Inverse Problems and Zero Forcing for Graphs

302 Pages·2022·7.556 MB·English

by  Leslie Hogben, Jephian C.-H. Lin, Bryan L. Shader

Mathematical
Surveys
and
Monographs
Volume 270
Inverse Problems
and Zero Forcing
for Graphs
Leslie Hogben
Jephian C.-H. Lin
Bryan L. Shader
Inverse Problems
and Zero Forcing
for Graphs

Mathematical
Surveys
and
Monographs
Volume 270
Inverse Problems
and Zero Forcing
for Graphs
Leslie Hogben
Jephian C.-H. Lin
Bryan L. Shader
EDITORIAL COMMITTEE
Ana Caraiani Natasa Sesum
Michael A. Hill Constantin Teleman
Bryna Kra (chair) Anna-Karin Tornberg
2020 Mathematics Subject Classification. Primary 05C50, 15-02, 05-02, 15A29, 15A18,
05C69, 05C22, 15B57.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-270
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DOI:https://doi.org/10.1090/surv/270
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Contents
Preface ix
Part 1. Introduction to the Inverse Eigenvalue Problem of a Graph
and Zero Forcing 1
Chapter 1. Introduction to and Motivation for the IEP-G 3
1.1. Forward Problems 3
1.2. Inverse Problems 4
1.3. Matrices, Forward Problems, and Structure 4
1.4. Inverse Eigenvalue Problems and Matrices with a Given Graph 8
1.5. Initial Results for the IEP-G 11
1.6. Connections of the IEP-G to Nodal Domains 13
Chapter 2. Zero Forcing and Maximum Eigenvalue Multiplicity 17
2.1. Introduction to Maximum Nullity and Minimum Rank 17
2.2. Introduction to Zero Forcing 19
2.3. Historical Background to the Minimum Rank Problem 22
2.4. Further Properties of Minimum Rank 26
2.5. Minimum Positive Semidefinite Rank and Zero Forcing Number 30
2.6. Colin de Verdi`ere Type Parameters 32
2.7. The δ-Conjecture and the Graph Complement Conjecture 36
Part 2. Strong Properties, Theory, and Consequences 39
Chapter 3. Implicit Function Theorem and Strong Properties 41
3.1. Implicit Function Theorem 41
3.2. Strong Properties 45
3.3. Strong Properties for Sign Patterns 49
Chapter 4. Consequences of the Strong Properties 53
4.1. Number of Distinct Eigenvalues 53
4.2. Augmentation Lemma 56
4.3. The IEP-G for Small Graphs 57
4.4. Verification Matrices 61
4.5. Matrix Liberation Lemma 64
4.6. Minor Monotonicity 67
4.7. Guaranteed Strong Properties 70
Chapter 5. Theoretical Underpinnings of the Strong Properties 75
5.1. Spaces of Matrices 75
5.2. Manifolds and Tangent Spaces 76
v
vi CONTENTS
5.3. Implicit Function Theorem Revisited 83
5.4. Strong Properties and Supergraph Lemma Revisited 86
5.5. Bifurcation Lemma 89
5.6. Tangent Space Matrix 90
5.7. Matrix Liberation Lemma Revisited 91
5.8. Future Work 92
Part 3. Further Discussion of Ancillary Problems 95
Chapter 6. Ordered Multiplicity Lists of a Graph 97
6.1. Multiplicity Lists for Special Families of Graphs 97
6.2. Constructive Techniques 102
6.3. Related Graph Parameters 108
6.4. Path Removal and Multiplicities for Trees 109
Chapter 7. Rigid Linkages 115
7.1. Rigid Linkages 115
7.2. Rigid Shortest Linkages 119
Chapter 8. Minimum Number of Distinct Eigenvalues 123
8.1. Number of Distinct Eigenvalues for Adjacency Matrices 123
8.2. Basic Results 124
8.3. Strong Properties and q 132
8.4. Joins and Graphs with Small q 134
8.5. A Nordhaus-Gaddum Conjecture for q 138
8.6. Minimum Number of Distinct Eigenvalues for Trees 140
Part 4. Zero Forcing, Propagation Time, and Throttling 147
Chapter 9. Zero Forcing, Variants, and Related Parameters 151
9.1. Standard Zero Forcing, Z(G) 151
9.2. Universal Definitions for Zero Forcing 160
9.3. Positive Semidefinite Zero Forcing, Z (G) 160
+
9.4. Skew Forcing, Z (G) 167
−
9.5. Connected and Total Zero Forcing 179
9.6. Additional Zero Forcing Parameters Related to the IEP-G 181
9.7. Rigid Linkage Zero Forcing 183
9.8. Minor Monotone Floors of Zero Forcing Parameters 187
9.9. Power Domination, γ (G) 191
P
9.10. Cops and Robbers, c(G) 196
9.11. Average Values and Random Graphs 200
9.12. Topics Not Covered 201
Chapter 10. Propagation Time and Capture Time 203
10.1. Universal Definitions for Forcing Propagation Time 203
10.2. Z-Propagation Time 206
10.3. Z -Propagation Time 212
+
10.4. Z -Propagation Time 216
−
10.5. Propagation Time for Power Domination 222
10.6. Capture Time for Cops and Robbers 225
CONTENTS vii
10.7. Expected Propagation Time for Probabilistic Zero Forcing 227
10.8. Topics Not Covered 229
Chapter 11. Throttling 231
11.1. Universal Definitions for Forcing Throttling 231
11.2. Z-throttling 235
11.3. Z -throttling 240
+
11.4. Z -throttling 244
−
11.5. Throttling for Power Domination 248
11.6. Throttling for Cops and Robbers 251
11.7. Product Throttling 255
11.8. Topics Not Covered 260
Appendix A. Graph Terminology and Notation 261
Bibliography 269
Index 281

