CHEMICAL GRAPH THEORY

Transcription

1 CHEMICAL GRAPH THEORY SECOND EDITION Nenad Trinajstic, Ph.D. Professor of Chemistry The Rugjer Boskovic Institute Zagreb The Republic of Croatia CRC Press Boca Raton Ann Arbor London Tokyo

2 TABLE OF CONTENTS Chapter 1 Introduction 1 References 3 Chapter 2 Elements of Graph Theory I. The Definition of a Graph 5 II. Isomorphic Graphs and Graph Automorphism 8 III. Walks, Trails, Paths, Distances, and Valencies in Graphs 9 IV. Subgraphs 12 V. Regulär Graphs 13 VI. Trees 13 VII. Planar Graphs 14 VIII. The Story of the Königsberg Bridge Problem and Eulerian Graphs 17 IX. Hamiltonian Graphs 19 X. Line Graphs 20 XI. Vertex Coloring of a Graph 21 References? 22 Chapter 3 Chemical Graphs I. The Concept of a Chemical Graph 25 II. Molecular Topology 25 III. Hückel Graphs 28 A. Kekule Graphs 29 IV Polyhexes and Benzenoid Graphs 30 A. The Dual and the Inner Dual of a Polyhex 31 B. The Dualist of a Polyhex 32 C. Factor Graphs 33 V. Weighted Graphs 35 A. Vertex- and Edge-Weighted Graphs 35 B. Möbius Graphs 36 References 37 Chapter 4 Graph-Theoretical Matrices I. The Adjacency Matrix 41 A. The Adjacency Matrix of a Bipartite Graph 47 B. The Relationship Between the Adjacency Matrix and the Number of Walks in a Graph 48

3 C. The Determinant of the Adjacency Matrix 48 D. The Permanent of the Adjacency Matrix 50 E. The Inverse of the Adjacency Matrix 52 II. The Distance Matrix 52 References 57 Chapter 5 The Characteristic Polynomial of a Graph I. The Definition of the Characteristic Polynomial 61 A. The Uses of the Characteristic Polynomial 62 B. Computational Methods for the Characteristic Polynomial 63 II. The Method of Sachs for Computing the Characteristic Polynomial 64 A. The Application of the Method of Sachs to Simple Graphs 65 B. The Extension of the Sachs Formula to Möbius Systems 66 C. The Extension of the Sachs Formula to Weighted Graphs 67 D. Summary of Some Results Obtained by the Use of the Sachs Formuta 70 III. The Characteristic Polynomials of Some Classes of Simple Graphs 72 A. Chains 72 B. Trees 73 C. Cycles 74 D. Möbius Cycles 74 IV. The Le Verrier-Faddeev-Frame Method for Computing the Characteristic Polynomial 76 References 80 Chapter 6 Topological Aspects of Hückel Theory I. Elements of Hückel Theory 85 II. Isomorphism of Hückel Theory and Graph Spectral Theory 88 III. The Hückel Spectrum 90 A. A Method for the Enumeration of NBMOs 92 B. The Enumeration of N 0 and N + from the Characteristic Polynomial 94 C. A Graph-Theoretical Classification of Conjugated Hydrocarbons Based on Their Spectral Characteristics 95

4 . IV. Charge Densities and Bond Orders in Conjugated Systems 96 V. The Two-Color Problem in Hückel Theory 97 A. Properties of Alternant Hydrocarbons 100 VI. Eigenvalues of Linear Polyenes 103 VII. Eigenvalues of Annulenes 105 VIII. Eigenvalues of Möbius Annulenes 108 IX. A Classification Scheme for Monocyclic Systems 110 X. Total Tr-Electron Energy 113 A. The Fundamental Identity for E,, 114 B. Relations between E, the Adjacency Matrix, and the Charge Density-Bond Order Matrix 116 C. The McClelland Formula for E 117 References 119! Chapter 7 Topological Resonance Energy I. Hückel Resonance Energy 125 II. Dewar Resonance Energy 127 III. The Concept of Topological Resonance Energy 131 A. Topological Resonance Energy 133 IV Computation of the Acyclic Polynomial 137 A. Connection between the Characteristic Polynomial and the Acyclic Polynomial 139 V Applications of the TRE Model* 142 A. Hückel Annulenes 143 B. Relationship between TREs and Ring Currents of [4m + 2] ir-electron Annulenes 145 C. Möbius Annulenes 148 D. Conjugated Hydrocarbons and Heterocyclics 150 E. Conjugated Ions and Radicals 150 F. Homoaromatic Systems 152 G. Aromaticity in the Lowest Excited State of Annulenes 154 H. Failures of the TRE Model 156 References 156 Chapter 8 Enumeration of Kekule Valence Structures I. The Role of Kekule Valence Structures in Chemistry 161 II. The Identification of Kekule Systems 163 III. Methods for the Enumeration of Kekule Structures 164 A. The Empirical Method 165 B. The Method of Fragmentation 165

5 C. Methods Based on Graphic Polynomials The Characteristic Polynomial The Acyclic Polynomial The Permanental Polynomial 167 D. The Method Based on the Coefficients of Nonbonding Molecular Orbitals 168 E. The Method of Gordon and Davison 170 F. The Numeral-in-Hexagon Method 171 G. The Generalized Gordon-Davison Method 171 H. Methods Based on the Lattice Structure of Fused Benzenoids The Method of Yen The Numeral-in-Hexagon Method 177 I. The Two-Step Fragmentation Method 177 J. The Path Counting Method 178 K. The Matrix Method of Hall 181 L. The Transfer-Matrix Method 185 M. The Method of Dewar and Longuet-Higgins 187 N. The Computational Me*hod Based on the Eigenvalue Spectrum 188 IV. The Concept of Parity of Kekule Structures 188 References 195 Chapter 9 The Conjugated-Circuit Model I. The Concept of Conjugated Circuits 199 II. The ir-resonance Energy Expression 203 III. Selection of the Parameters 205 A. The R n Parameters 205 B. A Parametrization Scheme for Other Parameters The Q n Parameters The HJ. Parameters The H'^ Parameters 207 IV. Computational Procedure 208 V Applications of the Conjugated-Circuit Model 208 A. Benzenoid Hydrocarbons 208 B. Nonalternant Hydrocarbons 213 C. Aromaticity Postulate 215 VI. Parity of Conjugated Circuits 219 References 221 Chapter 10 Topological Indices I. Definitions of Topological Indices 225 A. Topological Indices Based on Connectivity 226

6 1. The Zagreb Group Indices The Connectivity Index 226 a. The Relationship Between the ir-electronic Energy and the Connectivity Index 231 b. The Degeneracy of the Connectivity Index The Connectivity ID Number The Prime ID Number The Largest Eigenvalue as a Topological Index The Z-Index 239 B. Topological Indices Based on Distances The Wiener Number The Platt Number The Gordon-Scantlebury Index The Balaban Index The Information-Theoretic Indices The Bertz Index The Centric Index The Weighted ID Number The Schultz Index The Determinant of the Adjacency-Plus- Distance Matrix as a Topological Index 262 II. The Three-Dimensional Wiener Number 262 References 269 Chapter 11 Isomer Enumeration I. The Cayley Generating Functions 275 A. Enumeration of Trees 275 B. Enumeration of Alkanes 280 II. The Henze-Blair Approach 281 III. The Pölya Enumeration Method 286 IV. The Enumeration Method Based on the N-Tuple Code 291 References 298 Index 303