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El. knyga: Inverse Problems and Zero Forcing for Graphs

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This book provides an introduction to the inverse eigenvalue problem for graphs (IEP-$G$) and the related area of zero forcing, propagation, and throttling. The IEP-$G$ grew from the intersection of linear algebra and combinatorics and has given rise to both a rich set of deep problems in that area as well as a breadth of ""ancillary'' problems in related areas.

The IEP-$G$ asks a fundamental mathematical question expressed in terms of linear algebra and graph theory, but the significance of such questions goes beyond these two areas, as particular instances of the IEP-$G$ also appear as major research problems in other fields of mathematics, sciences and engineering. One approach to the IEP-$G$ is through rank minimization, a relevant problem in itself and with a large number of applications. During the past 10 years, important developments on the rank minimization problem, particularly in relation to zero forcing, have led to significant advances in the IEP-$G$.

The monograph serves as an entry point and valuable resource that will stimulate future developments in this active and mathematically diverse research area.
Preface ix
Part 1 Introduction to the Inverse Eigenvalue Problem of a Graph and Zero Forcing
1(38)
Chapter 1 Introduction to and Motivation for the IEP-G
3(14)
1.1 Forward Problems
3(1)
1.2 Inverse Problems
4(1)
1.3 Matrices, Forward Problems, and Structure
4(4)
1.4 Inverse Eigenvalue Problems and Matrices with a Given Graph
8(3)
1.5 Initial Results for the IEP-G
11(2)
1.6 Connections of the IEP-G to Nodal Domains
13(4)
Chapter 2 Zero Forcing and Maximum Eigenvalue Multiplicity
17(22)
2.1 Introduction to Maximum Nullity and Minimum Rank
17(2)
2.2 Introduction to Zero Forcing
19(3)
2.3 Historical Background to the Minimum Rank Problem
22(4)
2.4 Further Properties of Minimum Rank
26(4)
2.5 Minimum Positive Semidefinite Rank and Zero Forcing Number
30(2)
2.6 Colin de Verdiere Type Parameters
32(4)
2.7 The δ-Conjecture and the Graph Complement Conjecture
36(3)
Part 2 Strong Properties, Theory, and Consequences
39(56)
Chapter 3 Implicit Function Theorem and Strong Properties
41(12)
3.1 Implicit Function Theorem
41(4)
3.2 Strong Properties
45(4)
3.3 Strong Properties for Sign Patterns
49(4)
Chapter 4 Consequences of the Strong Properties
53(22)
4.1 Number of Distinct Eigenvalues
53(3)
4.2 Augmentation Lemma
56(1)
4.3 The IEP-G for Small Graphs
57(4)
4.4 Verification Matrices
61(3)
4.5 Matrix Liberation Lemma
64(3)
4.6 Minor Monotonicity
67(3)
4.7 Guaranteed Strong Properties
70(5)
Chapter 5 Theoretical Underpinnings of the Strong Properties
75(20)
5.1 Spaces of Matrices
75(1)
5.2 Manifolds and Tangent Spaces
76(7)
5.3 Implicit Function Theorem Revisited
83(3)
5.4 Strong Properties and Supergraph Lemma Revisited
86(3)
5.5 Bifurcation Lemma
89(1)
5.6 Tangent Space Matrix
90(1)
5.7 Matrix Liberation Lemma Revisited
91(1)
5.8 Future Work
92(3)
Part 3 Further Discussion of Ancillary Problems
95(52)
Chapter 6 Ordered Multiplicity Lists of a Graph
97(18)
6.1 Multiplicity Lists for Special Families of Graphs
97(5)
6.2 Constructive Techniques
102(6)
6.3 Related Graph Parameters
108(1)
6.4 Path Removal and Multiplicities for Trees
109(6)
Chapter 7 Rigid Linkages
115(8)
7.1 Rigid Linkages
115(4)
7.2 Rigid Shortest Linkages
119(4)
Chapter 8 Minimum Number of Distinct Eigenvalues
123(24)
8.1 Number of Distinct Eigenvalues for Adjacency Matrices
123(1)
8.2 Basic Results
124(8)
8.3 Strong Properties and q
132(2)
8.4 Joins and Graphs with Small q
134(4)
8.5 A Nordhaus-Gaddum Conjecture for q
138(2)
8.6 Minimum Number of Distinct Eigenvalues for Trees
140(7)
Part 4 Zero Forcing, Propagation Time, and Throttling
147(114)
Chapter 9 Zero Forcing, Variants, and Related Parameters
151(52)
9.1 Standard Zero Forcing, Z(G)
151(9)
9.2 Universal Definitions for Zero Forcing
160(1)
9.3 Positive Semidefinite Zero Forcing, Z+(G)
160(7)
9.4 Skew Forcing, Z_(G)
167(12)
9.5 Connected and Total Zero Forcing
179(2)
9.6 Additional Zero Forcing Parameters Related to the IEP-G
181(2)
9.7 Rigid Linkage Zero Forcing
183(4)
9.8 Minor Monotone Floors of Zero Forcing Parameters
187(4)
9.9 Power Domination, 7p(G)
191(5)
9.10 Cops and Robbers, c(G)
196(4)
9.11 Average Values and Random Graphs
200(1)
9.12 Topics Not Covered
201(2)
Chapter 10 Propagation Time and Capture Time
203(28)
10.1 Universal Definitions for Forcing Propagation Time
203(3)
10.2 Z-Propagation Time
206(6)
10.3 Z+-Propagation Time
212(4)
10.4 Z_-Propagation Time
216(6)
10.5 Propagation Time for Power Domination
222(3)
10.6 Capture Time for Cops and Robbers
225(2)
10.7 Expected Propagation Time for Probabilistic Zero Forcing
227(2)
10.8 Topics Not Covered
229(2)
Chapter 11 Throttling
231(30)
11.1 Universal Definitions for Forcing Throttling
231(4)
11.2 Z-throttling
235(5)
11.3 Z+-throttling
240(4)
11.4 Z_-throttling
244(4)
11.5 Throttling for Power Domination
248(3)
11.6 Throttling for Cops and Robbers
251(4)
11.7 Product Throttling
255(5)
11.8 Topics Not Covered
260(1)
Appendix A Graph Terminology and Notation 261(8)
Bibliography 269(12)
Index 281
Leslie Hogben, Iowa State University, Ames, IA, and American Institute of Mathematics, San Jose, CA.

Jephian C.-H. Lin, National Sun Yat-sen University, Kaohsiung, Taiwan.

Bryan L. Shader, University of Wyoming, Laramie, WY.
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