Combinatorial Matrix Theory

1. Front Matter

   Pages i-xxiv

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2. Some Combinatorially Defined Matrix Classes

   • Richard A. Brualdi

   Pages 1-46

3. Sign Pattern Matrices

   • P. van den Driessche

   Pages 47-82

4. Spectral Radius of Graphs

   • Dragan Stevanović

   Pages 83-130

5. The Group Inverse of the Laplacian Matrix of a Graph

   • Stephen Kirkland

   Pages 131-171

6. Boundary Value Problems on Finite Networks

   • Ángeles Carmona

   Pages 173-217

7. Back Matter

   Pages 219-219

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This 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.

• permutation matrices
• tournament matrices
• sign pattern matrices
• generalized inverses
• adjacency
• laplacian
• eigenvalues and eigenvectors
• discrete potential theory
• combinatorics
• matrix theory
• ordinary differential equations

“A very excellent treatment of the subject for mathematicians interested in the intersection between matrix theory and combinatorics. … it is appropriate for graduate students and others interested in the latest developments in this rich and diverse field.” (MAA Reviews, March 1, 2020)

• Departament de Matemàtiques, Escola d'Enginyeria de Barcelona Est Departament de Matemàtiques, Barcelona, Spain

  Andrés M. Encinas, Margarida Mitjana

• Mathematics Department, University of Wisconsin Mathematics Department, Madison, USA

  Richard A. Brualdi

• Departament de Matemàtiques, Universitat Politècnica de Catalunya, Barcelona, Spain

  Ángeles Carmona

• Department of Mathematics and Statistics, University of Victoria, Victoria, Canada

  P. van den Driessche

• Department of Mathematics, University of Manitoba, Winnipeg, Canada

  Stephen Kirkland

• Mathematical Institute, Serbian Academy of Sciences and Arts, Belgrade, Serbia

  Dragan Stevanović

Richard A. Brualdi is an Emeritus Professor at the University of Wisconsin in Madison, WI, USA.

Pauline van den Driessche is an Emeritus Professor at the University of Victoria, Canada.

Dragan Stevanović is a Full Research Professor at the Serbian Academy of Sciences and Arts in Belgrade, Serbia.

Stephen Kirkland is a Professor at the University of Manitoba in Winnipeg, Canada.

Ángeles Carmona is an Associate Professor at the Universitat Politècnica de Catalunya in Barcelona, Spain. 

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