Capra, leapfrog and other related operations on maps

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1 Capra, leapfrog and other related operations on maps Mircea V. Diudea Faculty of Chemistry and Chemical Engineering Babes-Bolyai Bolyai University 484 Cluj, ROMANIA 1

2 Contents Basic operations in a map Composite operations Capra operation Lattices with negative curvature π-electronic implications References 2

3 Relations in a map * A map M is a combinatorial representation of a closed surface. 1 The graph associated to the map is called regular if all its vertices have the same degree. * Basic relations in M : Σdv d = 2e Σsf s = 2e v - e + f = χ(m)= 2(1- g) Euler 2 T. Pisanski, and M. Randić,, in Geometry at Work,, M. A. A. Notes, 2, 53,, L. Euler, Comment. Acad. Sci. I. Petropolitanae, 1736, 8,,

4 Basic operations in a map Stellation 1 St (capping, triangulation): add a new vertex in the center of a face and connect it with each boundary vertex. St (M ): v = v + f e = 3 3e f = 2 2e 1. T. Pisanski, and M. Randić,, in Geometry at Work,, M. A. A. Notes, 2, 53,,

5 Stellation St C 36 :15 1 (C 2h ) St (C 36 :15) (C P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes,, Oxford University Press, Oxford, U.K.,

6 Basic operations in a map Dual 1 Du (Poincaré dual ): locate a point in the center of each face and join two such points if their corresponding faces share a common edge. Du (Du (M)) = M Du (M ): v = f e = e f = v 1. T. Pisanski, and M. Randić,, in Geometry at Work,, M. A. A. Notes, 2, 53,,

7 Dual Du C 4 :39 1 (D 5d ) Du (C 4 :39) (C 4 1. P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes,, Oxford University Press, Oxford, U.K.,

8 Basic operations in a map Medial 1 Me : put the new vertices as the midpoints of the original edges. Join two vertices if and only if the original edges span an angle. Me (M)) = Me (Du (M)) (a 4-valent 4 graph) Me(M ): v = e e = 2 2e f = f + v 1. T. Pisanski, and M. Randić,, in Geometry at Work,, M. A. A. Notes, 2, 53,,

9 Medial Me C 5 :271 1 (C s ) Me (C 5 :271) (C 5 1. P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes,, Oxford University Press, Oxford, U.K.,

10 Basic operations in a map Truncation 1 Tr : cut of the neighborhood of each vertex by a plane close to the vertex, such that it intersects each edge incident to the vertex. Truncation is similar to Me (M) Tr (M ): v = d v e = 3 3e f = f + v 1. T. Pisanski, and M. Randić,, in Geometry at Work,, M. A. A. Notes, 2, 53,,

11 Truncation Tr Icosahedron (I h ) Tr (Icosahedron)=C6 (PC PC) )=C 6 1. P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes,, Oxford University Press, Oxford, U.K.,

12 Composite operations Leapfrog 1 Le : Le (M ) = Du (St (M )) = Tr (Du (M )) Le(M ): ): v = d v e = 3 = 3e f = f + v It rotates the parent faces by π/s 1. P. W. Fowler, Phys. Lett., 1986, 131,

13 Leapfrog Le Le (Cube) Le (Le (Cube)) 13

14 Leapfrog Le C 3 :1 ( (C s ) Le (C 3 :1) = C 9 :1 ( (C s ) 14

15 Leapfrog Le - electronic implications In simple Hückel theory, 1 the energy of the i th π -molecular orbital is calculated on the grounds of A(G ) E i = α + βλ i E HOMO E LUMO = gap Study of eigenvalue spectra provided some rules of thumb for the stability of fullerenes. 1. E. Hückel, Z. Phys.,, 1931, 7,,

16 π -Electronic Structure Relation GAP shell symbol 1 λ N/2 > λ N/2+1 properly closed PC 2 λ N/2 > λ N/2+1 > pseudo closed PSC 3 λ N/2 > λ N/2+1 meta closed MC 4 λ N/2 = λ N/2+1 open OP 16

17 Leapfrog Le - electronic implications Leapfrog rule 1 LER : ( PC ) N Le Le = m ; (m = 3(2 + 2m) 1) In a-tubulenesa C 12k,k-V[2 V[2k,n]-[6] [6] ; ( PC ) N Le = 12k + 2 2k 3m m =, 1, 2,,, ( (k = 4 to 7) 1. P. W. Fowler and J. I. Steer, J. Chem. Soc., Chem. Commun., 1987,

