## Recent Results in the Theory of Graph SpectraThe 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. |

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### Contents

1 | |

21 | |

Chapter 3 Miscellaneous Results from the Theory of Graph Spectra | 41 |

Chapter 4 The Matching Polynomial and Other Graph Polynomials | 103 |

Chapter 5 Applications to Chemistry and Other Branches of Science | 131 |

Chapter 6 Spectra of Infinite Graphs | 155 |

Spectra of Graphs with Seven Vertices | 175 |

Bibliography | 233 |

291 | |

301 | |

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Acta Berl adjacency matrix automorphism group Beograd bipartite graph bound Characteristic polynomial coefficients characterized Chem Chim circuit coefficients and spectral combinatorial complete graph connected graphs construction COROLLARY corresponding cospectral graphs CVDSA1 CVETKović defined denote diameter digraph Discrete Math distance-regular graphs distance-transitive distinct eigenvalues DooB edges Eigenvalues Characteristic polynomial eigenvalues of G eigenvectors equal FARRELL E. J. finite G1 and G2 girth given GODSIL C. D. GRAOVAC graph G graph invariants graph of degree graph spectra GUTMAN Hückel induced subgraph infinite graphs Inst integers isomorphism largest eigenvalue least eigenvalue Let G line graphs matching polynomial McKAY MoHAR molecular graph molecular orbital Moore graph multiplicity NEPS NEUMAIER Number Eigenvalues Characteristic number of spanning number of vertices number of walks parameters PETRović Phys problem properties Publ root system Section spanning trees spectral graph theory spectral moments spectrum strongly regular graph symmetric TORGASEV tournament Univ vectors vertex set