9780521873079

Geometry of Chemical Graphs: Polycycles and Two-faced Maps

by
  • ISBN13:

    9780521873079

  • ISBN10:

    052187307X

  • Edition: 1st
  • Format: Hardcover
  • Copyright: 2008-07-14
  • Publisher: Cambridge University Press
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Summary

Polycycles and symmetric polyhedra appear as generalizations of graphs in the modeling of molecular structures, such as the Nobel prizewinning fullerenes, occurring in chemistry and crystallography. Chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules.

Table of Contents

Prefacep. ix
Introductionp. 1
Graphsp. 1
Topological notionsp. 2
Representation of mapsp. 9
Symmetry groups of mapsp. 12
Types of regularity of mapsp. 18
Operations on mapsp. 21
Two-faced mapsp. 24
The Goldberg-Coxeter constructionp. 28
Description of the classesp. 31
Computer generation of the classesp. 36
Fullerenes as tilings of surfacesp. 38
Classification of finite fullerenesp. 38
Toroidal and Klein bottle fullerenesp. 39
Projective fullerenesp. 41
Plane 3-fullerenesp. 42
Polycyclesp. 43
(r, q)-polycyclesp. 43
Examplesp. 45
Cell-homomorphism and structure of (r, q)-polycyclesp. 48
Angles and curvaturep. 51
Polycycles on surfacesp. 53
Polycycles with given boundaryp. 56
The problem of uniqueness of (r, q)-fillingsp. 56
(r, 3)-filling algorithmsp. 61
Symmetries of polycyclesp. 64
Automorphism group of (r, q)-polycyclesp. 64
Isohedral and isogonal (r, q)-polycyclesp. 65
Isohedral and isogonal (r, q)[subscript gen]-polyclclesp. 71
Elementary polycyclesp. 73
Decomposition of polycyclesp. 73
Parabolic and hyperbolic elementary (R, q)[subscript gen]-polycyclesp. 76
Kernel-elementary polycyclesp. 79
Classification of elementary ({2, 3, 4, 5}, 3)[subscript gen]-polycyclesp. 89
Classification of elementary ({2, 3}, 4)[subscript gen]-polycyclesp. 89
Classification of elementary ({2, 3}, 5)[subscript gen]-polycyclesp. 90
Appendix 1: 204 sporadic elementary ({2, 3, 4, 5}, 3)-polycyclesp. 93
Appendix 2: 57 sporadic elementary ({2, 3}, 5)-polycyclesp. 102
Applications of elementary decompositions to (r, q)-polycyclesp. 107
Extremal polycyclesp. 108
Non-extensible polycyclesp. 116
2-embeddable polycyclesp. 121
Strictly face-regular spheres and torip. 125
Strictly face-regular spheresp. 126
Non-polyhedral strictly face-regular ({a, b}, k)-spheresp. 136
Strictly face-regular ({a, b}, k)-planesp. 143
Parabolic weakly face-regular spheresp. 168
Face-regular ({2, 6}, 3)-spheresp. 168
Face-regular ({3, 6}, 3)-spheresp. 169
Face-regular ({4, 6}, 3)-spheresp. 169
Face-regular ({5, 6}, 3)-spheres (fullerenes)p. 170
Face-regular ({3, 4}, 4)-spheresp. 177
Face-regular ({2, 3}, 6)-spheresp. 179
General properties of 3-valent face-regular mapsp. 181
General ({a, b}, 3)-mapsp. 184
Remaining questionsp. 186
Spheres and tori that are aR[subscript i]p. 187
Maps aR[subscript 0]p. 187
Maps 4R[subscript 1]p. 189
Maps 4R[subscript 2]p. 195
Maps 5R[subscript 2]p. 203
Maps 5R[subscript 3]p. 204
Frank-Kasper spheres and torip. 218
Euler formula for ({a, b}, 3)-maps bR[subscript 0]p. 218
The major skeleton, elementary polycycles, and classification resultsp. 219
Spheres and tori that are bR[subscript 1]p. 225
Euler formula for ({a, b}, 3)-maps bR[subscript 1]p. 225
Elementary polycyclesp. 229
Spheres and tori that are bR[subscript 2]p. 234
({a, b}, 3)-maps bR[subscript 2]p. 234
({5, b}, 3)-tori bR[subscript 2]p. 237
({a, b}, 3)-spheres with a cycle of b-gonsp. 239
Spheres and tori that are bR[subscript 3]p. 246
Classification of ({4, b}, 3)-maps bR[subscript 3]p. 246
({5, b}, 3)-maps bR[subscript 3]p. 252
Spheres and tori that are bR[subscript 4]p. 256
({4, b}, 3)-maps bR[subscript 4]p. 256
({5, b}, 3)-maps bR[subscript 4]p. 270
Spheres and tori that are bR[subscript j] for j [greater than or equal] 5p. 274
Maps bR[subscript 5]p. 274
Maps bR[subscript b]p. 281
Icosahedral fulleroidsp. 284
Construction of I-fulleroids and infinite seriesp. 285
Restrictions on the p-vectorsp. 288
From the p-vectors to the structuresp. 291
Referencesp. 295
Indexp. 304
Table of Contents provided by Ingram. All Rights Reserved.

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