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Combinatorial Matrix Theory 1st ed. 2018 [Minkštas viršelis]

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  • Formatas: Paperback / softback, 219 pages, aukštis x plotis: 240x168 mm, weight: 454 g, XI, 219 p., 1 Paperback / softback
  • Serija: Advanced Courses in Mathematics - CRM Barcelona
  • Išleidimo metai: 13-Apr-2018
  • Leidėjas: Birkhauser Verlag AG
  • ISBN-10: 3319709526
  • ISBN-13: 9783319709529
Kitos knygos pagal šią temą:
Combinatorial Matrix Theory 1st ed. 2018Didesnis paveikslėlis
  • Formatas: Paperback / softback, 219 pages, aukštis x plotis: 240x168 mm, weight: 454 g, XI, 219 p., 1 Paperback / softback
  • Serija: Advanced Courses in Mathematics - CRM Barcelona
  • Išleidimo metai: 13-Apr-2018
  • Leidėjas: Birkhauser Verlag AG
  • ISBN-10: 3319709526
  • ISBN-13: 9783319709529
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783319709529.html
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 Stevanovic 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.

Recenzijos

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)

Foreword v
1 Some Combinatorially Defined Matrix Classes
1(46)
Richard A. Brualdi
1.1 Permutations and Permutation Matrices
1(14)
1.1.1 Basic Properties
1(2)
1.1.2 Generation
3(1)
1.1.3 Bruhat Order
3(2)
1.1.4 Matrix Bruhat Decomposition
5(3)
1.1.5 Flags
8(3)
1.1.6 Involutions and Symmetric Integral Matrices
11(4)
1.2 Alternating Sign Matrices
15(17)
1.2.1 Basic Properties
15(1)
1.2.2 Other Views of ASMs
16(3)
1.2.3 The A-determinant
19(1)
1.2.4 Maximal ASMs
20(1)
1.2.5 Generation
21(1)
1.2.6 MacNeille Completion and the Bruhat Order
22(3)
1.2.7 Bruhat Order Revisited
25(6)
1.2.8 Spectral Radius of ASMs
31(1)
1.3 Tournaments and Tournament Matrices
32(13)
1.3.1 The Inverse Problem
33(1)
1.3.2 Generation
34(2)
1.3.3 Loopy Tournaments and Their Generation
36(2)
1.3.4 Hankel Tournaments and Their Generation
38(2)
1.3.5 Combinatorially Skew-Hankel Tournaments and Their Generation
40(5)
Bibliography
45(2)
2 Sign Pattern Matrices
47(36)
P. van den Driessche
2.1 Introduction to Sign Pattern Matrices
47(1)
2.1.1 Notation and Definitions
47(1)
2.2 Potential Stability of Sign Patterns
48(9)
2.2.1 Stability Definitions
48(1)
2.2.2 Stability of a Dynamical System
49(1)
2.2.3 Characterization of Sign Stability
50(1)
2.2.4 Basic Facts for Potential Stability
51(1)
2.2.5 Known Results on Potential Stability for Small Orders
52(1)
2.2.6 Sufficient Condition for Potential Stability
52(2)
2.2.7 Construction of Higher-Order Potentially Stable Sign Patterns
54(2)
2.2.8 Number of Nonzero Entries
56(1)
2.2.9 Open Problems Related to Potential Stability
57(1)
2.3 Spectrally Arbitrary Sign Patterns
57(7)
2.3.1 Some Definitions Relating to Spectra of Sign Patterns
57(2)
2.3.2 A Family of Spectrally Arbitrary Sign Patterns
59(2)
2.3.3 Minimal Spectrally Arbitrary Patterns and Number of Nonzero Entries
61(1)
2.3.4 Reducible Spectrally Arbitrary Sign Patterns
62(1)
2.3.5 Some Results on Potentially Nilpotent Sign Patterns
63(1)
2.3.6 Some Open Problems Concerning SAPs
64(1)
2.4 Refined Inertia of Sign Patterns
64(9)
2.4.1 Definition and Maximum Number of Refined Inertias
64(1)
2.4.2 The Set of Refined Inertias Hn
65(1)
2.4.3 Sign Patterns of Order 3 and H3
66(1)
2.4.4 Sign Patterns of Order 4 and H4
66(1)
2.4.5 Sign Patterns with All Diagonal Entries Negative
67(1)
2.4.6 Detecting Periodic Solutions in Dynamical Systems
68(4)
2.4.7 Some Open Problems Concerning Hn
72(1)
2.5 Inertially Arbitrary Sign Patterns
73(6)
2.5.1 Definition and Relation to Other Properties
73(1)
2.5.2 Generalization of the Nilpotent-Jacobian Method
74(2)
2.5.3 Reducible IAPs
76(1)
2.5.4 A Glimpse at Zero-Nonzero Patterns
76(1)
2.5.5 A Taste of More General Patterns
77(1)
2.5.6 Some Open Problems Concerning IAPs
78(1)
Bibliography
79(4)
3 Spectral Radius of Graphs
83(48)
Dragan Stevanovic
3.1 Graph-Theoretical Definitions
83(2)
3.2 The Adjacency Matrix and Its Spectral Properties
85(4)
3.3 The Big Gun Approach
89(6)
3.4 The Eigenvector Approach
95(10)
3.5 The Characteristic Polynomial Approach
105(5)
3.6 Walk Counting
110(17)
Bibliography
127(4)
4 The Group Inverse of the Laplacian Matrix of a Graph
131(42)
Stephen Kirkland
4.1 Introduction
131(1)
4.2 The Laplacian Matrix
132(6)
4.3 The Group Inverse
138(3)
4.4 L# and the Bottleneck Matrix
141(3)
4.5 L# for Weighted Trees
144(4)
4.6 Algebraic Connectivity
148(4)
4.7 Joins
152(2)
4.8 Resistance Distance
154(6)
4.9 Computational Considerations
160(8)
4.10 Closing Remarks
168(3)
Bibliography
171(2)
5 Boundary Value Problems on Finite Networks
173
Angeles Carmona
5.1 Introduction
173(1)
5.2 The M-Matrix Inverse Problem
174(2)
5.3 Difference Operators on Networks
176(8)
5.3.1 Schrodinger Operators
181(3)
5.4 Glossary
184(1)
5.5 Networks with Boundaries
185(3)
5.6 Self-Adjoint Boundary Value Problems
188(5)
5.7 Monotonicity and the Minimum Principle
193(2)
5.8 Green and Poisson Kernels
195(4)
5.9 The Dirichlet-to-Robin Map
199(2)
5.10 Characterization of Symmetric M-Matrices as Resistive Inverses
201(6)
5.10.1 The Kirchhoff Index and Effective Resistances
202(2)
5.10.2 Characterization
204(3)
5.11 Distance-regular Graphs with the M-Property
207(8)
5.11.1 Strongly Regular Graphs
209(1)
5.11.2 Distance-regular Graphs with Diameter 3
210(5)
Bibliography
215
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 Stevanovi 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.

Įngeles Carmona is an Associate Professor at the Universitat Politčcnica de Catalunya in Barcelona, Spain.