Polycycles and symmetric polyhedra appear as generalizations of graphs in the modeling of molecular structures, such as the Nobel prize winning fullerenes, occurring in chemistry and crystallography. The chemistry has inspired and informed many interesting questions in mathematics and computer science, which in turn have suggested directions for synthesis of molecules. Here the authors give access to new results in the theory of polycycles and two-faced maps together with the relevant background material and mathematical tools for their study. Organized so that, after reading the introductory chapter, each chapter can be read independently from the others, the book should be accessible to researchers and students in graph theory, discrete geometry, and combinatorics, as well as to those in more applied areas such as mathematical chemistry and crystallography. Many of the results in the subject require the use of computer enumeration; the corresponding programs are available from the author's website.
Author(s): Michel Deza, Mathieu Dutour Sikiric
Edition: 1
Year: 2008
Language: English
Pages: 221
Cover......Page 1
About......Page 2
Encyclopedia of mathematics and its applications......Page 3
Geometry of Chemical Graphs: Polycycles and Two-faced Maps......Page 4
052187307X......Page 5
Contents......Page 6
Preface......Page 10
1.1 Graphs ......Page 12
1.2 Topological notions ......Page 13
1.3 Representation of maps ......Page 20
1.4 Symmetry groups of maps ......Page 23
1.5 Types of regularity of maps ......Page 29
1.6 Operations on maps ......Page 32
2 Two-faced maps ......Page 35
2.1 The Goldberg-Coxeter construction ......Page 39
2.2 Description of the classes ......Page 42
2.3 Computer generation of the classes ......Page 47
3.1 Classification of finite fullerenes ......Page 49
3.2 Toroidal and Klein bottle fullerenes ......Page 50
3.3 Projective fullerenes ......Page 52
3.4 Plane 3-fullerenes ......Page 53
4.1 (r, q)-polycycles ......Page 54
4.2 Examples ......Page 56
4.3 Cell-homomorphism and structure of (r, q)-polycycles ......Page 59
4.4 Angles and curvature ......Page 62
4.5 Polycycles on surfaces ......Page 64
5.1 The problem of uniqueness of (r, q)-fillings ......Page 67
5.2 (r, 3)-filling algorithms ......Page 72
6.1 Automorphism group of (r, q)-polycycles ......Page 75
6.2 Isohedral and isogonal (r, q)-polycycles ......Page 76
6.3 Isohedral and isogonal (r, q)_{gen}-polycycles......Page 82
7.1 Decomposition of polycycles ......Page 84
7.2 Parabolic and hyperbolic elementary (R, q)_{gen}-polycycles......Page 87
7.3 Kernel-elementary polycycles ......Page 90
7.4 Classification of elementary ({2, 3, 4, 5}, 3)_{gen}-polycycles......Page 94
7.5 Classification of elementary ({2, 3}, 4)_{gen}-polycycles......Page 100
7.6 Classification of elementary ({2, 3}, 5)_{gen}-polycycles......Page 101
7.7 Appendix 1: 204 sporadic elementary ({2, 3, 4, 5}, 3)-polycycles ......Page 104
7.8 Appendix 2: 57 sporadic elementary ({2, 3}, 5)-polycycles ......Page 113
8 Applications of elementary decompositions to (r, q)-polycycles ......Page 118
8.1 Extremal polycycles ......Page 119
8.2 Non-extensible polycycles ......Page 127
8.3 2-embeddable polycycles ......Page 132
9 Strictly face-regular spheres and tori ......Page 136
9.1 Strictly face-regular spheres ......Page 137
9.2 Non-polyhedral strictly face-regular ({a, b}, k)-spheres ......Page 147
9.3 Strictly face-regular ({a, b}, k)-planes ......Page 154
10.1 Face-regular ({2, 6}, 3)-spheres ......Page 179
10.3 Face-regular ({4, 6}, 3)-spheres ......Page 180
10.4 Face-regular ({5, 6}, 3)-spheres (fullerenes) ......Page 181
10.5 Face-regular ({3, 4}, 4)-spheres ......Page 188
10.6 Face-regular ({2, 3}, 6)-spheres ......Page 190
11 General properties of 3-valent face-regular maps ......Page 192
11.1 General ({a, b}, 3)-maps ......Page 195
11.2 Remaining questions ......Page 197
12.1 Maps aR_0......Page 198
12.2 Maps 4R_1......Page 200
12.3 Maps 4R_2......Page 206
12.4 Maps 5R_2......Page 214
12.5 Maps 5R_3......Page 215
13.1 Euler formula for ({a, b}, 3)-maps bR_0......Page 229
13.2 The major skeleton, elementary polycycles, and classification results ......Page 230
14.1 Euler formula for ({a, b}, 3)-maps bR_1......Page 236
14.2 Elementary polycycles ......Page 240
15.1 ({a, b}, 3)-maps bR_2......Page 245
15.2 ({5, b}, 3)-tori bR_2......Page 248
15.3 ({a, b}, 3)-spheres with a cycle of b-gons ......Page 250
16.1 Classification of ({4, b}, 3)-maps bR_3......Page 257
16.2 ({5, b}, 3)-maps bR_3......Page 263
17.1 ({4, b}, 3)-maps bR_4......Page 267
17.2 ({5, b}, 3)-maps bR_4......Page 281
18.1 Maps bR_5......Page 285
18.2 Maps bR_6......Page 292
19 Icosahedral fulleroids ......Page 295
19.1 Construction of I -fulleroids and infinite series ......Page 296
19.2 Restrictions on the p-vectors ......Page 299
19.3 From the p-vectors to the structures ......Page 302
References ......Page 306
Index ......Page 315
