## Chemical Graph Theory: Introduction and Fundamentals"This volume presents the fundamentals of graph theory and then goes on to discuss specific chemical applications. Chapter 1 provides a historical setting for the current upsurge of interest in chemical graph theory. chapter 2 gives a full background of the basic ideas and mathematical formalism of graph theory and includes such chemically relevant notions as connectedness, graph matrix representations, metric properties, symmetry and operations on graphs. This is followed by a discussion on chemical nomenclature and the ... Show more This volume presents the fundamentals of graph theory and then goes on to discuss specific chemical applications. Chapter 1 provides a historical setting for the current upsurge of interest in chemical graph theory. chapter 2 gives a full background of the basic ideas and mathematical formalism of graph theory and includes such chemically relevant notions as connectedness, graph matrix representations, metric properties, symmetry and operations on graphs. This is followed by a discussion on chemical nomenclature and the trends in its rationalization by using graph theory, which has important implications for the storage and retrieval of chemical information. This volume also contains a detailed discussion of the relevance of graph-theoretical polynomials; it describes methodologies for the enumeration of isomers, incorporating the classical Polya method, as well as more recent approaches. The final chapter considers the interplay between graph theory and molecular orbital theory from the standpoint of graph spectral theory, highlighting the concept of topographical resonance in molecular species"--Back cover. |

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### Contents

THE ORIGINS OF CHEMICAL GRAPH THEORY Dennis H Rouvray | 1 |

The First Use of Chemical Graphs | 3 |

The Emergence of Structure Theory | 5 |

The Concept of Valence | 10 |

The Growth of Chemical Graph Theory | 16 |

Isomer Enumeration Techniques | 18 |

Early Additivity Studies | 22 |

The Introduction of Topological Indices | 26 |

Polynomials | 136 |

The Characteristic Polynomial | 137 |

The Matching Polynomial | 152 |

More Graphic Polynomials | 164 |

References | 169 |

ENUMERATION OF ISOMERS Alexandru T Balaban | 177 |

Introduction | 178 |

Definitions and Mathematical Background | 179 |

Elementary Bonding Theory | 30 |

Conclusion | 33 |

References | 34 |

ELEMENTS OF GRAPH THEORY FOR CHEMISTS Oskar E Polansky | 41 |

What is a Graph and What Kinds of Graph Exist? | 42 |

Some Graphtheoretical Terms | 45 |

Connectedness of Graphs | 47 |

Partitioning of a Graph | 54 |

Planarity of Graphs | 57 |

Line Graphs | 62 |

Operations on Graphs | 63 |

The Automorphism Group of a Graph | 67 |

Matrix Representation and Eigenvalue Problems of Undirected Graphs | 78 |

The Matrix Representation of Digraphs | 83 |

Distances in Graphs and Digraphs | 85 |

Metric and Topological Spaces for Simple Graphs | 89 |

Graphs in Quantum Chemistry | 93 |

Bibliography | 94 |

Acknowledgment | 96 |

NOMENCLATURE OF CHEMICAL COMPOUNDS Alan L Goodson | 97 |

Development of Chemical Nomenclature | 101 |

Development of Chemical Line Notations | 105 |

Development of Graph Theory | 107 |

Summary | 124 |

References and Notes | 125 |

POLYNOMIALS IN GRAPH THEORY Ivan Gutman | 133 |

Historical | 184 |

Pólyas Theorem | 187 |

Generalized Pólya Theorem | 197 |

Ruchs Double Coset Form ism | 200 |

De BruijnHararyPalmer Power Group Theory | 202 |

Valence Isomers | 203 |

Polyhexes | 210 |

Diamond Hydrocarbons and Staggered Alkane Rotamers | 215 |

Diastereomeric Annulenes | 216 |

Isomers and Computer Programs for Their Generation | 219 |

Isomerism and Reaction Graphs | 222 |

Conclusion | 224 |

References | 226 |

GRAPH THEORY AND MOLECULAR ORBITALS Nenad Trinajstić | 235 |

Introduction | 236 |

Elements of Graph Spectral Theory | 237 |

The Essence of Hückel Theory | 244 |

Isomorphism of Hückel Theory and Graph Spectral Theory | 248 |

The Spectrum of a Hückel Graph | 249 |

The Number Nonbonding Molecular Orbitals | 251 |

Total 7Electron Energy | 256 |

Topological Resonance Energy 262 | 262 |

Concluding Remarks | 272 |

References | 273 |

281 | |

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### Common terms and phrases

A.T. Balaban Acta acyclic addition alkanes applications atoms benzene bonds branched called carbon atoms chapter characteristic polynomial Chem chemical chemical graph chemical structure chemistry Chim classes complete components compounds concept connected considered constitutional contain corresponding counting cycle defined definition denoted derived described determinant developed diagrams directed discussed distance edges electrons elements energy enumeration example exist extensive Figure formula four given graph G graph theory Gutman Hückel hydrocarbons illustrated important independent indices introduced isomerism isomers Kekulé known later matching polynomial Math mathematical methods molecular molecule nomenclature notations obtained operations orbitals organic pair path permutation Phys possible present problem properties proposed published recently reference represent respectively result ring rules Sachs shown simple stereoisomers structure symmetry Table theorem topological trees Trinajstić valence values vertex vertices weighted zeros