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Full Description
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.
Contents
Introduction to the inverse eigenvalue problem of a graph and zero forcing: Introduction to an motivation for the IEP-$G$
Zero forcing and maximum eigenvalue multiplicity
Strong properties, theory, and consequences: Implicit function theorem and strong properties
Consequences of the strong properties
Theoretical underpinnings of the strong properties
Further discussion of ancillary problems: Ordered multiplicity lists of a graph
Rigid linkages
Minimum number of district eigenvalues
Zero forcing, propagation time, and throttling: Zero forcing, variants, and related parameters
Propagation time and capture time
Throttling
Appendix A. Graph terminology and notation
Bibliography
Index
