887n + 1798 (4)

ABCve(VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v) minus 2


1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888

Table 3 Edge Partition of vanadium carbide

Edge (dve(u) dve(v)) Frequency

E1 (4 8) 1E2 (5 11) 2E3 (6 10) 3E4 (6 14) m + n minus 4E5 (9 8) 2E6 (10 11) 2E7 (11 11) 2E8 (10 13) 2m + 2n minus 10E9 (9 13) 2E10 (11 14) 2E11 (12 10) 2E12 (12 14) 2m + 2n minus 10E13 (8 14) 1E14 (17 13) m + n minus 6E15 (14 13) 2E16 (16 13) 2m + 2n minus 8E17 (16 11) 2E18 (17 11) 2E19 (17 14) 2E20 (18 14) 3m + 3n minus 14E21 (16 18) m + n minus 3E22 (17 18) 2m + 2n minus 10E23 (18 10) 1E24 (18 18) 6mn minus 15m minus 15n + 37


+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])

2dve(u) times dve(v)

1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve

(VC[m n]) 1113944uvisinE(VC[mn])

2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857minus 12

(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) times dve(v)( 1113857minus 12

(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)


4 Numerical Results and Discussion

Topological indices are used as vital tools for the analysisof chemicals given the essential topology of chemicalstructures Zagreb-type indices are used to calculate thetotal πminus electronic energy of molecules [32] e Randichindex is commonly used to determine the chemicalsimilarity of molecular compounds as well as to calculate

the boiling point and Kovaz constants of molecules [33]e atom-bond connectivity index (ABC) provides a verygood correlation for calculating strain energies as well asfor the stability of linear and branched chemical structures[34] It can be seen from Table 4 and 5 and Figures 2ndash6 ofindices an increase in the value of m and n raises thevalues of topological descriptors for vanadium carbidestructure

Table 4 Numerical results of indices for vanadium carbide

[m n] Mev(H) Rev(H) Mαve1 (H) M

βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818


[m n] ABCve(H) GAve(H) Hve(H) χve(H) Rve(H)

[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)

Figure 2 3D plot of (a) Mev(VC[m n]) (b) Rev(VC[m n])


16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)

Figure 3 3D plot of (a) Mβve1 (VC[m n]) (b) Mβve

2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)

Figure 4 3D plot of (a) ABCve(VC[m n]) (b) GAve(VC[m n])

353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)

Figure 5 3D plot of (a) Hve(VC[m n]) (b) χve(VC[m n])


5 Conclusion

Graphs invariants are calculated by some well-known to-pological indices which are important tools for resemblingand forecasting the properties of chemical compounds inQSPRs and the QSARs e TI is a numerical measure thatrepresents the biological physical and chemical propertiesof molecules such as boiling melting and flickering pointmoisture and forming heat In this paper we have com-puted the ev degree and ve degree-based topological indiceswith graphical representations for the molecular structure ofvanadium carbide for a better understanding of pharma-ceutical physical chemical and biological properties

Data Availability

No data were used to support this study

Disclosure

is research was carried out as a part of employment of theauthors

Conflicts of Interest

e authors hereby declare that there are no conflicts ofinterest regarding the publication of this paper

References

[1] B Sahin and S Ediz ldquoOn ev-degree and ve-degree topo-logical indicesrdquo Iranian Journal of Mathematical Chemistryvol 9 no 4 pp 263ndash277 2018

[2] S Hosamani D Perigidad S Jamagoud Y Maled andS Gavade ldquoQSPR analysis of certain degree based topologicalindicesrdquo Journal of Statistics Applications amp Probability vol 6no 2 pp 361ndash371 2017

[3] S Ediz ldquoPredicting some physicochemical properties of oc-tane isomers a topological approach using ev-degree and ve-degree Zagreb indicesrdquo 2017 httpsuiadsabsharvardedulink_gateway2017arXiv170102859EarXiv170102859

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

[5] M Randic ldquoCharacterization of molecular branchingrdquoJournal of the American Chemical Society vol 97 no 23pp 6609ndash6615 1975

[6] 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

[7] S Ediz ldquoPredicting some physicochemical properties ofoctane isomers a topological approach using ev-degree andve-degree Zagreb indicesrdquo 2017

[8] Y-X Li A Rauf M Naeem M Ahsan Binyamin andA Aslam ldquordquoValency-based topological properties of linearhexagonal chain and hammer-like benzenoidrdquo Complexityp 2021 2021

[9] S-B Chen A Rauf M Ishtiaq M Naeem and A Aslam ldquoOnve-degree- and ev-degree-based topological properties ofcrystallographic structure of cuprite Cu2Ordquo Open Chemistryvol 19 no 1 pp 576ndash585 2021

[10] S Shirakol M Kalyanshetti M Sunilkumar and HosamanldquoQSPR analysis of certain distance based topological indicesrdquoApplied Mathematics and Nonlinear Sciences vol 4 no 2pp 371ndash386 2019

[11] S J Patel D Ng and M S Mannan ldquoQSPR flash pointprediction of solvents using topological indices for applicationin computer aided molecular designrdquo Industrial amp Engi-neering Chemistry Research vol 48 no 15 pp 7378ndash73872009

[12] J C Dearden ldquoe use of topological indices in QSAR andQSPR modelingrdquo in Advances in QSAR Modeling Challengesand Advances in Computational Chemistry and Physicspp 57ndash88 Springer Cham Switzerland 2017

[13] W Xing F Meng and R Yu ldquoA new type of vanadiumcarbide V5C3 and its hardening by tuning Fermi energyrdquoScientific Reports vol 6 no 1 pp 21794ndash21799 2016

[14] A VahidMohammadi A Hadjikhani S Shahbazmohamadiand M Beidaghi ldquoTwo-dimensional vanadium carbide(MXene) as a high-capacity cathode material for rechargeablealuminum batteriesrdquo ACS Nano vol 11 no 11pp 11135ndash11144 2017

[15] Y Guan S Jiang Y Cong et al ldquoA hydrofluoric acid-freesynthesis of 2D vanadium carbide (V2C) MXene for

70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn

Figure 6 Graphical representation of Rve(VC[m n])


supercapacitor electrodesrdquo 2D Materials vol 7 no 2 ArticleID 025010 2020

[16] D Huang Y Xie D Lu et al ldquoDemonstration of a white laserwith V2C MXene-based quantum dotsrdquo Advanced Materialsvol 31 pp 1901117ndash24 2019

