Capra, leapfrog and other related operations on maps Mircea V. Diudea Faculty of Chemistry and Chemical Engineering Babes-Bolyai Bolyai University 484 Cluj, ROMANIA diudea@chem.ubbcluj.ro 1
Contents Basic operations in a map Composite operations Capra operation Lattices with negative curvature π-electronic implications References 2
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,, 174-194. 194. L. Euler, Comment. Acad. Sci. I. Petropolitanae, 1736, 8,, 128-14 14 3
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,, 174-194. 194. 4
Stellation St C 36 :15 1 (C 2h ) St (C 36 :15) (C 36 1. P. W. Fowler and D. E. Manolopolous, An atlas of fullerenes,, Oxford University Press, Oxford, U.K., 1995. 5
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,, 174-194. 194. 6
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., 1995. 7
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,, 174-194. 194. 8
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., 1995. 9
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,, 174-194. 194. 1
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., 1995. 11
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, 444. 12
Leapfrog Le Le (Cube) Le (Le (Cube)) 13
Leapfrog Le C 3 :1 ( (C s ) Le (C 3 :1) = C 9 :1 ( (C s ) 14
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,, 24. 15
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
Leapfrog Le - electronic implications Leapfrog rule 1 LER : ( PC ) N Le Le = 6 + 6 6m ; (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, 143-145 145. 17
Composite operations Leapfrog............. M St(M) Du(St(M)) 18
Composite operations Leapfrog: bounding polygon has s = 2 2d (a) (b) 19
Leapfrog of a 4-valent 4 net TUC 4 [8,4] TURC 4 C 8 [8,4] 2
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
Leapfrog Le TUZ[8,4] Le (TUZ[8,4]) = TUA[8,9] 22
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, 57-85. 23
Leapfrog of planar benzenoids C 16 Le (C Le (C 16 ) 24
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,, 153-159. 159. 25
Dual of Stellation of Medial M Me(M) Du(St(Me(M))) 26
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,, 41-8. 27
Quadrupling Q (M )........ M. V. Diudea, P. E. John, A. Graovac, M. Primorac, and T. Pisanski, Croat. Chem. Acta, 23, 76, 153-159. 28
Quadrupling Q (M ) Q (Cube)=Chamfering Chamfering 1 Q (Q (Cube)) 1. M. Goldberg, Tôhoku Math. J., 1934, 4, 226-236 29
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
Quadrupling of planar benzenoids C 16 Q (C 16 ) 31
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, 3-213 32
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
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, 14-18. 34
Capra Ca (M)............ E2(M) Pe(E2(M)) Trr(Pe(E2(M))) M. V. Diudea, Studia Univ. Babes-Bolyai Bolyai,, 23, 48, 3-213 35
Capra Ca (M) E2 (Cube) Pe (E2 (Cube)) 36
Capra Ca (M) Trr(Pe(E2(Cube))) (Cube))) Ca (Cube) Schlegel project. 37
Capra Ca (M) Ca (Dodecahedron) = C 14 Ca (Icosahedron) = C 132 38
A racemic pair of Ca -transformed TUZ[8,3 [8,3] 39