Preface
The primary driver for writing this book is the desire to provide an on-ramp
to the new, developing, and mathematically diverse research related to the inverse
eigenvalue problem for graphs (IEP-G) and the related area of zero forcing, prop-
agation, and throttling. Inverse problems play a central role in mathematics and
naturally arise in applications. In many instances the inverse problem reduces to a
question about the existence of a matrix with a prescribed set of eigenvalues and
prescribed structure. The IEP-G studies such questions.
Due to the lack of effective tools for the IEP-G, much early research focused
on ancillary problems (subquestions) that could lead to progress on the IEP-G.
The most important of these is the study of the maximum possible multiplicity of
an eigenvalue among matrices described by G, or equivalently maximum possible
nullity or minimum possible rank. Since the birth of the minimum rank problem
in a 1997 paper by Peter Nylen, the IEP-G and related questions have proven to
be intriguing, but difficult problems. The 2006 American Institute of Mathematics
Workshop ‘Spectra of families of matrices described by graphs, digraphs, and sign
patterns’ and the small research group that grew out of it, ‘Minimum rank of
symmetric matrices described by a graph,’ catalyzed two new research areas, which
we call Strong Properties of Matrices and Zero-Forcing of Graphs, related to the
IEP-G. Since then there has been a rapid, expansive development of the areas,
which has resulted in many deep results for the IEP-G. Because of the growth of
these emerging areas, we believe that there is a need for a book like this which
provides the essential concepts, techniques and results in a unified way, and which
suggests topics for future research.
A pleasing aspect of these new topics is their interplay with other areas of
mathematics and applications. The topic of Strong Properties of Matrices is closely
related to the Implicit Function Theorem and the transversal intersection of mani-
folds. It provides algebraic conditions on a matrix with a certain spectral property
and graph that guarantee the existence of a matrix with the same spectral property
for each graph in a related family of graphs. Additionally, strong properties provide
interesting combinatorial matrix theoretic and graph minor problems.
Zero forcing originated independently in both the IEP-G setting and in the
study of control of quantum systems (where it is often referred to as graph infec-
tion). Zero forcing is a game on a graph in which vertices are initially filled or
unfilled, and at each stage certain vertices change from unfilled to filled accord-
ing to some rule. The goal of the game is to fill all vertices. Zero forcing gives a
graph-theoretic method to bound the multiplicity of an eigenvalue of a symmetric
matrix with a given zero-nonzero structure. Zero forcing is closely related to the
power domination process that provides a model for placing monitoring units in
ix