[17] D B West Introduction to Graph eory Vol 2 Prentice-Hall Upper Saddle River NJ USA 2001

[18] M Chellali T W Haynes S T Hedetniemi and T M LewisldquoOn ve-degrees and ev-degrees in graphsrdquo Discrete Mathe-matics vol 340 no 2 pp 31ndash38 2017

[19] A N A Koam A Ahmad and A A Ahamdini ldquoCompu-tation of vertex-edge degree based topological descriptors forhex-derived networksrdquo IEEE Access 2021

[20] E A Refaee and A Ahmad ldquoA study of hexagon star networkwith vertex-edge-based topological descriptorsrdquo Complexityvol 2021 2021

[21] A Ahmad ldquoComparative study of ve-degree and ev-degreetopological descriptors for benzene ring embedded in P-Type-Surface in 2D networkrdquo Polycyclic Aromatic Compoundspp 1ndash10 2020

[22] J Zhang M K Siddiqui A Rauf and M Ishtiaq ldquoOn ve-degree and ev-degree based topological properties of singlewalled titanium dioxide nanotuberdquo Journal of Cluster Sciencepp 1ndash12 2020

[23] B Horoldagva K Ch Das and T-A Selenge ldquoOn ve-Degreeand ev-Degree of Graphsrdquo Discrete Optimization vol 31pp 1ndash7 2019

[24] Te-H Fang S-R Jian and D-S Chuu ldquoNanomechanicalproperties of TiC TiN and TiCN thin films using scanningprobe microscopy and nanoindentationrdquo Applied SurfaceScience vol 228 no 1ndash4 pp 365ndash372 2004

[25] H G Prengel W R Pfouts and A T Santhanam ldquoState ofthe art in hard coatings for carbide cutting toolsrdquo Surface andCoatings Technology vol 102 no 3 pp 183ndash190 1998

[26] K H Lo F T Cheng C T Kwok and H Chung ManldquoMprovement of cavitation erosion resistance of AISI 316stainless steel by laser surface alloying using fineWC powderrdquoSurface and Coatings Technology vol 165 no 3 pp 258ndash2672003

[27] B Chicco W E Borbidge and E Summerville ldquoExperi-mental study of vanadium carbide and carbonitride coatingsrdquoMaterials Science and Engineering A vol 266 no 1-2pp 62ndash72 1999

[28] S A Hassanzadeh-Tabrizi ldquoHoma Hosseini Badr and SaraAlizadeh rdquoIn situ synthesis of vanadium carbide-coppernanocomposite by a modified mechanochemical combustionmethodrdquo Ceramics International vol 42 no 8pp 9371ndash9374 2016

[29] D Ferro J V Rau A Generosi V Rossi Albertini A Latiniand S M Barinov ldquoElectron beam deposited VC and NbCthin films on titanium hardness and energy-dispersive X-raydiffraction studyrdquo Surface and Coatings Technology vol 202no 10 pp 2162ndash2168 2008

[30] L BWang Z F Chen Y Zhang andW PWu InternationalJournal of Refractory Metals and Hard Materials p 20042009

[31] A Aouni P Weisbecker T Huu Loi and E Bauer-GrosseldquoSearch for new materials in sputtered V1-xCx filmsrdquo inSolid Films vol 469 pp 315ndash321 2004

[32] I Gutman B Ruscic N Trinajstic and C F Wilcox JrldquoGraph theory and molecular orbitals XII Acyclic polyenesrdquoe Journal of Chemical Physics vol 62 no 9 pp 3399ndash34051975

[33] N Nikolova and J Jaworska ldquoApproaches to measurechemical similarityndasha reviewrdquo QSAR amp Combinatorial Sci-ence vol 22 no 9-10 pp 1006ndash1026 2003

[34] E Estrada L Torres L Rodriguez and I Gutman ldquoAn atom-bond connectivity index modelling the enthalpy of formationof alkanesrdquo 1998


cup v be the closed neighborhood of v To read more aboutthe basic concepts related to graph see [17]

M Chellali et al [18] first introduced the concept of evdegree of an edge e and ve degree of a vertex v e ev degreeof an edge e is denoted by dev(e) and is defined as the totalnumber of vertices in the closed neighborhoods of the endvertices of an edge e e ve degree of a vertex v is denoted bydve(v) and is the total number of edges that are adjacent withv and the first neighbor of v Ediz [3] first introduced theconcept of ve degree and ev degree Zagreb and Randicindices e mathematical formulas of these indices arepresented in Table 1 ese newly defined indices werecompared with Zagreb Weiner and Randic indices bymodeling some of the physicalchemical properties of octaneisomers ese indices have been observed to provide bettercorrelation than the Randic Weiner and Zagreb indices forpredicting some specific physical and chemical properties ofoctane isomers Recently a lot of work is done in the di-rection of computing newly defined ve degree and ev degree-based indices [19ndash23]

2 Vanadium Carbide

Vanadium carbide belongs to the family of group IV to VItransition metal carbides and shows homogeneity to metalnitrides monocarbides and carbonitrides ey possessunique associations between physio-chemical propertiessuch as high melting points high temperature resistivitystrength and hardness which are associated with goodelectrical and thermal conductivityese rare combinationsof properties make such compounds very interesting for theresearchers ese materials can be used as wear-resistant

hard alloys and as hard coatings for protection purposes dueto their nanochemical properties [24 25]

Vanadium carbide is the hardest inorganic metal-car-bide with the formula VC VC is an incredibly hard re-fractory ceramic with exceptional wear resistance highmodulus of elasticity (400GPa) and good strength retentioneven at high temperatures [26ndash28] VC coatings are used incorrosion prevention cutting tool application machiningdrilling and dyeing Some industrial uses of VC are given in[27 29ndash31] We denote the crystallographic structure ofvanadium carbide by VC[m n] e molecular structure ofvanadium carbide for m 5 n is depicted in Figure 1 estructure of VC[m n] has total number of 3mn + m + n

vertices and 6mn minus m minus n edges Let Vi denote the vertex setcontaining the vertices of VC[m n] of degree i en thevertex set V(VC[m n]) can be partitioned into six sets with|V1| m + n + 2 |V2| 2m + 2n minus 4 |V3| 4m minus 2n + 2|V4| 1 |V5| m + n minus 2 and |V6| mn minus m minus n + 1 LetE(ij) denote the edge set containing the edges of VC[m n]

with end vertices of degree i and degree j e edge set ofVC[m n] can be partitioned based on the degree of endvertices as follows E(14) with 1 edge E(15) with 2 edgesE(16) with m + n minus 1 edges E(24) with 2 edges E(25) with2m + 2n minus 4 edges E(26) with 2m + 2n minus 6 edges E(34) with1 edge E(35) with 3m + 3n minus 8 edges and E(3 6) having6mn minus 9m minus 9n + 13 edges Ineorem 1 we compute the evdegree Randic and ev degree Zagreb index of VC[m n]