Two successive Ca-operations CaS (CaS (Cube)) CaR (CaS (Cube)) 4
Capra Ca (M) Negative curvature lattices Ca(Cube) [7] Ca(Cube) [7] (optimized) 41
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,, 87-1. 42
Positive and Negative Curvature Lattices Ca (Ca (Tetrahedron)); N = 196 Ca (Ca (Tetrahedron) [7] ); N = 232 43
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
θ POAV1 Strain p Energy Angle (deg) Deviation θ p SE 1 2 3 (deg) (deg) (kcal/mol) C 3,6,6 Ca (Ca (Tetrahedron)) 18.17 18.78 6.6 27.967 31.692 61.189 18.9 18.26 6. 27.961 31.687 61.171 18.71 18.17 59.995 27.972 31.695 61.22 average 27.967 31.691 61.188 C 7,6,6 Ca (Ca (Tetrahedron) [7] ) 117.475 125.63 114.694.923 5.529 1.862 118.227 117.678 12.442 1.218 6.378 2.478 114.883 117.99 126.795.111 1.96.221 114.15 116.541 129.236.39 1.133.78 117.824 117.818 122.78.76 5.27 1.539 116.736 119.16 123.226.293 3.112.59 114.422 114.92 13.22.218 2.664.432 average.59 3.678 1.29 45
Capra of planar benzenoids Ca (C 6 ) Ca 2 (C 6 ) 46
VARIOUS NANOTUBES TUAC 6 [8,16]V1,2 Geodesic projection 47
VARIOUS NANOTUBES TUZC 6 [16,8]H1,2 Geodesic projection 48
VARIOUS NANOTUBES RC 4 C 8 [16,8] Geodesic projection 49
VARIOUS NANOTUBES SC 4 C 8 [16,8] Geodesic projection 5
VARIOUS NANOTUBES SC 5 C 7 [16,8] Geodesic projection 51
VARIOUS NANOTUBES HC 5 C 7 [16,8] Geodesic projection 52
VARIOUS NANOTUBES VC 5 C 7 [16,8] Geodesic projection 53
VARIOUS NANOTUBES VAC 5 C 7 [16,8] Geodesic projection 54
VARIOUS NANOTUBES HAC 5 C 7 [16,8] Geodesic projection 55
VARIOUS NANOTUBES VAC 5 C 6 C 7 [16,8] Geodesic projection 56
VARIOUS NANOTUBES HAC 5 C 6 C 7 [16,8] Geodesic projection 57
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
Capra of VARIOUS NANOTUBES Ca (ZC 6 [16,8]); N=832 Ca (AC 6 [8,16]); N=832 59
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
VARIOUS NANOTUBES Le (HAC 5 C 6 C 7 [16,8]); N=328 Q (HAC 5 C 7 [16,8]); N=44 61
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
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] 32.75 -.75 OP 2 Ca(Tuz[8,4]) 192.273.17 -.17 -.273.34 PC 3 Tuz[1,4] 4.95.95 -.95 -.95.19 PC 4 Ca(Tuz[1,4]) 24.64.64 -.64 -.64.129 PC 5 Tua[4,8] 32.532 -.532 OP 6 Ca(Tua[4,8]) 192.265.24 -.24 -.265.48 PC 7 Tua[4,1] 4.31.169 -.169 -.31.338 PC 8 Ca(Tua[4,1]) 248.132.95 -.95 -.132.189 PC 9 Z[8,1] 8.414.414 -.414 -.414.828 PC 1 Ca(Z[8,1]) 56.169.169 -.169 -.169.337 PC 11 A[8,12] 96 M 12 Ca(A[8,12]) 672 M 63
Spectral data of various types of open nanotubes TUBE HOMO LUMO GAP E π x+ x x- SHELL 1 TUH[16,8] 1.527 63 2 63 OP 2 HC 5 C 7 [16,8].139.139 1.49 67 1 6 OP 3 HAC 5 C 7 [16,8].191.191 1.487 64 3 61 PSC 4 HAC 5 C 6 C 7 [16,8] 1.517 63 2 63 OP 5 RC 4 C 8 [16,8] 1.425 62 4 62 M 6 SC 4 C 8 [16,8].63 -.63.126 1.446 64 64 PC 7 SC 5 C 7 [16,8].134.68.66 1.497 65 1 62 PSC 8 TUV[8,16].19 -.19.217 1.537 64 64 PC 9 VC 5 C 7 [16,8].244.153.91 1.474 67 1 6 PSC 1 VAC 5 C 7 [16,8].147.128.19 1.477 65 1 62 PSC 11 VAC 5 C 6 C 7 [16,8].174.138.36 1.495 65 63 PSC 64
