## Graph Spectra for Complex NetworksAnalyzing the behavior of complex networks is an important element in the design of new man-made structures such as communication systems and biologically engineered molecules. Because any complex network can be represented by a graph, and therefore in turn by a matrix, graph theory has become a powerful tool in the investigation of network performance. This self-contained 2010 book provides a concise introduction to the theory of graph spectra and its applications to the study of complex networks. Covering a range of types of graphs and topics important to the analysis of complex systems, this guide provides the mathematical foundation needed to understand and apply spectral insight to real-world systems. In particular, the general properties of both the adjacency and Laplacian spectrum of graphs are derived and applied to complex networks. An ideal resource for researchers and students in communications networking as well as in physics and mathematics. |

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### Contents

Algebraic graph theory | 13 |

Eigenvalues of the adjacency matrix | 29 |

Eigenvalues of the Laplacian Q | 67 |

Spectra of special types of graphs | 115 |

Density function of the eigenvalues | 159 |

Spectra of complex networks | 179 |

Eigensystem of a matrix | 211 |

Polynomials with real coefficients | 263 |

Orthogonal polynomials | 313 |

339 | |

345 | |

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### Common terms and phrases

adjacency matrix algebraic bipartite graph Cauchy Cauchy index characteristic polynomial clique coefficients column complete bipartite graph complete graph complex networks components computed connected graph corresponding deduced degree denote density function derived eigenvector belonging equation example fc=i fc=l fe=i graph G Hence hopcount implies incidence matrix inequality integer interlacing Laplacian eigenvalues Laplacian matrix Laplacian Q largest eigenvalue Lemma line graph linear lower bound Mieghem modularity multiplicity negative neighbors nodes non-negative non-zero number of links number of nodes obtain orthogonal matrix orthogonal polynomials partition Perron-Frobenius Theorem polynomial pn Proof quotient matrix random graph Rayleigh real zeros recursion regular graph relabeling rewiring Section sequence shows smallest eigenvalue spectral gap spectrum subgraph submatrix symmetric matrix total number upper bound vector yields zero eigenvalue zeros of pn