3 Main Results


Mev

(VC[m n]) 486mn minus 262m minus 262n + 130

Rev

(VC[m n]) 17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m

+17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n +15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889

(1)

Proof To compute the ev degree Zagreb and ev degreeRandic index of VC[m n] we need to compute the ev degreeof the edges in each partition set E(ij) is calculation is

presented in Table 2 Now using the information presentedin Table 2 and the definition of ev degree Zagreb and evdegree Randic index we get

Mev


dev(e)2

(5)2

E(14)

11138681113868111386811138681113868111386811138681113868 +(6)

2E(15)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(16)

11138681113868111386811138681113868111386811138681113868 +(6)

2E(24)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(25)

11138681113868111386811138681113868111386811138681113868

+(8)2

E(26)

11138681113868111386811138681113868111386811138681113868 +(7)

2E(34)

11138681113868111386811138681113868111386811138681113868 +(8)

2E(35)

11138681113868111386811138681113868111386811138681113868 +(9)

2E(36)

11138681113868111386811138681113868111386811138681113868

(5)2(1) +(6)

2(2) +(7)

2(m + n minus l) +(6)

2(2) +(7)

2(2m + 2n minus 4)


+(8)2(2m + 2n minus 6) +(9)

2(1) +(7)

2(3m + 3n minus 8)



Rev


dev(e)minus 12

(5)minus 12

E(14)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(15)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(16)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(24)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(25)

11138681113868111386811138681113868111386811138681113868

+(8)minus 12

E(26)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(34)

11138681113868111386811138681113868111386811138681113868 +(8)

minus 12E(35)

11138681113868111386811138681113868111386811138681113868 +(9)

minus 12E(36)

11138681113868111386811138681113868111386811138681113868









1113968)(dve(u) + dve(v))



1113968

n = 5

n = 4

n = 3

n = 2

n = 1

m = 1m = 2m = 3m = 4m = 5







(5)minus 12

(1) +(6)minus 12

(2) +(7)minus 12


(2) +(7)minus 12

(2m + 2n minus 4)

+(8)minus 12


(1) +(7)minus 12

(3m + 3n minus 8)

+(8)minus 12


17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n

+15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889 (2)


Mβve1 (VC[m n]) 216mn + 76m + 20n + 54


ABCve

(VC[m n]) 634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888

+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970 22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889

GAve

(VC[m n]) 12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

301113888

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

36⎞⎠

Hve

(VC[m n]) 13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(3a)

χve(Z) mn +

120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m +120

radic1113888

+223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic 1113889n +112

radic +216

radic +316

radic1113888

minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic minus1026

radic +122

radic minus630

radic +227

radic


minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic minus1035

radic +128

radic +3736

radic 1113889

Rve

(VC[m n]) 6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic +2121

radic1113888

minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic +2182

radic minus8208

radic

+2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic +1180

radic +37324

radic 1113889 (3b)



Mβve1 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857

(12) E11113868111386811138681113868

1113868111386811138681113868 +(16) E21113868111386811138681113868

1113868111386811138681113868 +(16) E31113868111386811138681113868

1113868111386811138681113868 +(20) E41113868111386811138681113868

1113868111386811138681113868 +(17) E51113868111386811138681113868

1113868111386811138681113868 +(21) E61113868111386811138681113868

1113868111386811138681113868

+(22) E71113868111386811138681113868

1113868111386811138681113868 +(23) E81113868111386811138681113868

1113868111386811138681113868 +(22) E91113868111386811138681113868

1113868111386811138681113868 +(25) E101113868111386811138681113868

1113868111386811138681113868 +(22) E111113868111386811138681113868

1113868111386811138681113868 +(26) E121113868111386811138681113868

1113868111386811138681113868 +(22) E131113868111386811138681113868

1113868111386811138681113868

+(30) E141113868111386811138681113868

1113868111386811138681113868 +(27) E151113868111386811138681113868

1113868111386811138681113868 +(29) E161113868111386811138681113868

1113868111386811138681113868 +(27) E171113868111386811138681113868

1113868111386811138681113868 +(28) E181113868111386811138681113868

1113868111386811138681113868

+(31) E191113868111386811138681113868

1113868111386811138681113868 +(32) E201113868111386811138681113868

1113868111386811138681113868 +(34) E211113868111386811138681113868

1113868111386811138681113868 +(35) E221113868111386811138681113868

1113868111386811138681113868

+(28) E231113868111386811138681113868

1113868111386811138681113868 +(36) E241113868111386811138681113868

1113868111386811138681113868

(12)(1) +(16)(2) +(16)(3) +(20)(m + n minus 4) +(17)(2) +(21)(2) +(22)(2)

+(23)(2m + 2n minus 10) +(22)(2) +(25)(2) +(22)(2) +(26)(2m + 2n minus 10)

+(22)(1) +(30)(m + n minus 6) +(27)(2) +(29)(2m + 2n minus 8) +(27)(2) +(28)(2)


+(28)(1) +(36)(6mn minus 15m minus 15n + 37)

216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1113944

uvisinE(VC[mn])