18 Composite operations Leapfrog M St(M) Du(St(M)) 18

19 Composite operations Leapfrog: bounding polygon has s = 2 2d (a) (b) 19

20 Leapfrog of a 4-valent 4 net TUC 4 [8,4] TURC 4 C 8 [8,4] 2

21 Leapfrog of a 4-valent 4 net TUHRC 4 [8,4] = Me (TUC 4 [8,4]) Le (TUHRC 4 [8,4] )=TUVSC 4 C 8 [8,12] 21

22 Leapfrog Le TUZ[8,4] Le (TUZ[8,4]) = TUA[8,9] 22

23 Schlegel version 1 of Le (M) Dodecahedron = C 2 Le (Dodecahedron)= C 6 1. J. R. Dias, From benzenoid hydrocarbons to fullerene carbons. MATCH Commun. Math. Chem. Comput. 1996, 33,

24 Leapfrog of planar benzenoids C 16 Le (C Le (C 16 ) 24

25 Composite operations Dual of Stellation of Medial = DSM (M ) 1 DSM (M ) = Du (St (Me (M ))) = Le (Me (M))) DSM (M ): v = 2 = 2d v = 4 4e (d = = 4; s = 4) e = 6 6e f = f + e + v It involves two rotations by π/s = no rotation 1. M. V. Diudea, P. E. John, A. Graovac, M. Primorac, and T. Pisanski, ski, Croat. Chem. Acta,, 23, 76,,

26 Dual of Stellation of Medial M Me(M) Du(St(Me(M))) 26

27 Composite operations Quadrupling Q (M ) Chamfering 1 Q (M ) = Du (Str (Me (M ))) Q (M ): v = ( = (d +1)v e = 4 4e f = f + e It involves two rotations by π/s = no rotation 1. A. Deza, M. Deza and V. P. Grishukhin, Discrete Math., 1998, 192,,

28 Quadrupling Q (M ) M. V. Diudea, P. E. John, A. Graovac, M. Primorac, and T. Pisanski, Croat. Chem. Acta, 23, 76,

29 Quadrupling Q (M ) Q (Cube)=Chamfering Chamfering 1 Q (Q (Cube)) 1. M. Goldberg, Tôhoku Math. J., 1934, 4,

30 Quadrupling Q (M ) C 2 Q (C 2 ) 1 = C 8 (OP OP) 1. P. W. Fowler, J. E. Cremona, and J. I. Steer, Theor. Chim. Acta, 1988, 73, 1 3

31 Quadrupling of planar benzenoids C 16 Q (C 16 ) 31

32 Capra Ca (M) Ca (M ) 1 Romanian Leapfrog Ca (M ) = Trr (Pe (E2 (M ))) Ca(M ): v = (2 = (2d +1)v =v + 2 2e + s f e = 7 7e = 3 3e + 2 2s f f = (s +1)f It involves rotation by π/2s of the parent faces 1. M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 23, 48,

33 Capra Ca (M) Ca n (M ); Iterative operation v n = 8 8v n -1 7v n -2; n 2 v n = 7 n v ; d = 3 e n = 7 n e = 7 n 3v /2 f n = f + (7 n -1) 1) v /2 33

34 Capra Ca (M) Goldberg 1 relation: m = ( (a 2 + ab + b 2 ); a b ; a + b > Le : (1, 1); m = 3 Q : (2, ); m = 4 Ca : (2, 1); m = 7 1. M. Goldberg, Tohoku Math. J., 1937, 43,

35 Capra Ca (M) E2(M) Pe(E2(M)) Trr(Pe(E2(M))) M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 23, 48,

36 Capra Ca (M) E2 (Cube) Pe (E2 (Cube)) 36

37 Capra Ca (M) Trr(Pe(E2(Cube))) (Cube))) Ca (Cube) Schlegel project. 37

38 Capra Ca (M) Ca (Dodecahedron) = C 14 Ca (Icosahedron) = C

39 A racemic pair of Ca -transformed TUZ[8,3 [8,3] 39

40 Two successive Ca-operations CaS (CaS (Cube)) CaR (CaS (Cube)) 4

41 Capra Ca (M) Negative curvature lattices Ca(Cube) [7] Ca(Cube) [7] (optimized) 41

42 Negative and Positive Curvature Lattices Ca (C (C 2 ) [7] [7] ; N = 2 C 26 (I h ) Fowler 1 1. A. Dress and G. Brinkmann, MATCH- Commun. Math. Comput. Chem., 1996, 33,,

