Recent results in the theory of graph spectra

Media type: E-Book

Title: Recent results in the theory of graph spectra

Contains: Front Cover; Recent Results in the Theory of Graph Spectra; Copyright Page; Contents; Introduction; Chapter 1. Characterizations of Graphs by their Spectra; 1.1. Generalized Line Graphs and Graphs with Least Eigenvalue -2; 1.2. Other Graph Characterizations; 1.3. Cospectral Constructions; 1.4. Hereditary Characterizations; Chapter 2. Distance-Regular and Similar Graphs; 2.1. The Bose-Mesner Algebra; 2.2. Moore Graphs and their Generalizations; 2.3. Distance-Transitive Graphs; 2.4. Distance-Regular Graphs and other Combinatorial Objects; 2.5. Root Systems and Distance-Regular Graphs
Chapter 3. Miscellaneous Results from the Theory of Graph Spectra3.1. The Sachs Theorem; 3.2. Spectra of Graphs Derived by Operations and Transformations; 3.3. Constructions of Graphs Using Spectra; 3.4. The Automorphism Group and the Spectrum of a Graph; 3.5. Identification and Reconstruction of Graphs; 3.6. The Shannon Capacity and Spectral Bounds for Graph Invariants; 3.7. Spectra of Random Graphs; 3.8. The Number of Walks in a Graph; 3.9. The Number of Spanning Trees in a Graph; 3.10. The Use of Spectra to Solve Graph Equations; 3.11. Spectra of Tournaments; 3.12. Other Results
Chapter 4. The Matching Polynomial and Other Graph Polynomials4.1. The Matching Polynomial; 4.2. The Matching Polynomial of Weighted Graphs; 4.3. The Rook Polynomial; 4.4. The Independence Polynomial; 4.5. The F-Polynomial; 4.6. The Permanental Polynomial; 4.7. Polynomials and the Admittance Matrix; 4.8. The Distance Polynomial; 4.9. Miscellaneous Results; Chapter 5. Applications to Chemistry and Other Branches of Science; 5.1. On Hückel Molecular Orbital Theory; 5.2. The Characteristic Polynomial; 5.3. Cospectral Molecular Graphs
5.4. The Spectrum and the Automorphism Group of Molecular Graphs5.5. The Energy of a Graph; 5.6. S- and T- Isomers; 5.7. Circuits and the Energy of a Graph; 5.8. Charge and Bond Order; 5.9. HOMO-LUMO Separation; 5.10. The Determinant of the Adjacency Matrix; 5.11. The Magnetic Properties of Conjugated Hydrocarbons; 5.12. The Topological Resonance Energy; 5.13. Some Spectral Properties of Hexagonal Systems; 5.14. Miscellaneous HMO Results; 5.15. Molecular Orbital Approaches Other than the HMO Model
5.16. Applications of Graph Eigenvalues in Physics and Chemistry Other than Molecular Orbital Models5.17. Graph Eigenvalues in Geography and the Social Sciences; Chapter 6. Spectra of Infinite Graphs; 6.1. General Properties; 6.2. Spectral Properties of some Classes of Infinite Graphs; 6.3. The Characteristic Function of an Infinite Graph; 6.4. Graphs with a Finite Spectrum; 6.5. Operations on Infinite Graphs; 6.6. The Automorphism Group of an Infinite Graph; 6.7. Infinite Generalized Line Graphs; 6.8. The D-spectrum of Infinite Graphs; 6.9. Graphs with Uniformly Bounded Spectra
6.10. Another Approach to the Spectrum of an Infinite Graph
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

Contributor: Cvetković, Dragoš M. [Other]

imprint: Amsterdam; New York: North-Holland, 1988

Extent: Online-Ressource

Language: English

ISBN: 0080867766; 1281793159; 9780080867762; 9781281793157; 0444703616; 9780444703613

RVK notation: SK 890 : Ganzzahlige und kombinatorische Optimierung, Graphentheorie

Keywords: Graphentheorie

Reproductino series: Elsevier e-book collection on ScienceDirect

Origination:

Footnote: Includes indexes

Description: 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 objec

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