(32) E11113868111386811138681113868

1113868111386811138681113868 +(55) E21113868111386811138681113868

1113868111386811138681113868 +(60) E31113868111386811138681113868

1113868111386811138681113868 +(84) E41113868111386811138681113868

1113868111386811138681113868 +(72) E51113868111386811138681113868

1113868111386811138681113868 +(110) E61113868111386811138681113868

1113868111386811138681113868

+(168) E121113868111386811138681113868

1113868111386811138681113868 +(112) E131113868111386811138681113868

1113868111386811138681113868 +(121) E71113868111386811138681113868

1113868111386811138681113868 +(130) E81113868111386811138681113868

1113868111386811138681113868 +(117) E91113868111386811138681113868

1113868111386811138681113868

+(154) E101113868111386811138681113868

1113868111386811138681113868 +(120) E111113868111386811138681113868

1113868111386811138681113868 +(121) E141113868111386811138681113868

1113868111386811138681113868 +(182) E151113868111386811138681113868

1113868111386811138681113868 +(208) E161113868111386811138681113868

1113868111386811138681113868

+(176) E171113868111386811138681113868

1113868111386811138681113868 +(187) E181113868111386811138681113868

1113868111386811138681113868 +(238) E191113868111386811138681113868

1113868111386811138681113868 +(252) E201113868111386811138681113868

1113868111386811138681113868 +(288) E211113868111386811138681113868

1113868111386811138681113868


+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)


1944mn minus 51m minus 1887n + 1798 (4)


uvisinE(VC[mn])



1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888





+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























+(8)2(2m + 2n minus 6) +(9)

2(1) +(7)

2(3m + 3n minus 8)



Rev


dev(e)minus 12

(5)minus 12

E(14)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(15)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(16)

11138681113868111386811138681113868111386811138681113868 +(6)

minus 12E(24)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(25)

11138681113868111386811138681113868111386811138681113868

+(8)minus 12

E(26)

11138681113868111386811138681113868111386811138681113868 +(7)

minus 12E(34)

11138681113868111386811138681113868111386811138681113868 +(8)

minus 12E(35)

11138681113868111386811138681113868111386811138681113868 +(9)

minus 12E(36)

11138681113868111386811138681113868111386811138681113868









1113968)(dve(u) + dve(v))



1113968

n = 5

n = 4

n = 3

n = 2

n = 1

m = 1m = 2m = 3m = 4m = 5







(5)minus 12

(1) +(6)minus 12

(2) +(7)minus 12


(2) +(7)minus 12

(2m + 2n minus 4)

+(8)minus 12


(1) +(7)minus 12

(3m + 3n minus 8)

+(8)minus 12


17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n

+15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889 (2)


Mβve1 (VC[m n]) 216mn + 76m + 20n + 54


ABCve

(VC[m n]) 634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888

+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970 22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889

GAve

(VC[m n]) 12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

301113888

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

36⎞⎠

Hve

(VC[m n]) 13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(3a)

χve(Z) mn +

120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m +120

radic1113888

+223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic 1113889n +112

radic +216

radic +316

radic1113888

minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic minus1026

radic +122

radic minus630

radic +227

radic


minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic minus1035

radic +128

radic +3736

radic 1113889

Rve

(VC[m n]) 6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic +2121

radic1113888

minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic +2182

radic minus8208

radic

+2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic +1180

radic +37324

radic 1113889 (3b)



Mβve1 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857

(12) E11113868111386811138681113868

1113868111386811138681113868 +(16) E21113868111386811138681113868

1113868111386811138681113868 +(16) E31113868111386811138681113868

1113868111386811138681113868 +(20) E41113868111386811138681113868

1113868111386811138681113868 +(17) E51113868111386811138681113868

1113868111386811138681113868 +(21) E61113868111386811138681113868

1113868111386811138681113868

+(22) E71113868111386811138681113868

1113868111386811138681113868 +(23) E81113868111386811138681113868

1113868111386811138681113868 +(22) E91113868111386811138681113868

1113868111386811138681113868 +(25) E101113868111386811138681113868

1113868111386811138681113868 +(22) E111113868111386811138681113868

1113868111386811138681113868 +(26) E121113868111386811138681113868

1113868111386811138681113868 +(22) E131113868111386811138681113868

1113868111386811138681113868

+(30) E141113868111386811138681113868

1113868111386811138681113868 +(27) E151113868111386811138681113868

1113868111386811138681113868 +(29) E161113868111386811138681113868

1113868111386811138681113868 +(27) E171113868111386811138681113868

1113868111386811138681113868 +(28) E181113868111386811138681113868

1113868111386811138681113868

+(31) E191113868111386811138681113868

1113868111386811138681113868 +(32) E201113868111386811138681113868

1113868111386811138681113868 +(34) E211113868111386811138681113868

1113868111386811138681113868 +(35) E221113868111386811138681113868

1113868111386811138681113868

+(28) E231113868111386811138681113868

1113868111386811138681113868 +(36) E241113868111386811138681113868

1113868111386811138681113868

(12)(1) +(16)(2) +(16)(3) +(20)(m + n minus 4) +(17)(2) +(21)(2) +(22)(2)

+(23)(2m + 2n minus 10) +(22)(2) +(25)(2) +(22)(2) +(26)(2m + 2n minus 10)

+(22)(1) +(30)(m + n minus 6) +(27)(2) +(29)(2m + 2n minus 8) +(27)(2) +(28)(2)


+(28)(1) +(36)(6mn minus 15m minus 15n + 37)

216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1113944

uvisinE(VC[mn])


(32) E11113868111386811138681113868

1113868111386811138681113868 +(55) E21113868111386811138681113868

1113868111386811138681113868 +(60) E31113868111386811138681113868

1113868111386811138681113868 +(84) E41113868111386811138681113868

1113868111386811138681113868 +(72) E51113868111386811138681113868

1113868111386811138681113868 +(110) E61113868111386811138681113868

1113868111386811138681113868

+(168) E121113868111386811138681113868

1113868111386811138681113868 +(112) E131113868111386811138681113868

1113868111386811138681113868 +(121) E71113868111386811138681113868

1113868111386811138681113868 +(130) E81113868111386811138681113868

1113868111386811138681113868 +(117) E91113868111386811138681113868

1113868111386811138681113868

+(154) E101113868111386811138681113868

1113868111386811138681113868 +(120) E111113868111386811138681113868

1113868111386811138681113868 +(121) E141113868111386811138681113868

1113868111386811138681113868 +(182) E151113868111386811138681113868

1113868111386811138681113868 +(208) E161113868111386811138681113868

1113868111386811138681113868

+(176) E171113868111386811138681113868

1113868111386811138681113868 +(187) E181113868111386811138681113868

1113868111386811138681113868 +(238) E191113868111386811138681113868

1113868111386811138681113868 +(252) E201113868111386811138681113868

1113868111386811138681113868 +(288) E211113868111386811138681113868

1113868111386811138681113868


+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)