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.81 -.81.163 1.548 168 168 PC 2 HC 5 C 7 [16,8] Le.211.23.9 1.519 168 16 PSC 3 HAC 5 C 7 [16,8 ] Le.23.158.45 1.519 168 16 PSC 4 HAC 5 C 6 C 7 [16,8] Le.183.17.14 1.53 167 161 PSC 5 RC 4 C 8 [16,8] Le 1.464 152 16 152 M 6 SC 4 C 8 [16,8] Le 1.464 152 16 152 M 7 SC 5 C 7 [16,8] Le.77.27.5 1.531 169 167 PSC 8 TUV[8,16] Le.18 -.18.217 1.547 168 168 PC 9 VC 5 C 7 [16,8] Le.256.21.55 1.57 157 151 PSC 1 VAC 5 C 7 [16,8] Le.23.124.79 1.56 156 152 PSC 11 VAC 5 C 6 C 7 [16,8] Le.239.147.92 1.514 156 152 PSC 65
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 1.547 223 2 223 OP 2 HC 5 C 7 [16,8]Q.115.115 1.528 227 213 OP 3 HAC 5 C 7 [16,8 ]Q.99.99 1.53 223 1 216 OP 4 HAC 5 C 6 C 7 [16,8]Q.3.3 1.538 221 219 OP 5 RC 4 C 8 [16,8]Q 1.486 213 6 213 M 6 SC 4 C 8 [16,8]Q 1.55 221 6 221 M 7 SC 5 C 7 [16,8]Q.55.55 1.536 224 1 223 PSC 8 TUV[8,16]Q.78 -.78.156 1.553 224 224 PC 9 VC 5 C 7 [16,8]Q.82.82 1.517 28 1 23 OP 1 VAC 5 C 7 [16,8]Q.13.13 1.52 28 24 OP 11 VAC 5 C 6 C 7 [16,8]Q.78.78 1.527 28 24 OP 66
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 1.557 416 416 OP/PC 2 HC 5 C 7 [16,8]Ca -.6 -.16.11 1.547 411 413 MC 3 HAC 5 C 7 [16,8 ]Ca.75.75 1.547 416 48 OP 4 HAC 5 C 6 C 7 [16,8]Ca.77.71.7 1.551 416 48 PSC 5 RC 4 C 8 [16,8]Ca.7 -.7.14 1.528 48 48 PC 6 SC 4 C 8 [16,8]Ca.9 -.9.17 1.536 416 416 PC 7 SC 5 C 7 [16,8]Ca.1 -.12.113 1.55 416 416 PC 8 TUV[8,16]Ca.57 -.57.114 1.558 416 416 PC 9 VC 5 C 7 [16,8]Ca.92.92 1.539 4 396 OP 1 VAC 5 C 7 [16,8]Ca.82.79.3 1.539 4 396 PSC 11 VAC 5 C 6 C 7 [16,8]Ca.17.17 1.543 399 397 OP 67
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 3-1 -1-1 OP Ca(M) [7].154.154.154 -.614 OP CaS(CaS(M) [7] ) -.5 -.5 -.5 -.36 OP 4 CaR(CaS(M) [7] ) -.1 -.1 -.1 -.325 OP 4 2 M = Cube 1 1-1 -1 2 PC Ca(M) [7] -.188 OP CaS(CaS(M) [7] ) -.1 -.1 -.127 -.127.117 MC 6 CaR(CaS(M) [7] ) -.4 -.4 -.141 -.141.137 MC 6 3 M = Dodecahedron 1 OP Ca(M) [7] -.165 OP CaS(CaS(M) [7] ) -.69 -.69 -.69 -.131 OP 6 CaR(CaS(M) [7] ) -.7 -.7 -.7 -.138 OP 6 4 M = Octahedron -2 OP Ca(M) [7].222 OP CaS(CaS(M) [7] ).8 -.44 -.44 -.44 OP 2 CaR(CaS(M) [7] ).21 -.58 -.58 -.58 OP 2 5 M = Icosahedron -1-1 -1-1 OP Ca(M) [7].11.11.11.11 OP CaS(CaS(M) [7] ) -.22 -.22 -.22 -.22 OP 6 CaR(CaS(M) [7] ) -.29 -.29 -.29 -.29 OP 6 68
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, 529-532 69
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
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), 3-22 71
Negative curvature lattices O1CaM(7) C-Ca1-7-Ca2 Ca2 = M; N = 464 72
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), 35-46 73
Negative curvature lattices CUM; N = 176 LCUMj 74
Negative curvature lattices UCUM; N = 14 LUCUMj 75
Negative curvature lattices CUCUM; N = 14 LCUCUMj 76
Negative curvature lattices SUM; N = 152 LSUMi 77
Negative curvature lattices SCUCUM; N = 8 LSCUCUMi 78
Negative curvature lattices PLSUM; N = 2 PLSCUCUM; N = 14 79
Negative curvature lattices SCUM; N = 152 LSCUMi 8
Negative curvature lattices PLSCUM; N = 224 O2CaM(8) 81
Negative curvature lattices PLSCUMO=O1QM; N = 56 DYCK GRAPH LPLSCUMOj 82
