## Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue -2Line graphs have the property that their least eigenvalue is greater than or equal to -2, a property shared by generalized line graphs and a finite number of so-called exceptional graphs. This book deals with all these families of graphs in the context of their spectral properties. The authors discuss the three principal techniques that have been employed, namely 'forbidden subgraphs', 'root systems' and 'star complements'. They bring together the major results in the area, including the recent construction of all the maximal exceptional graphs. Technical descriptions of these graphs are included in the appendices, while the bibliography provides over 250 references. This will be an important resource for all researchers with an interest in algebraic graph theory. |

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### Contents

Introduction | 1 |

Forbidden subgraphs | 25 |

Root systems | 64 |

Regular graphs | 88 |

Star Complements | 112 |

The maximal exCeptional graphs | 139 |

MisCellaneous results | 164 |

Appendix | 193 |

Table A3 Regular exceptional graphs and their spectra | 213 |

Table A4 A construction of the 68 connected regular graphs which | 228 |

Table A5 Onevertex extensions of exceptional star complements | 243 |

Table A6 The maximal exceptional graphs | 249 |

Table A7 The index and vertex degrees of the maximal exceptional | 273 |

Bibliography | 281 |

295 | |

### Common terms and phrases

adjacency matrix Algebra Chang graphs characteristic polynomial characterized cliques cocktail party graph Combinatorics connected graph Corollary corresponding cospectral cubic graphs CvDSa Cvetkovic D. K. Ray-Chaudhuri denotes determined distinct eigenvalues eigenspace eigenvalue of G eigenvector equation exceptional star complements following result GCPs graph G graph obtained Graph Theory graphs of order graphs of type graphs with least incidence matrix induced subgraph integral graphs isomorphic layer least eigenvalue greater Lemma Let G Let H line graph line system line-regular Math maximal exceptional graphs maximal graph minimal forbidden subgraphs multigraph number of edges number of vertices odd cycle orthogonal pair parameters pendant edge petal Petersen graph Proposition regular exceptional graphs representation root graph root multigraph root system Schlafli graph Section Seidel Simic spectral spectrum star set star-closed strongly regular graph switching with respect Table A3 Theorem vectors of type vertex degrees vertex of type vertices of degree