Recent Results in the Theory of Graph Spectra (eBook)

D.M. Cvetkovic, M. Doob, I. Gutman, A. Torgasev (Autoren)

eBook Download: PDF

1988 | 1. Auflage
305 Seiten
Elsevier Science (Verlag)
978-0-08-086776-2 (ISBN)

Infos

The purpose of this volume is to review the results in spectral graph theory which have appeared since 1978.

The problem of characterizing graphs with least eigenvalue -2 was one of the original problems of spectral graph theory. The techniques used in the investigation of this problem have continued to be useful in other contexts including forbidden subgraph techniques as well as geometric methods involving root systems. In the meantime, the particular problem giving rise to these methods has been solved almost completely. This is indicated in Chapter 1.

The study of various combinatorial objects (including distance regular and distance transitive graphs, association schemes, and block designs) have made use of eigenvalue techniques, usually as a method to show the nonexistence of objects with certain parameters. The basic method is to construct a graph which contains the structure of the combinatorial object and then to use the properties of the eigenvalues of the graph. Methods of this type are given in Chapter 2.

Several topics have been included in Chapter 3, including the relationships between the spectrum and automorphism group of a graph, the graph isomorphism and the graph reconstruction problem, spectra of random graphs, and the Shannon capacity problem. Some graph polynomials related to the characteristic polynomial are described in Chapter 4. These include the matching, distance, and permanental polynomials. Applications of the theory of graph spectra to Chemistry and other branches of science are described from a mathematical viewpoint in Chapter 5. The last chapter is devoted to the extension of the theory of graph spectra to infinite graphs.

Inhalt

Front Cover 1
Recent Results in the Theory of Graph Spectra 4
Copyright Page 5
Contents 10
Introduction 6
Chapter 1. Characterizations of Graphs by their Spectra 14
1.1. Generalized Line Graphs and Graphs with Least Eigenvalue –2 14
1.2. Other Graph Characterizations 20
1.3. Cospectral Constructions 26
1.4. Hereditary Characterizations 30
Chapter 2. Distance-Regular and Similar Graphs 34
2.1. The Bose-Mesner Algebra 34
2.2. Moore Graphs and their Generalizations 41
2.3. Distance-Transitive Graphs 45
2.4. Distance-Regular Graphs and other Combinatorial Objects 48
2.5. Root Systems and Distance-Regular Graphs 52
Chapter 3. Miscellaneous Results from the Theory of Graph Spectra 54
3.1. The Sachs Theorem 55
3.2. Spectra of Graphs Derived by Operations and Transformations 60
3.3. Constructions of Graphs Using Spectra 67
3.4. The Automorphism Group and the Spectrum of a Graph 74
3.5. Identification and Reconstruction of Graphs 79
3.6. The Shannon Capacity and Spectral Bounds for Graph Invariants 86
3.7. Spectra of Random Graphs 92
3.8. The Number of Walks in a Graph 95
3.9. The Number of Spanning Trees in a Graph 98
3.10. The Use of Spectra to Solve Graph Equations 102
3.11. Spectra of Tournaments 107
3.12. Other Results 109
Chapter 4. The Matching Polynomial and Other Graph Polynomials 116
4.1. The Matching Polynomial 116
4.2. The Matching Polynomial of Weighted Graphs 132
4.3. The Rook Polynomial 133
4.4. The Independence Polynomial 135
4.5. The F-Polynomial 136
4.6. The Permanental Polynomial 137
4.7. Polynomials and the Admittance Matrix 138
4.8. The Distance Polynomial 139
4.9. Miscellaneous Results 142
Chapter 5. Applications to Chemistry and Other Branches of Science 144
5.1. On Hückel Molecular Orbital Theory 145
5.2. The Characteristic Polynomial 148
5.3. Cospectral Molecular Graphs 148
5.4. The Spectrum and the Automorphism Group of Molecular Graphs 149
5.5. The Energy of a Graph 149
5.6. S– and T– Isomers 152
5.7. Circuits and the Energy of a Graph 154
5.8. Charge and Bond Order 154
5.9. HOMO-LUMO Separation 155
5.10. The Determinant of the Adjacency Matrix 156
5.11. The Magnetic Properties of Conjugated Hydrocarbons 158
5.12. The Topological Resonance Energy 159
5.13. Some Spectral Properties of Hexagonal Systems 160
5.14. Miscellaneous HMO Results 163
5.15. Molecular Orbital Approaches Other than the HMO Model 163
5.16. Applications of Graph Eigenvalues in Physics and Chemistry Other than Molecular Orbital Models 164
5.17. Graph Eigenvalues in Geography and the Social Sciences 165
Chapter 6. Spectra of Infinite Graphs 168
6.1. General Properties 169
6.2. Spectral Properties of some Classes of Infinite Graphs 172
6.3. The Characteristic Function of an Infinite Graph 174
6.4. Graphs with a Finite Spectrum 176
6.5. Operations on Infinite Graphs 177
6.6. The Automorphism Group of an Infinite Graph 178
6.7. Infinite Generalized Line Graphs 179
6.8. The D-spectrum of Infinite Graphs 181
6.9. Graphs with Uniformly Bounded Spectra 183
6.10. Another Approach to the Spectrum of an Infinite Graph 184
Spectra of Graphs with Seven Vertices 188
Bibliography 246
Bibliographic Index 304
Index 314

Produkt-Details

• Erscheint lt. Verlag: 1.1.1988
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
• Themenwelt: Technik
• ISBN-10: 0-08-086776-6 / 0080867766
• ISBN-13: 978-0-08-086776-2 / 9780080867762