## Combinatorial Matrix TheoryThis book contains the notes of the lectures delivered at an Advanced Course on Combinatorial Matrix Theory held at Centre de Recerca Matemàtica (CRM) in Barcelona. These notes correspond to five series of lectures. The first series is dedicated to the study of several matrix classes defined combinatorially, and was delivered by Richard A. Brualdi. The second one, given by Pauline van den Driessche, is concerned with the study of spectral properties of matrices with a given sign pattern. Dragan Stevanović delivered the third one, devoted to describing the spectral radius of a graph as a tool to provide bounds of parameters related with properties of a graph. The fourth lecture was delivered by Stephen Kirkland and is dedicated to the applications of the Group Inverse of the Laplacian matrix. The last one, given by Ángeles Carmona, focuses on boundary value problems on finite networks with special in-depth on the M-matrix inverse problem. |

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### Contents

1 | |

Chapter 2 Sign Pattern Matrices | 47 |

Chapter 3 Spectral Radius of Graphs | 83 |

Chapter 4 The Group Inverse of the Laplacian Matrix of a Graph | 131 |

Chapter 5 Boundary Value Problems on Finite Networks | 173 |

### Other editions - View all

Combinatorial Matrix Theory Richard A. Brualdi,Herbert J. Ryser,Brualdi Richard a. Limited preview - 1991 |

### Common terms and phrases

adjacency matrix bottleneck matrix boundary value problem Bruhat order C(FU H1 characteristic polynomial closed walks connected graph Corollary corresponding define denote distance-regular graph Driessche edge eigenvalue eigenvector equal Example given graph G group inverse Hankel diagonal hence inequality irreducible Laplacian matrix Lemma Let G Linear Algebra Linear Algebra Appl lower triangular M-matrix M-property main diagonal matrix theory Moreover n x n nilpotent nonnegative integers nonsingular nonzero entries numbers of walks permutation matrix potentially nilpotent potentially stable sign principal eigenvector Proof Proposition resistance distance result row and column score vector sequence sign pattern ſº spectral radius spectrally arbitrary stable sign pattern strongly regular graph subgraph submatrix superpattern symmetric Theorem tree sign patterns u e C(V unique unweighted vertices walks of length weighted graph zero zero-nonzero pattern