1944mn minus 51m minus 1887n + 1798 (4)


uvisinE(VC[mn])



1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888





+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























(5)minus 12

(1) +(6)minus 12

(2) +(7)minus 12


(2) +(7)minus 12

(2m + 2n minus 4)

+(8)minus 12


(1) +(7)minus 12

(3m + 3n minus 8)

+(8)minus 12


17

radic mn +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889m +17

radic +27

radic +28

radic +38

radic minus99

radic1113888 1113889n

+15

radic +46

radic minus47

radic minus68

radic minus88

radic +13

9

radic1113888 1113889 (2)


Mβve1 (VC[m n]) 216mn + 76m + 20n + 54


ABCve

(VC[m n]) 634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888

+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970 22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889

GAve

(VC[m n]) 12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

301113888

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

36⎞⎠

Hve

(VC[m n]) 13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(3a)

χve(Z) mn +

120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m +120

radic1113888

+223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic 1113889n +112

radic +216

radic +316

radic1113888

minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic minus1026

radic +122

radic minus630

radic +227

radic


minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic minus1035

radic +128

radic +3736

radic 1113889

Rve

(VC[m n]) 6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic +2121

radic1113888

minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic +2182

radic minus8208

radic

+2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic +1180

radic +37324

radic 1113889 (3b)



Mβve1 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857

(12) E11113868111386811138681113868

1113868111386811138681113868 +(16) E21113868111386811138681113868

1113868111386811138681113868 +(16) E31113868111386811138681113868

1113868111386811138681113868 +(20) E41113868111386811138681113868

1113868111386811138681113868 +(17) E51113868111386811138681113868

1113868111386811138681113868 +(21) E61113868111386811138681113868

1113868111386811138681113868

+(22) E71113868111386811138681113868

1113868111386811138681113868 +(23) E81113868111386811138681113868

1113868111386811138681113868 +(22) E91113868111386811138681113868

1113868111386811138681113868 +(25) E101113868111386811138681113868

1113868111386811138681113868 +(22) E111113868111386811138681113868

1113868111386811138681113868 +(26) E121113868111386811138681113868

1113868111386811138681113868 +(22) E131113868111386811138681113868

1113868111386811138681113868

+(30) E141113868111386811138681113868

1113868111386811138681113868 +(27) E151113868111386811138681113868

1113868111386811138681113868 +(29) E161113868111386811138681113868

1113868111386811138681113868 +(27) E171113868111386811138681113868

1113868111386811138681113868 +(28) E181113868111386811138681113868

1113868111386811138681113868

+(31) E191113868111386811138681113868

1113868111386811138681113868 +(32) E201113868111386811138681113868

1113868111386811138681113868 +(34) E211113868111386811138681113868

1113868111386811138681113868 +(35) E221113868111386811138681113868

1113868111386811138681113868

+(28) E231113868111386811138681113868

1113868111386811138681113868 +(36) E241113868111386811138681113868

1113868111386811138681113868

(12)(1) +(16)(2) +(16)(3) +(20)(m + n minus 4) +(17)(2) +(21)(2) +(22)(2)

+(23)(2m + 2n minus 10) +(22)(2) +(25)(2) +(22)(2) +(26)(2m + 2n minus 10)

+(22)(1) +(30)(m + n minus 6) +(27)(2) +(29)(2m + 2n minus 8) +(27)(2) +(28)(2)


+(28)(1) +(36)(6mn minus 15m minus 15n + 37)

216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1113944

uvisinE(VC[mn])


(32) E11113868111386811138681113868

1113868111386811138681113868 +(55) E21113868111386811138681113868

1113868111386811138681113868 +(60) E31113868111386811138681113868

1113868111386811138681113868 +(84) E41113868111386811138681113868

1113868111386811138681113868 +(72) E51113868111386811138681113868

1113868111386811138681113868 +(110) E61113868111386811138681113868

1113868111386811138681113868

+(168) E121113868111386811138681113868

1113868111386811138681113868 +(112) E131113868111386811138681113868

1113868111386811138681113868 +(121) E71113868111386811138681113868

1113868111386811138681113868 +(130) E81113868111386811138681113868

1113868111386811138681113868 +(117) E91113868111386811138681113868

1113868111386811138681113868

+(154) E101113868111386811138681113868

1113868111386811138681113868 +(120) E111113868111386811138681113868

1113868111386811138681113868 +(121) E141113868111386811138681113868

1113868111386811138681113868 +(182) E151113868111386811138681113868

1113868111386811138681113868 +(208) E161113868111386811138681113868

1113868111386811138681113868

+(176) E171113868111386811138681113868

1113868111386811138681113868 +(187) E181113868111386811138681113868

1113868111386811138681113868 +(238) E191113868111386811138681113868

1113868111386811138681113868 +(252) E201113868111386811138681113868

1113868111386811138681113868 +(288) E211113868111386811138681113868

1113868111386811138681113868


+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)


1944mn minus 51m minus 1887n + 1798 (4)


uvisinE(VC[mn])



1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888





+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic minus1035

radic +128

radic +3736

radic 1113889

Rve

(VC[m n]) 6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic +2121

radic1113888

minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic +2182

radic minus8208

radic

+2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic +1180

radic +37324

radic 1113889 (3b)



Mβve1 (VC[m n]) 1113944

uvisinE(VC[mn])

dve(u) + dve(v)( 1113857

(12) E11113868111386811138681113868

1113868111386811138681113868 +(16) E21113868111386811138681113868

1113868111386811138681113868 +(16) E31113868111386811138681113868

1113868111386811138681113868 +(20) E41113868111386811138681113868

1113868111386811138681113868 +(17) E51113868111386811138681113868

1113868111386811138681113868 +(21) E61113868111386811138681113868

1113868111386811138681113868

+(22) E71113868111386811138681113868

1113868111386811138681113868 +(23) E81113868111386811138681113868

1113868111386811138681113868 +(22) E91113868111386811138681113868

1113868111386811138681113868 +(25) E101113868111386811138681113868

1113868111386811138681113868 +(22) E111113868111386811138681113868

1113868111386811138681113868 +(26) E121113868111386811138681113868

1113868111386811138681113868 +(22) E131113868111386811138681113868

1113868111386811138681113868

+(30) E141113868111386811138681113868

1113868111386811138681113868 +(27) E151113868111386811138681113868

1113868111386811138681113868 +(29) E161113868111386811138681113868

1113868111386811138681113868 +(27) E171113868111386811138681113868

1113868111386811138681113868 +(28) E181113868111386811138681113868

1113868111386811138681113868

+(31) E191113868111386811138681113868

1113868111386811138681113868 +(32) E201113868111386811138681113868

1113868111386811138681113868 +(34) E211113868111386811138681113868

1113868111386811138681113868 +(35) E221113868111386811138681113868

1113868111386811138681113868

+(28) E231113868111386811138681113868

1113868111386811138681113868 +(36) E241113868111386811138681113868

1113868111386811138681113868

(12)(1) +(16)(2) +(16)(3) +(20)(m + n minus 4) +(17)(2) +(21)(2) +(22)(2)

