A Chemistry and Topology.- 1 Topological Aspects in Chemistry.- 1.1 Topology in Chemistry.- 1.2 Abstraction in Science and How Far One Can Go.- 2 Molecular Topology.- 2.1 What is Molecular Topology?.- 2.2 Geometry, Symmetry, Topology.- 2.3 Definition of Molecular Topology.- B Chemistry and Graph Theory.- 3 Chemical Graphs.- 4 Fundamentals of Graph Theory.- 4.1 The Definition of a Graph.- 4.1.1 Relations.- 4.1.2 The First Definition of a Graph.- 4.1.3 The Second Definition of a Graph.- 4.1.4 Vertices and Edges.- 4.1.5 Isomorphic Graphs and Graph Automorphisms.- 4.1.6 Special Graphs.- 4.2 Subgraphs.- 4.2.1 Sachs Graphs.- 4.2.2 Matchings.- 4.3 Graph Spectral Theory.- 4.3.1 The Adjacency Matrix.- 4.3.2 The Spectrum of a Graph.- 4.3.3 The Sachs Theorem.- 4.3.4 The ?-Polynomial.- 4.4 Graph Operations.- 5 Graph Theory and Molecular Orbitals.- 6 Special Molecular Graphs.- 6.1 Acyclic Molecules.- 6.1.1 Trees.- 6.1.2 The Path and the Star.- 6.1.3 The Characteristic Polynomial of Trees.- 6.1.4 Trees with Greatest Number of Matchings.- 6.1.5 The Spectrum of the Path.- 6.2 The Cycle.- 6.3 Alternant Molecules.- 6.3.1 Bipartite Graphs.- 6.3.2 The Pairing Theorem.- 6.3.3 Some Consequences of the Pairing Theorem.- 6.4 Benzenoid Molecules.- 6.4.1 Benzenoid Graphs.- 6.4.2 The Characteristic Polynomial of Benzenoid Graphs.- 6.5 Hydrocarbons and Molecules with Heteroatoms.- 6.5.1 On the Question of the Molecular Graph.- 6.5.2 The Characteristic Polynomial of Weighted Graphs.- 6.5.3 Some Regularities in the Electronic Structure of Heteroconjugated Molecules.- C Chemistry and Group Theory.- 7 Fundamentals of Group Theory.- 7.1 The Symmetry Group of an Equilateral Triangle.- 7.2 Order, Classes and Representations of a Group.- 7.3 Reducible and Irreducible Representations.- 7.4 Characters and Reduction of a Reducible Representation.- 7.5 Subgroups and Sidegroups — Products of Groups.- 7.6 Abelian Groups.- 7.7 Abstract Groups and Group Isomorphism.- 8 Symmetry Groups.- 8.1 Notation of Symmetry Elements and Representations.- 8.2 Some Symmetry Groups.- 8.2.1 Rotation Groups.- 8.2.2 Groups with More than One n-Fold Axis, n > 2.- 8.2.3 Groups of Collinear Molecules.- 8.3 Transformation Properties and Direct Products of Irreducible Representations.- 8.3.1 Transformation Properties.- 8.3.2 Rules Concerning the Direct Product of Irreducible Representations.- 8.4 Some Applications of Symmetry Groups.- 8.4.1 Electric Dipole Moment.- 8.4.2 Polarizability.- 8.4.3 Motions of Atomic Nuclei: Translations, Rotations and Vibrations.- 8.4.4 Transition Probabilities for the Absorption of Light.- 8.4.5 Transition Probabilities in Raman Spectra.- 8.4.6 Group Theory and Quantum Chemistry.- 8.4.7 Orbital and State Correlations.- 9 Automorphism Groups.- 9.1 Automorphism of a Graph.- 9.2 The Automorphism Group A(G1).- 9.3 Cycle Structure of Permutations.- 9.4 Isomorphism of Graphs and of Automorphism Groups 112..- 9.5 Notation of some Permutation Groups.- 9.6 Direct Product and Wreath Product.- 9.7 The Representation of Automorphism Groups as Group Products.- 10 Some Interrelations between Symmetry and Automorphism Groups.- 10.1 The Idea of Rigid Molecules.- 10.2 Local Symmetries.- 10.3 Non-Rigid Molecules.- 10.4 What Determines the Respective Orders of the Symmetry and the Automorphism Group of a Given Molecule?.- D Special Topics.- 11 Topological Indices.- 11.1 Indices Based on the Distance Matrix.- 11.1.1 The Wiener Number and Related Quantities.- 11.1.2 Applications of the Wiener Number.- 11.2 Hosoya’s Topological Index.- 11.2.1 Definition and Chemical Applications of Hosoya’s Index.- 11.2.2 Mathematical Properties of Hosoya’s Index.- 11.2.3 Example: Hosoya’s Index of the Path and the Cycle.- 11.2.4 Some Inequalities for Hosoya’s Index.- 12 Thermodynamic Stability of Conjugated Molecules.- 12.1 Total ?-Electron Energy and Thermodynamic Stability of Conjugated Molecules.- 12.2 Total ?-Electron Energy and Molecular Topology.- 12.3 The Energy of a Graph.- 12.4 The Coulson Integral Formula.- 12.5 Some Further Applications of the Coulson Integral Formula.- 12.6 Bounds for Total ?-Electron Energy.- 12.7 More on the McClelland Formula.- 12.8 Conclusion: Factors Determining the Total ?-Electron Energy.- 12.9 Use of Total ?-Electron Energy in Chemistry.- 13 Topological Effect on Molecular Orbitals.- 13.1 Topologically Related Isomers.- 13.2 Interlacing Rule.- 13.3 PE Spectra of Topomers.- 13.4 TEMO and a-Electron Systems.- 13.5 TEMO and Symmetry.- Appendices.- Appendix 1 Matrices.- Appendix 2 Determinants.- Appendix 3 Eigenvalues and Eigenvectors.- Appendix 4 Polynomials.- Appendix 5 Characters of Irreducible Representations of Symmetry Groups.- Appendix 6 The Symbols Used.- Literature.- References.
