Geometry of Chemical Graphs: Polycycles and Two-faced Maps

by Michel Deza , Mathieu Dutour Sikirić

• ISBN13: 9780521873079
• ISBN10: 052187307X
• Edition: 1st
• Format: Hardcover
• Copyright: 2008-07-14
• Publisher: Cambridge University Press

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

Preface	p. ix
Introduction	p. 1
Graphs	p. 1
Topological notions	p. 2
Representation of maps	p. 9
Symmetry groups of maps	p. 12
Types of regularity of maps	p. 18
Operations on maps	p. 21
Two-faced maps	p. 24
The Goldberg-Coxeter construction	p. 28
Description of the classes	p. 31
Computer generation of the classes	p. 36
Fullerenes as tilings of surfaces	p. 38
Classification of finite fullerenes	p. 38
Toroidal and Klein bottle fullerenes	p. 39
Projective fullerenes	p. 41
Plane 3-fullerenes	p. 42
Polycycles	p. 43
(r, q)-polycycles	p. 43
Examples	p. 45
Cell-homomorphism and structure of (r, q)-polycycles	p. 48
Angles and curvature	p. 51
Polycycles on surfaces	p. 53
Polycycles with given boundary	p. 56
The problem of uniqueness of (r, q)-fillings	p. 56
(r, 3)-filling algorithms	p. 61
Symmetries of polycycles	p. 64
Automorphism group of (r, q)-polycycles	p. 64
Isohedral and isogonal (r, q)-polycycles	p. 65
Isohedral and isogonal (r, q)[subscript gen]-polyclcles	p. 71
Elementary polycycles	p. 73
Decomposition of polycycles	p. 73
Parabolic and hyperbolic elementary (R, q)[subscript gen]-polycycles	p. 76
Kernel-elementary polycycles	p. 79
Classification of elementary ({2, 3, 4, 5}, 3)[subscript gen]-polycycles	p. 89
Classification of elementary ({2, 3}, 4)[subscript gen]-polycycles	p. 89
Classification of elementary ({2, 3}, 5)[subscript gen]-polycycles	p. 90
Appendix 1: 204 sporadic elementary ({2, 3, 4, 5}, 3)-polycycles	p. 93
Appendix 2: 57 sporadic elementary ({2, 3}, 5)-polycycles	p. 102
Applications of elementary decompositions to (r, q)-polycycles	p. 107
Extremal polycycles	p. 108
Non-extensible polycycles	p. 116
2-embeddable polycycles	p. 121
Strictly face-regular spheres and tori	p. 125
Strictly face-regular spheres	p. 126
Non-polyhedral strictly face-regular ({a, b}, k)-spheres	p. 136
Strictly face-regular ({a, b}, k)-planes	p. 143
Parabolic weakly face-regular spheres	p. 168
Face-regular ({2, 6}, 3)-spheres	p. 168
Face-regular ({3, 6}, 3)-spheres	p. 169
Face-regular ({4, 6}, 3)-spheres	p. 169
Face-regular ({5, 6}, 3)-spheres (fullerenes)	p. 170
Face-regular ({3, 4}, 4)-spheres	p. 177
Face-regular ({2, 3}, 6)-spheres	p. 179
General properties of 3-valent face-regular maps	p. 181
General ({a, b}, 3)-maps	p. 184
Remaining questions	p. 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 tori	p. 218
Euler formula for ({a, b}, 3)-maps bR[subscript 0]	p. 218
The major skeleton, elementary polycycles, and classification results	p. 219
Spheres and tori that are bR[subscript 1]	p. 225
Euler formula for ({a, b}, 3)-maps bR[subscript 1]	p. 225
Elementary polycycles	p. 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-gons	p. 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] 5	p. 274
Maps bR[subscript 5]	p. 274
Maps bR[subscript b]	p. 281
Icosahedral fulleroids	p. 284
Construction of I-fulleroids and infinite series	p. 285
Restrictions on the p-vectors	p. 288
From the p-vectors to the structures	p. 291
References	p. 295
Index	p. 304
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