Geometric structure of chemistry-relevant graphs. Zigzags and central circuits.

Abstract

The central theme of the present book is zigzags and central-circuits of three- or four-regular plane graphs, which allow a double covering or covering of the edgeset to be obtained. The book presents zigzag and central circuit structures of geometric fullerenes and several other classes of graph of interest in the fields of chemistry and mathematics. It also discusses the symmetries, parameterization and the Goldberg?Coxeter construction for those graphs. It is the first book on this subject, presenting full structure theory of such graphs. While many previous publications only addressed particular questions about selected graphs, this book is based on numerous computations and presents extensive data (tables and figures), as well as algorithmic and computational information. It will be of interest to researchers and students of discrete geometry, mathematical chemistry and combinatorics, as well as to lay mathematicians. Chapter 1. Introduction: main ZC-notions -- Chapter 2. Zigzags of fullerenes and c-disk-fullerenes -- Chapter 3. Zigzags and railroads of spheres 3_v and 4_v -- Chapter 4. ZC-circuits of 4-regular and self-dual {2,3,4}-spheres -- Chapter 5. ZC-circuits of 5- and 6-regular spheres -- Chapter 6. Goldberg?Coxeter construction and parametrization -- Chapter 7. ZC-circuits of Goldberg?Coxeter construction -- Chapter 8. Zigzags of polytopes and complexes. Machine generated contents note: 1. Introduction: Main ZC-Notions -- 1.1. Graphs -- 1.2. Symmetries -- 1.3. Zigzag and Central Circuits -- 1.4. Curvature of Faces -- 1.5. Bifaced Maps -- 1.6. Computer Generation of the Families -- References -- 2. Zigzags of Fullerenes and c-Disk-Fullerenes -- 2.1. Zigzags Statistics for Small Fullerenes -- 2.2. Kekule Graphs -- 2.3. z-knot Fullerenes -- 2.4. Railroads in Fullerenes -- 2.5. Fullerenes 5v with Simple Zigzags -- 2.6. Tight Fullerenes -- 2.7. Disk-Fullerenes -- References -- 3. Zigzags and Railroads of Spheres 3v and 4v -- 3.1. General Results for Plane Graphs -- 3.2. Graphs 3v -- 3.3. Graphs 4v -- 3.4. Railroads and Pseudo-Roads -- References -- 4. ZC-Circuits of 4-Regular and Self-dual {2, 3, 4}-Spheres -- 4.1. Central Circuits of i-hedrites -- 4.2. Connectivity and Symmetries of i-hedrites -- 4.3. Tight i-hedrites and All Pure Tight i-hedrites -- 4.4. Enumeration and Generation of i-hedrites -- 4.4.1. 8-hedrites -- 4.4.2. 4-hedrites -- 4.4.3. 5-hedrites -- 4.4.4. 6-hedrites -- 4.4.5. 7-hedrites -- 4.4.6. Generation of i-hedrites -- 4.5. Self-dual {1, 2, 3, 4}-Spheres -- References -- 5. ZC-Circuits of 5- and 6-Regular Spheres -- 5.1. Icosahedrites -- 5.2. Generation Method of ({1, 2, 3}, 6)-Spheres -- 5.3. Symmetry Groups of the ({1, 2, 3}, 6)-Spheres -- 5.4. Goldberg--Coxeter Construction for 6-Regular Graphs -- 5.5. Zigzags and Central Circuits of 6-Regular Graphs -- References -- 6. Goldberg--Coxeter Construction and Parametrization -- 6.1. Complex Number Rings Z[ω] and Z[i] -- 6.2. GC-Construction for 3- and 4-Regular Graphs -- 6.3. Classes of Graphs -- 6.4. Triangulations of Oriented Maps -- 6.5. Two Parameters Constructions -- 6.6. General Case of Parameterization of Maps on Oriented Surfaces -- 6.7. Thurston's Theory for Maps of Positive Curvature -- 6.8. Extensions and Other Cases of Parameter Descriptions -- References -- 7. ZC-Circuits of Goldberg-Coxeter Construction -- 7.1. Directed Edge Formalism -- 7.2. ZC-Circuits of Inflation Graphs -- 7.3. Moving Group and the (k, l)-product -- 7.4. Stabilizer Group -- 7.5. GC-Construction on Basic Plane Graphs -- 7.6. Projections of ZC-Transitive GCk, l(G0) for Some Graphs Go -- 7.7. Zigzags of Other Parameter Constructions -- References -- 8. Zigzags of Polytopes and Complexes -- 8.1. Zigzags for d-Dimensional Complexes -- 8.2. Z-Structure of Some Generalizations of Regular d-Polytopes -- 8.3. Wythoff Kaleidoscope Construction -- 8.4. Wilson--Lins Triality -- References.