Geometry of Chemical Graphs: Polycycles and Two-faced Maps (Encyclopedia of Mathematics and its Applications, Vol. 119)

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. 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 Cover 1 About 2 Encyclopedia of mathematics and its applications 3 Geometry of Chemical Graphs: Polycycles and Two-faced Maps 4 Copyright 5 052187307X 5 Contents 6 Preface 10 1 Introduction 12 1.1 Graphs 12 1.2 Topological notions 13 1.3 Representation of maps 20 1.4 Symmetry groups of maps 23 1.5 Types of regularity of maps 29 1.6 Operations on maps 32 2 Two-faced maps 35 2.1 The Goldberg-Coxeter construction 39 2.2 Description of the classes 42 2.3 Computer generation of the classes 47 3 Fullerenes as tilings of surfaces 49 3.1 Classification of finite fullerenes 49 3.2 Toroidal and Klein bottle fullerenes 50 3.3 Projective fullerenes 52 3.4 Plane 3-fullerenes 53 4 Polycycles 54 4.1 (r, q)-polycycles 54 4.2 Examples 56 4.3 Cell-homomorphism and structure of (r, q)-polycycles 59 4.4 Angles and curvature 62 4.5 Polycycles on surfaces 64 5 Polycycles with given boundary 67 5.1 The problem of uniqueness of (r, q)-fillings 67 5.2 (r, 3)-filling algorithms 72 6 Symmetries of polycycles 75 6.1 Automorphism group of (r, q)-polycycles 75 6.2 Isohedral and isogonal (r, q)-polycycles 76 6.3 Isohedral and isogonal (r, q)_{gen}-polycycles 82 7 Elementary polycycles 84 7.1 Decomposition of polycycles 84 7.2 Parabolic and hyperbolic elementary (R, q)_{gen}-polycycles 87 7.3 Kernel-elementary polycycles 90 7.4 Classification of elementary ({2, 3, 4, 5}, 3)_{gen}-polycycles 94 7.5 Classification of elementary ({2, 3}, 4)_{gen}-polycycles 100 7.6 Classification of elementary ({2, 3}, 5)_{gen}-polycycles 101 7.7 Appendix 1: 204 sporadic elementary ({2, 3, 4, 5}, 3)-polycycles 104 7.8 Appendix 2: 57 sporadic elementary ({2, 3}, 5)-polycycles 113 8 Applications of elementary decompositions to (r, q)-polycycles 118 8.1 Extremal polycycles 119 8.2 Non-extensible polycycles 127 8.3 2-embeddable polycycles 132 9 Strictly face-regular spheres and tori 136 9.1 Strictly face-regular spheres 137 9.2 Non-polyhedral strictly face-regular ({a, b}, k)-spheres 147 9.3 Strictly face-regular ({a, b}, k)-planes 154 10 Parabolic weakly face-regular spheres 179 10.1 Face-regular ({2, 6}, 3)-spheres 179 10.2 Face-regular ({3, 6}, 3)-spheres 180 10.3 Face-regular ({4, 6}, 3)-spheres 180 10.4 Face-regular ({5, 6}, 3)-spheres (fullerenes) 181 10.5 Face-regular ({3, 4}, 4)-spheres 188 10.6 Face-regular ({2, 3}, 6)-spheres 190 11 General properties of 3-valent face-regular maps 192 11.1 General ({a, b}, 3)-maps 195 11.2 Remaining questions 197 12 Spheres and tori that are aR_i 198 12.1 Maps aR_0 198 12.2 Maps 4R_1 200 12.3 Maps 4R_2 206 12.4 Maps 5R_2 214 12.5 Maps 5R_3 215 13 Frank-Kasper spheres and tori 229 13.1 Euler formula for ({a, b}, 3)-maps bR_0 229 13.2 The major skeleton, elementary polycycles, and classification results 230 14 Spheres and tori that are bR_1 236 14.1 Euler formula for ({a, b}, 3)-maps bR_1 236 14.2 Elementary polycycles 240 15 Spheres and tori that are bR_2 245 15.1 ({a, b}, 3)-maps bR_2 245 15.2 ({5, b}, 3)-tori bR_2 248 15.3 ({a, b}, 3)-spheres with a cycle of b-gons 250 16 Spheres and tori that are bR_3 257 16.1 Classification of ({4, b}, 3)-maps bR_3 257 16.2 ({5, b}, 3)-maps bR_3 263 17 Spheres and tori that are bR_4 267 17.1 ({4, b}, 3)-maps bR_4 267 17.2 ({5, b}, 3)-maps bR_4 281 18 Spheres and tori that are bR_j for j≥5 285 18.1 Maps bR_5 285 18.2 Maps bR_6 292 19 Icosahedral fulleroids 295 19.1 Construction of I -fulleroids and infinite series 296 19.2 Restrictions on the p-vectors 299 19.3 From the p-vectors to the structures 302 References 306 Index 315 9780521873079 Cambridge University Press 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. Organised 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.--jacket. 1. Introduction -- 2. Two-faced Maps -- 3. Fullerenes As Tilings Of Surfaces -- 4. Polycycles -- 5. Polycycles With Given Boundary -- 6. Symmetries Of Polycycles -- 7. Elementary Polycycles -- 8. Applications Of Elementary Decompositions To (r,q)-polycycles -- 9. Strictly Face-regular Spheres And Tori -- 10. Parabolic Weakly Face-regular Spheres -- 11. General Properties Of 3-valent Face-regular Maps -- 12. Spheres And Tori That Are Ar[subscript I] -- 13. Frank-kasper Spheres And Tori -- 14. Spheres And Tori That Are Br[subscript 1] -- 15. Spheres And Tori That Are Br[subscript 2] -- 16. Spheres And Tori That Are Br[subscript 3] -- 17. Spheres And Tori That Are Br[subscript 4] -- 18. Spheres And Tori That Are Br[subscript J] For J[greater Than Or Equal To]5 -- 19. Icosahedral Fulleroids. Michel Deza, Mathieu Dutour Sikirić. Includes Bibliography (p. 295-303) And Index. Polycycles and symmetric polyhedra appear as generalisations of graphs in the modelling 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. Organised 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.