+(23)(2m + 2n minus 10) +(22)(2) +(25)(2) +(22)(2) +(26)(2m + 2n minus 10)

+(22)(1) +(30)(m + n minus 6) +(27)(2) +(29)(2m + 2n minus 8) +(27)(2) +(28)(2)


+(28)(1) +(36)(6mn minus 15m minus 15n + 37)

216mn + 76m + 20n + 54

Mβve2 (VC[m n]) 1113944

uvisinE(VC[mn])


(32) E11113868111386811138681113868

1113868111386811138681113868 +(55) E21113868111386811138681113868

1113868111386811138681113868 +(60) E31113868111386811138681113868

1113868111386811138681113868 +(84) E41113868111386811138681113868

1113868111386811138681113868 +(72) E51113868111386811138681113868

1113868111386811138681113868 +(110) E61113868111386811138681113868

1113868111386811138681113868

+(168) E121113868111386811138681113868

1113868111386811138681113868 +(112) E131113868111386811138681113868

1113868111386811138681113868 +(121) E71113868111386811138681113868

1113868111386811138681113868 +(130) E81113868111386811138681113868

1113868111386811138681113868 +(117) E91113868111386811138681113868

1113868111386811138681113868

+(154) E101113868111386811138681113868

1113868111386811138681113868 +(120) E111113868111386811138681113868

1113868111386811138681113868 +(121) E141113868111386811138681113868

1113868111386811138681113868 +(182) E151113868111386811138681113868

1113868111386811138681113868 +(208) E161113868111386811138681113868

1113868111386811138681113868

+(176) E171113868111386811138681113868

1113868111386811138681113868 +(187) E181113868111386811138681113868

1113868111386811138681113868 +(238) E191113868111386811138681113868

1113868111386811138681113868 +(252) E201113868111386811138681113868

1113868111386811138681113868 +(288) E211113868111386811138681113868

1113868111386811138681113868


+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)


1944mn minus 51m minus 1887n + 1798 (4)


uvisinE(VC[mn])



1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888





+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























+(306) E221113868111386811138681113868

1113868111386811138681113868 +(180) E231113868111386811138681113868

1113868111386811138681113868 +(324) E241113868111386811138681113868

1113868111386811138681113868

(32)(1) +(55)(2) +(60)(3) +(84)(m + n minus 4) +(72)(2) +(110)(2) +(121)(2)

+(130)(2m + 2n minus 10) +(117)(2) +(154)(2) +(120)(2) +(168)(2m + 2n minus 10)

+(112)(1) +(121)(m + n minus 6) +(182)(2) +(208)(2m + 2n minus 8) +(176)(2)

+(187)(2) +(238)(2) +(252)(3m + 3n minus 14) +(288)(m + n minus 3)


1944mn minus 51m minus 1887n + 1798 (4)


uvisinE(VC[mn])



1113971

1032

1113970

1113888 1113889 E11113868111386811138681113868

1113868111386811138681113868 +

1455

1113970

1113888 1113889 E21113868111386811138681113868

1113868111386811138681113868 +

1460

1113970

1113888 1113889 E31113868111386811138681113868

1113868111386811138681113868 +

1884

1113970

1113888 1113889 E41113868111386811138681113868

1113868111386811138681113868 +

21130

1113970

1113888 1113889 E81113868111386811138681113868

1113868111386811138681113868

+

1572

1113970

1113888 1113889 E51113868111386811138681113868

1113868111386811138681113868 +

19110

1113970

1113888 1113889 E61113868111386811138681113868

1113868111386811138681113868 +

20121

1113970

1113888 1113889 E71113868111386811138681113868

1113868111386811138681113868 +

22117

1113970

1113888 1113889 E91113868111386811138681113868

1113868111386811138681113868

+

23154

1113970

1113888 1113889 E101113868111386811138681113868

1113868111386811138681113868 +

20120

1113970

1113888 1113889 E111113868111386811138681113868

1113868111386811138681113868 +

24168

1113970

1113888 1113889 E121113868111386811138681113868

1113868111386811138681113868 +

22112

1113970

1113888 1113889 E131113868111386811138681113868

1113868111386811138681113868

+

28121

1113970

1113888 1113889 E141113868111386811138681113868

1113868111386811138681113868 +

25182

1113970

1113888 1113889 E151113868111386811138681113868

1113868111386811138681113868 +

27208

1113970

1113888 1113889 E161113868111386811138681113868

1113868111386811138681113868 +

25176

1113970

1113888 1113889 E171113868111386811138681113868

1113868111386811138681113868

+

26187

1113970

1113888 1113889 E181113868111386811138681113868

1113868111386811138681113868 +

29238

1113970

1113888 1113889 E191113868111386811138681113868

1113868111386811138681113868 +

30252

1113970

1113888 1113889 E201113868111386811138681113868

1113868111386811138681113868 +

32288

1113970

1113888 1113889 E211113868111386811138681113868

1113868111386811138681113868

+

33306

1113970

1113888 1113889 E221113868111386811138681113868

1113868111386811138681113868 +

26180

1113970

1113888 1113889 E231113868111386811138681113868

1113868111386811138681113868 +

34324

1113970

1113888 1113889 E241113868111386811138681113868

1113868111386811138681113868

634324

1113970

1113888 1113889mn +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

+ 227208

1113970

+ 330252

1113970

1113888





+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























+

32288

1113970

+ 833306

1113970

minus 1534324

1113970

1113889m +

1884

1113970

+ 221130

1113970

+ 224168

1113970

+

28121

1113970

1113888

+ 227208

1113970

+ 330252

1113970

+

32288

1113970

+ 233306

1113970

minus 1534324

1113970

1113889n +

1032

1113970

+ 21455

1113970

1113888

+ 31460

1113970

minus 41884

1113970

+ 21572

1113970

+ 219110

1113970

+ 220121

1113970

minus 1021130

1113970

+ 222117

1113970

+ 223154

1113970

+ 220120

1113970

minus 1024168

1113970

+

22112

1113970

minus 628121

1113970

+ 225182

1113970

minus 827208

1113970

+ 225176

1113970

+ 226187

1113970

+ 229238

1113970

minus 1430252

1113970

minus 332288

1113970

minus 1033306

1113970

+

26180

1113970

+ 3734324

1113970

1113889 (5a)