43 Positive and Negative Curvature Lattices Ca (Ca (Tetrahedron)); N = 196 Ca (Ca (Tetrahedron) [7] ); N =

44 POAV Strain Energy In the POAV1 theory 1,2 the π-orbital axis vector makes equal angles to the three σ-bonds of the sp 2 carbon: θ p = θ σπ - 9 o pyramidalization angle SE = 2(θ p ) 2 strain energy 12 - (1/3) Σθij deviation to planarity 1. R.C. Haddon, J. Am. Chem. Soc., 112, 3385 (199). 2. R.C. Haddon, J. Phys. Chem. A, 15, 4164 (21). 44

45 θ POAV1 Strain p Energy Angle (deg) Deviation θ p SE (deg) (deg) (kcal/mol) C 3,6,6 Ca (Ca (Tetrahedron)) average C 7,6,6 Ca (Ca (Tetrahedron) [7] ) average

46 Capra of planar benzenoids Ca (C 6 ) Ca 2 (C 6 ) 46

47 VARIOUS NANOTUBES TUAC 6 [8,16]V1,2 Geodesic projection 47

48 VARIOUS NANOTUBES TUZC 6 [16,8]H1,2 Geodesic projection 48

49 VARIOUS NANOTUBES RC 4 C 8 [16,8] Geodesic projection 49

50 VARIOUS NANOTUBES SC 4 C 8 [16,8] Geodesic projection 5

51 VARIOUS NANOTUBES SC 5 C 7 [16,8] Geodesic projection 51

52 VARIOUS NANOTUBES HC 5 C 7 [16,8] Geodesic projection 52

53 VARIOUS NANOTUBES VC 5 C 7 [16,8] Geodesic projection 53

54 VARIOUS NANOTUBES VAC 5 C 7 [16,8] Geodesic projection 54

55 VARIOUS NANOTUBES HAC 5 C 7 [16,8] Geodesic projection 55

56 VARIOUS NANOTUBES VAC 5 C 6 C 7 [16,8] Geodesic projection 56

57 VARIOUS NANOTUBES HAC 5 C 6 C 7 [16,8] Geodesic projection 57

58 VARIOUS TORI A C 6 like toroidal object HAC 5 C 6 C 7 [12,12]; N=144 HAC 5 C 7 [12,12]; N=144 58

59 Capra of VARIOUS NANOTUBES Ca (ZC 6 [16,8]); N=832 Ca (AC 6 [8,16]); N=832 59

60 Capra of VARIOUS NANOTUBES Ca (HAC 5 C 6 C 7 [16,8]); N=824 Ca (HAC 5 C 7 [16,8]) ; N=824 6

61 VARIOUS NANOTUBES Le (HAC 5 C 6 C 7 [16,8]); N=328 Q (HAC 5 C 7 [16,8]); N=44 61

62 Spectral implications of Ca operation Ca operation, applied to a finite structure, leaves unchanged its π-electronic shells. There exist exceptions, the most notably being the transformed Ca (Tuz/a[c,n]) of nanotubes, which all have PC shell disregarding the character of their parent shell. 62

63 Spectral data of open nanotubes Tuz/a[c,n] and tori Z/A[c,n] ] and their Ca -transforms STRUCTURE N HOMO -1 HOMO LUMO LUMO +1 Gap Shell 1 Tuz[8,4] OP 2 Ca(Tuz[8,4]) PC 3 Tuz[1,4] PC 4 Ca(Tuz[1,4]) PC 5 Tua[4,8] OP 6 Ca(Tua[4,8]) PC 7 Tua[4,1] PC 8 Ca(Tua[4,1]) PC 9 Z[8,1] PC 1 Ca(Z[8,1]) PC 11 A[8,12] 96 M 12 Ca(A[8,12]) 672 M 63

64 Spectral data of various types of open nanotubes TUBE HOMO LUMO GAP E π x+ x x- SHELL 1 TUH[16,8] OP 2 HC 5 C 7 [16,8] OP 3 HAC 5 C 7 [16,8] PSC 4 HAC 5 C 6 C 7 [16,8] OP 5 RC 4 C 8 [16,8] M 6 SC 4 C 8 [16,8] PC 7 SC 5 C 7 [16,8] PSC 8 TUV[8,16] PC 9 VC 5 C 7 [16,8] PSC 1 VAC 5 C 7 [16,8] PSC 11 VAC 5 C 6 C 7 [16,8] PSC 64