Negative curvature lattices DYCK TESSELLATION (8,3) 12 + 6 OCTAGONS KLEIN TESSELLATION (7,3) 24 HEPTAGONS + 6 OCTAGONS 83
Negative curvature lattices QM; N = 32 LEM; N = 24 84
Negative curvature lattices UO1LEM; N = 72 CUO1LEM; N = 42 85
Negative curvature lattices LUO1LEMi LCUO1LEMj 86
Negative curvature lattices TUM; N = 88 LTUM 87
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. 23 88
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
Peanut dimers; topology Me(C 14 14 ); ); 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
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
References 1. Diudea, M. V.; Graovac, A.; Generation and graph-theoretical properties of C4-tori. MATCH -Commun. Math. Comput. Chem., 21, 44, 93-12 2. Diudea, M. V.; Silaghi-Dumitrescu, I.; Parv, B. Toranes versus torenes. MATCH -Commun. Math. Comput. Chem., 21, 44, 117-133 3. Diudea, M. V.; John, P. E. Covering polyhedral tori. MATCH -Commun. Math. Comput. Chem., 21, 44, 13-116 4. Diudea, M. V.; Kirby, E. C. The energetic stability of tori and ssingle-wall tubes. Fullerene Sci. Technol. 21, 9, 445-465. 5. Diudea, M. V. Graphenes from 4-valent tori. Bull. Chem. Soc. Japan, 22, 75, 487-492. 92
References 6. Diudea, M. V.; Silaghi-Dumitrescu, I.; Pârv, Toroidal fullerenes. Ann. West Univ.Timisoara, 21, 1, 21-4 7. Diudea, M. V.; Silaghi-Dumitrescu, I.; Pârv, B. Toroidal fullerenes from square tiled tori. Internet Electronic Journal of Molecular Design. 22, 1, 1-22. 8. Diudea, M. V. Hosoya polynomial in tori. MATCH -Commun. Math. Comput. Chem., 22, 45, 19-122. 9. Diudea, M. V. Phenylenic and naphthylenic tori. Fullerenes, Nanotubes Carbon Nanostruct., 22, 1, 273-292. 1. Diudea, M. V.; Parv, B.; Kirby, E. C. Azulenic tori. Commun. Math. Comput. Chem. (MATCH), 23, 47, 53-7. 11. Diudea M. V., Periodic 4,7 cages. Bul. Stiint. Univ. Baia Mare Ser. B, 22, 18, 31-38 93
References 12. Diudea, M. V. Topology of naphthylenic tori. PCCP, 22, 4, 474-4746. 13. 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, 23-36 17. Diudea, M. V.; Nagy, C. L.; Ursu, O.; Balaban, T. S., C6 dimers revisited. Fullerenes, Nanotubes Carbon Nanostruct., 23, 11, 245-255. 94
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, 421 4214. 2. 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, 3-1. 23. Diudea, M.V.; Ursu, O., Parv, B., Hex tubes from square tubes, Studia Univ. Babes-Bolyai, 23, 48, 11-2. 95
References 24. Diudea, M.V.; Silaghi-Dumitrescu, I., Small fulleroids, Studia Univ. Babes-Bolyai, 23, 48, 21-3. 25. 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, 31-4. 27. 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, 153-159. 29. Diudea, M. V., Nanotube covering modification, Studia Univ. Babes-Bolyai, 23, 48, -. 3. Diudea, M. V., Capra- a Leapfrog related map operation, Studia Univ. Babes-Bolyai, 23, 48, -. 96
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
TOPO GROUP CLUJ, ROMANIA, EUROPE 98
TOPO GROUP CLUJ, ROMANIA, EUROPE 99
TOPO GROUP CLUJ, ROMANIA, EUROPE 1
TOPO GROUP CLUJ, ROMANIA, EUROPE 11