GAve(VC[m n]) 1113944

uvisinE(VC[mn])


1113968

dve(u) + dve(v)( 1113857

2

32

radic

121113888 1113889 E1

11138681113868111386811138681113868111386811138681113868 +

255

radic

161113888 1113889 E2

11138681113868111386811138681113868111386811138681113868 +

260

radic

161113888 1113889 E3

11138681113868111386811138681113868111386811138681113868 +

284

radic

201113888 1113889 E4

11138681113868111386811138681113868111386811138681113868 +

272

radic

171113888 1113889 E5

11138681113868111386811138681113868111386811138681113868

+2

110

radic

211113888 1113889 E6

11138681113868111386811138681113868111386811138681113868 +

2121

radic

221113888 1113889 E7

11138681113868111386811138681113868111386811138681113868 +

2130

radic

231113888 1113889 E8

11138681113868111386811138681113868111386811138681113868 +

2117

radic

221113888 1113889 E9

11138681113868111386811138681113868111386811138681113868

+2

154

radic

251113888 1113889 E10

11138681113868111386811138681113868111386811138681113868 +

2120

radic

221113888 1113889 E11

11138681113868111386811138681113868111386811138681113868 +

2168

radic

261113888 1113889 E12

11138681113868111386811138681113868111386811138681113868 +

2112

radic

221113888 1113889 E13

11138681113868111386811138681113868111386811138681113868

+2

121

radic

301113888 1113889 E14

11138681113868111386811138681113868111386811138681113868 +

2182

radic

271113888 1113889 E15

11138681113868111386811138681113868111386811138681113868 +

2208

radic

291113888 1113889 E16

11138681113868111386811138681113868111386811138681113868 +

2176

radic

271113888 1113889 E17

11138681113868111386811138681113868111386811138681113868

+2

187

radic

281113888 1113889 E18

11138681113868111386811138681113868111386811138681113868 +

2238

radic

311113888 1113889 E19

11138681113868111386811138681113868111386811138681113868 +

2252

radic

321113888 1113889 E20

11138681113868111386811138681113868111386811138681113868 +

2288

radic

341113888 1113889 E21

11138681113868111386811138681113868111386811138681113868

+2

306

radic

351113888 1113889 E22

11138681113868111386811138681113868111386811138681113868 +

2180

radic

281113888 1113889 E23

11138681113868111386811138681113868111386811138681113868 +

2324

radic

361113888 1113889 E24

11138681113868111386811138681113868111386811138681113868

12

324

radic

36mn +

284

radic

20+4

130

radic

23+4

168

radic

26+2

121

radic

30+4

208

radic

29+6

252

radic

321113888

+2

288

radic

34+16

306

radic

35minus30

324

radic

361113889m +

284

radic

201113888 +

4130

radic

23+4

168

radic

26+2

121

radic

30

+4

208

radic

29+6

252

radic

32+2

288

radic

34+4

306

radic

35minus30

324

radic

361113889n +

232

radic

12+4

55

radic

161113888

+6

60

radic

16minus8

84

radic

20+4

72

radic

17+4

110

radic

21+4

121

radic

22minus20

130

radic

23+4

117

radic

22+4

154

radic

25

+4

120

radic

22minus20

168

radic

26+2

112

radic

22minus12

121

radic

30+4

182

radic

27minus16

208

radic

29+4

176

radic

27

+4

187

radic

28+4

238

radic

31minus28

252

radic

32minus6

288

radic

34minus20

306

radic

35+2

180

radic

28+74

324

radic

361113889

(5b)

Hve


2dve(u) + dve(v)

212

1113874 1113875 E11113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E21113868111386811138681113868

1113868111386811138681113868 +216

1113874 1113875 E31113868111386811138681113868

1113868111386811138681113868 +220

1113874 1113875 E41113868111386811138681113868

1113868111386811138681113868 +217

1113874 1113875 E51113868111386811138681113868

1113868111386811138681113868 +221

1113874 1113875 E61113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E71113868111386811138681113868

1113868111386811138681113868 +223

1113874 1113875 E81113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E91113868111386811138681113868

1113868111386811138681113868 +225

1113874 1113875 E101113868111386811138681113868

1113868111386811138681113868 +222

1113874 1113875 E111113868111386811138681113868

1113868111386811138681113868 +226

1113874 1113875 E121113868111386811138681113868

1113868111386811138681113868

+222

1113874 1113875 E131113868111386811138681113868

1113868111386811138681113868 +230

1113874 1113875 E141113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E151113868111386811138681113868

1113868111386811138681113868 +229

1113874 1113875 E161113868111386811138681113868

1113868111386811138681113868 +227

1113874 1113875 E171113868111386811138681113868

1113868111386811138681113868


+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























+228

1113874 1113875 E181113868111386811138681113868

1113868111386811138681113868 +231

1113874 1113875 E191113868111386811138681113868

1113868111386811138681113868 +232

1113874 1113875 E201113868111386811138681113868

1113868111386811138681113868 +234

1113874 1113875 E211113868111386811138681113868

1113868111386811138681113868 +235

1113874 1113875 E221113868111386811138681113868

1113868111386811138681113868

+228

1113874 1113875 E231113868111386811138681113868

1113868111386811138681113868 +236

1113874 1113875 E241113868111386811138681113868

1113868111386811138681113868

13

mn +1097490593929469607440

m +39532069247643760

n +801297510323950023374300

(6a)

χve(VC[m n]) 1113944

uvisinE(VC[mn])


(12)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(16)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(20)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(17)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(21)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(23)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E91113868111386811138681113868

1113868111386811138681113868 +(25)minus 12

E101113868111386811138681113868

1113868111386811138681113868

+(22)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(26)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(22)minus 12