65 Spectral data of the Le transforms of various types of open nanotubes TUBE HOMO LUMO GAP E π x+ x x- SHELL 1 TUH[16,8] Le PC 2 HC 5 C 7 [16,8] Le PSC 3 HAC 5 C 7 [16,8 ] Le PSC 4 HAC 5 C 6 C 7 [16,8] Le PSC 5 RC 4 C 8 [16,8] Le M 6 SC 4 C 8 [16,8] Le M 7 SC 5 C 7 [16,8] Le PSC 8 TUV[8,16] Le PC 9 VC 5 C 7 [16,8] Le PSC 1 VAC 5 C 7 [16,8] Le PSC 11 VAC 5 C 6 C 7 [16,8] Le PSC 65

66 Spectral data of the Q transforms of various types of open nanotubes TUBE HOMO LUMO GAP E π x+ x x- SHELL 1 TUH[16,8]Q OP 2 HC 5 C 7 [16,8]Q OP 3 HAC 5 C 7 [16,8 ]Q OP 4 HAC 5 C 6 C 7 [16,8]Q OP 5 RC 4 C 8 [16,8]Q M 6 SC 4 C 8 [16,8]Q M 7 SC 5 C 7 [16,8]Q PSC 8 TUV[8,16]Q PC 9 VC 5 C 7 [16,8]Q OP 1 VAC 5 C 7 [16,8]Q OP 11 VAC 5 C 6 C 7 [16,8]Q OP 66

67 Spectral data of the Ca transforms of various types of open nanotubes TUBE HOMO LUMO GAP E π x+ x x- SHELL 1 TUH[16,8]Ca OP/PC 2 HC 5 C 7 [16,8]Ca MC 3 HAC 5 C 7 [16,8 ]Ca OP 4 HAC 5 C 6 C 7 [16,8]Ca PSC 5 RC 4 C 8 [16,8]Ca PC 6 SC 4 C 8 [16,8]Ca PC 7 SC 5 C 7 [16,8]Ca PC 8 TUV[8,16]Ca PC 9 VC 5 C 7 [16,8]Ca OP 1 VAC 5 C 7 [16,8]Ca PSC 11 VAC 5 C 6 C 7 [16,8]Ca OP 67

68 Spectral data of the Platonic polyhedra and their Ca [7] -transforms Ca [7] Structure HOMO -1 HOMO LUMO LUMO +1 Gap Shell Ex. e 1 M = Tetrahedron OP Ca(M) [7] OP CaS(CaS(M) [7] ) OP 4 CaR(CaS(M) [7] ) OP 4 2 M = Cube PC Ca(M) [7] OP CaS(CaS(M) [7] ) MC 6 CaR(CaS(M) [7] ) MC 6 3 M = Dodecahedron 1 OP Ca(M) [7] OP CaS(CaS(M) [7] ) OP 6 CaR(CaS(M) [7] ) OP 6 4 M = Octahedron -2 OP Ca(M) [7].222 OP CaS(CaS(M) [7] ) OP 2 CaR(CaS(M) [7] ) OP 2 5 M = Icosahedron OP Ca(M) [7] OP CaS(CaS(M) [7] ) OP 6 CaR(CaS(M) [7] ) OP 6 68

69 Negative curvature lattices GAUSS-BONET Theorem - relates the geometric curvature to the topology Euler characteristic κda = 2πχ χ S ( S ) = v e + f v - e + f = 2(1- g) ( S ) g = ( e v + 2) / 2 = f / 2 O. Bonnet, C. R. Acad. Sci. Paris, 1853, 37,

70 Negative curvature lattices The genus of Ca (M) [7] objects is calculable as: χ(m) [7] =v 1[7] - e 1[7] + 1[7] f = 2(1 - g)=v - e g = ( (e - v + 2)/2 = f /2 (spherical character of the parent polyhedron involves: v - e + f = 2) Lattices with g > 1 will have negative χ(m) [7] and consequently negative curvature. For the five Platonic solids, the genus of the corresponding Ca (M) [7] is: 2 (Tetrahedron); 3 (Cube); 4 (Octahedron); 6 (Dodecahedron) and 1 (Icosahedron( Icosahedron). 7

71 Negative curvature lattices Ca 2 (Cube) [7], P2D; N=176 Ca 2 (Cube) [7], P3D; ; N=3424 Diudea, M. V., Capra-a leapfrog related operation on maps, Studia Univ. Babes- Bolyai, 23, 48 (2),

72 Negative curvature lattices O1CaM(7) C-Ca1-7-Ca2 Ca2 = M; N =

73 Negative curvature lattices UM; N = 176 LUMj Nagy, Cs. L., Diudea, M. V., Carbon allotropes with negative curvature, Studia Univ. Babes-Bolyai, 23, 48 (2),

74 Negative curvature lattices CUM; N = 176 LCUMj 74

75 Negative curvature lattices UCUM; N = 14 LUCUMj 75

76 Negative curvature lattices CUCUM; N = 14 LCUCUMj 76

77 Negative curvature lattices SUM; N = 152 LSUMi 77

78 Negative curvature lattices SCUCUM; N = 8 LSCUCUMi 78

79 Negative curvature lattices PLSUM; N = 2 PLSCUCUM; N = 14 79

80 Negative curvature lattices SCUM; N = 152 LSCUMi 8

81 Negative curvature lattices PLSCUM; N = 224 O2CaM(8) 81

82 Negative curvature lattices PLSCUMO=O1QM; N = 56 DYCK GRAPH LPLSCUMOj 82

Negative curvature lattices DYCK TESSELLATION (8,3) 12 + 6

83 Negative curvature lattices DYCK TESSELLATION (8,3) OCTAGONS KLEIN TESSELLATION (7,3) 24 HEPTAGONS + 6 OCTAGONS 83

84 Negative curvature lattices QM; N = 32 LEM; N = 24 84

85 Negative curvature lattices UO1LEM; N = 72 CUO1LEM; N = 42 85

86 Negative curvature lattices LUO1LEMi LCUO1LEMj 86

87 Negative curvature lattices TUM; N = 88 LTUM 87

88 SOFTWARE TOPOCLUJ 2. - Calculations in MOLECULAR TOPOLOGY M. V. Diudea, O. Ursu and Cs. L. Nagy, B-B B B Univ. 22 CageVersatile 1.1 Operations on maps M. Stefu and M. V. Diudea, B-B B Univ

89 Peanut dimers; topology 1 C 2(7-5),5 5),5-H[1,1] H[1,1]-[7] [7] = C 14 (C 1 ) vertex orbits: 4{1}; 5{2} 1. Cs. L. Nagy, M. Stefu; M. V. Diudea and A. Dress, A. Mueler, C 7 Dimers - energetics and topology, Croat. Chem. Acta,, 23 (accepted). 89

90 Peanut dimers; topology Me(C ); ); edge orbits: 9{1}; 6{2} Du(C 14 ); face orbits: 9{1}; 6{2} [5]{2}; 2{1}; [6] 4{1}; [7] {1} 9

91 Conclusions A new operation on maps, called Capra Ca, was proposed and discussed in comparison with the well-known Leapfrog Le and Quadrupling Q operations. Ca-operation insulates each parent face by its own hexagons (i.e.,., coronene-like substructures), in contrast to Le and Q. The transformed constitutive parameters were given. The utility of this operation is in building of large cages that preserve the symmetry and spectral properties of the parent structures and in extremely facile access to several constructions with negative curvature. Clearly, many other authors have used such a transformation but no paper, in our best knowledge, has been devoted so far. 91

92 References 1. Diudea, M. V.; Graovac, A.; Generation and graph-theoretical properties of C4-tori. MATCH -Commun. Math. Comput. Chem., 21, 44, Diudea, M. V.; Silaghi-Dumitrescu, I.; Parv, B. Toranes versus torenes. MATCH -Commun. Math. Comput. Chem., 21, 44, Diudea, M. V.; John, P. E. Covering polyhedral tori. MATCH -Commun. Math. Comput. Chem., 21, 44, Diudea, M. V.; Kirby, E. C. The energetic stability of tori and ssingle-wall tubes. Fullerene Sci. Technol. 21, 9, Diudea, M. V. Graphenes from 4-valent tori. Bull. Chem. Soc. Japan, 22, 75,