E131113868111386811138681113868

1113868111386811138681113868 +(30)minus 12

E141113868111386811138681113868

1113868111386811138681113868

+(27)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(29)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(27)minus 12

E171113868111386811138681113868

1113868111386811138681113868 +(28)minus 12

E181113868111386811138681113868

1113868111386811138681113868

+(31)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(32)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(34)minus 12

E211113868111386811138681113868

1113868111386811138681113868 +(35)minus 12

E221113868111386811138681113868

1113868111386811138681113868

+(28)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(36)minus 12

E241113868111386811138681113868

1113868111386811138681113868

mn +120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +835

radic minus1536

radic1113888 1113889m

+120

radic +223

radic +226

radic +130

radic +229

radic +332

radic +134

radic +235

radic minus1536

radic1113888 1113889n

+112

radic +216

radic +316

radic minus420

radic +217

radic +221

radic +222

radic minus1023

radic +222

radic +225

radic1113888

minus1026

radic +122

radic minus630

radic +227

radic minus829

radic +227

radic +228

radic +231

radic minus1432

radic minus334

radic

minus1035

radic +128

radic +3736

radic 1113889

(6b)

Rve(VC[m n]) 1113944

uvisinE(VC[mn])


(32)minus 12

E11113868111386811138681113868

1113868111386811138681113868 +(55)minus 12

E21113868111386811138681113868

1113868111386811138681113868 +(60)minus 12

E31113868111386811138681113868

1113868111386811138681113868 +(84)minus 12

E41113868111386811138681113868

1113868111386811138681113868 +(72)minus 12

E51113868111386811138681113868

1113868111386811138681113868

+(110)minus 12

E61113868111386811138681113868

1113868111386811138681113868 +(121)minus 12

E71113868111386811138681113868

1113868111386811138681113868 +(130)minus 12

E81113868111386811138681113868

1113868111386811138681113868 +(117)minus 12

E91113868111386811138681113868

1113868111386811138681113868

+(154)minus 12

E101113868111386811138681113868

1113868111386811138681113868 +(120)minus 12

E111113868111386811138681113868

1113868111386811138681113868 +(168)minus 12

E121113868111386811138681113868

1113868111386811138681113868 +(112)minus 12

E131113868111386811138681113868

1113868111386811138681113868

+(121)minus 12

E141113868111386811138681113868

1113868111386811138681113868 +(182)minus 12

E151113868111386811138681113868

1113868111386811138681113868 +(208)minus 12

E161113868111386811138681113868

1113868111386811138681113868 +(176)minus 12

E171113868111386811138681113868

1113868111386811138681113868

+(187)minus 12

E181113868111386811138681113868

1113868111386811138681113868 +(238)minus 12

E191113868111386811138681113868

1113868111386811138681113868 +(252)minus 12

E201113868111386811138681113868

1113868111386811138681113868 +(288)minus 12

E211113868111386811138681113868

1113868111386811138681113868

+(306)minus 12

E221113868111386811138681113868

1113868111386811138681113868 +(180)minus 12

E231113868111386811138681113868

1113868111386811138681113868 +(324)minus 12

E241113868111386811138681113868

1113868111386811138681113868

6324

radic mn +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic +1288

radic1113888

+8306

radic minus15324

radic 1113889m +184

radic +2130

radic +2168

radic +1221

radic +2208

radic +3252

radic1113888

+1288

radic +2306

radic minus15324

radic 1113889n +132

radic +255

radic +360

radic minus472

radic +2110

radic1113888

+2121

radic minus484

radic minus10130

radic +2117

radic +2154

radic minus10168

radic +1112

radic +6221

radic

+2182

radic minus8208

radic +2176

radic +2187

radic +2238

radic minus14252

radic minus3288

radic minus10300

radic

+1180

radic +37324

radic 1113889

(7)







βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn





























βve1 (H) M

βve2 (H)

[1 1] 92 0133 2 1324 2110 1804[2 2] 1026 1070 4 2732 2854 5698[3 3] 2932 2763 6 4788 4030 13 480[4 4] 5810 5212 6 7492 5638 25150[5 5] 9660 8417 7 10 844 7678 40 708[6 6] 14 482 12378 6 14 844 10150 60154[7 7] 20 276 17095 4 19 492 13 054 83 488[8 8] 27 042 22568 2 24 788 16 390 110 710[9 9] 34 780 28796 9 30 732 20158 141 820[10 10] 43 490 35781 6 37 324 24 358 176 818



[1 1] 4707 4 10857 0 1708 8 1855 0 1031 9[2 2] 13289 6 32034 6 3240 9 6122 7 2722 3[3 3] 25759 0 65212 2 5439 6 12390 3 5079 3[4 4] 42115 7 110389 8 8305 0 20657 9 8103 0[5 5] 62359 7 167567 4 11837 0 30925 5 11793 3[6 6] 86491 0 236745 0 16035 7 43193 2 16150 3[7 7] 114509 6 317922 6 20901 1 57460 8 21174 0[8 8] 146415 5 411100 2 26433 2 73728 4 26864 3[9 9] 182208 7 516277 9 32631 9 919961 33221 3[10 10] 221889 3 633455 5 39497 3 112263 7 40245 0

14000

12000

10000

8000

6000

4000

2000

Mev

1 2 3 4 5 6 5 4 3 2 1

mn

(a)

20

15

10

5

Rev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(b)



16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























16000

14000

12000

10000

80001

6000

4000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1

mn

(a)

100000

80000

60000

40000

20000

Mβev

1 2 3 4 5 6 7 8 7 6 5 4 3 2 1n m

2

(b)


2 (VC[m n])

200

150

100

50

ABCv

e

1 2 3 4 5 6 7 8 910 9 8 7 6 5 4 3 2 1

mn

(a)

600

500

400

300

200

100

GAve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)


353025201510

5

Hve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(a)

100

80

60

40

20

xve

1 2 3 4 5 6 7 8 9 10 9 8 7 6 5 4 3 2 1

mn

(b)



5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























5 Conclusion


Data Availability


Disclosure




References
















70

60

50

40

30

20

10

Rve

2 4 6 8 10 12 14 12 10 8 6 4 2

mn
























Research Article Study of Vanadium Carbide ... - Hindawi.com

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Transcript of Research Article Study of Vanadium Carbide ... - Hindawi.com