93 References 6. Diudea, M. V.; Silaghi-Dumitrescu, I.; Pârv, Toroidal fullerenes. Ann. West Univ.Timisoara, 21, 1, Diudea, M. V.; Silaghi-Dumitrescu, I.; Pârv, B. Toroidal fullerenes from square tiled tori. Internet Electronic Journal of Molecular Design. 22, 1, Diudea, M. V. Hosoya polynomial in tori. MATCH -Commun. Math. Comput. Chem., 22, 45, Diudea, M. V. Phenylenic and naphthylenic tori. Fullerenes, Nanotubes Carbon Nanostruct., 22, 1, Diudea, M. V.; Parv, B.; Kirby, E. C. Azulenic tori. Commun. Math. Comput. Chem. (MATCH), 23, 47, Diudea M. V., Periodic 4,7 cages. Bul. Stiint. Univ. Baia Mare Ser. B, 22, 18,

94 References 12. Diudea, M. V. Topology of naphthylenic tori. PCCP, 22, 4, John, E. P.; Diudea, M. V., Wiener index of zig-zag polyhex nanotubes, Croat. Chem. Acta, 23, (in press). 14. Diudea, M. V.; Stefu, M.; Parv, B.; John, P. J., Wiener index of armchair polyhex nanotubes Croat. Chem. Acta, 23, (in press). 15. Diudea, M. V.; Fowler, P. W. π-electronic structure of polyhex tori originating in square tori. PCCP, 23 (submitted). 16. Diudea, M. V.; Parv, B.; John, E. P.; Ursu, O.; Graovac, A., Distance counting in tori. MATCH Commun. Math. Comput. Chem., 23, 49, Diudea, M. V.; Nagy, C. L.; Ursu, O.; Balaban, T. S., C6 dimers revisited. Fullerenes, Nanotubes Carbon Nanostruct., 23, 11,

95 References 18. Nagy, C. L.; Stefu, M.; Diudea, M. V.; Dress, A., Muller,A., Structure and energetics of C7 dimers, Croat. Chem. Acta, 23, (accepted). 19. Diudea, M. V.; Balaban, T. S.; Kirby, E.C.; Graovac, A. Energetics and topology of polyhex nanotubes. Phys. Chem.,Chem.Phys., 23, 5, Diudea, M.V., Periodic fulleroids, Int. J. Nanostruct., 23, (in press). 21. Diudea, M.V., Silaghi-Dumitrescu, I.; Graovac, A. Periodic cages. Croat. Chem. Acta, 23, (submitted). 22. Diudea, M.V.; Parv, B.; Ursu, O., Hex tori from square tori, Studia Univ. Babes-Bolyai, 23, 48, Diudea, M.V.; Ursu, O., Parv, B., Hex tubes from square tubes, Studia Univ. Babes-Bolyai, 23, 48,

96 References 24. Diudea, M.V.; Silaghi-Dumitrescu, I., Small fulleroids, Studia Univ. Babes-Bolyai, 23, 48, Stefu, M.; Diudea, M.V.; Wiener index of C4C8 nanotubes. MATCH - Commun. Math. Comput. Chem., 23 (in press) 26. M.V. Diudea, The zig-zag cylinder rule, Studia Univ. Babes- Bolyai, 23, 48, M.V. Diudea, Stability of tubulenes, Phys. Chem., Chem.Phys., 23, (submitted). 28. Diudea, M. V.; John, P. E.; Graovac, A.; Primorac, M., Pisanski, T. Leapfrog and Related Operations on Toroidal Fullerenes. Croat. Chem. Acta, 23, 76, Diudea, M. V., Nanotube covering modification, Studia Univ. Babes-Bolyai, 23, 48, Diudea, M. V., Capra- a Leapfrog related map operation, Studia Univ. Babes-Bolyai, 23, 48, -. 96

97 TOPO GROUP CLUJ, ROMANIA Mircea V. Diudea Gabriel Katona Oleg Ursu Csaba L. Nagy Monica Stefu Crina V. Veres Mihaela Caprioara Cristina D. Moldovan Ioana Florea 97

98 TOPO GROUP CLUJ, ROMANIA, EUROPE 98

99 TOPO GROUP CLUJ, ROMANIA, EUROPE 99

100 TOPO GROUP CLUJ, ROMANIA, EUROPE 1

101 TOPO GROUP CLUJ, ROMANIA, EUROPE 11

THE GENERALIZED ZAGREB INDEX OF CAPRA-DESIGNED PLANAR BENZENOID SERIES Ca k (C 6 )

Open J Math Sci, Vol 1(2017), No 1, pp 44-51 ISSN 2523-0212 Website: http://wwwopenmathsciencecom THE GENERALIZED ZAGREB INDEX OF CAPRA-DESIGNED PLANAR BENZENOID SERIES Ca k (C 6 ) MUHAMMAD S SARDAR, SOHAIL