Campus Units
Mathematics
Document Type
Book Chapter
Publication Version
Accepted Manuscript
Publication Date
2020
Journal or Book Title
50 Years of Combinatorics, Graph Theory, and Computing
First Page
239
Last Page
262
Abstract
Historically, matrix theory and combinatorics have enjoyed a powerful, mutually beneficial relationship. Examples include:
- Perron-Frobenius theory describes the relationship between the combinatorial arrangement of the entries of a nonnegative matrix and the properties of its eigenvalues and eigenvectors (see [53, Chapter 8]).
- The theory of vibrations (e.g., of a system of masses connected by strings) provides many inverse problems (e.g., can the stiffness of the springs be prescribed to achieve a system with a given set of fundamental vibrations?) whose resolution intimately depends upon the families of matrices with a common graph (see [46, Chapter 7]).
The Inverse Eigenvalue Problem of a graph (IEP-G), which is the focus of this chapter, is another such example of this relationship. The IEP-G is rooted in the 1960s work of Gantmacher, Krein, Parter and Fielder, but new concepts and techniques introduced in the last decade have advanced the subject significantly and catalyzed several mathematically rich lines of inquiry and application. We hope that this chapter will highlight these new ideas, while serving as a tutorial for those desiring to contribute to this expanding area.
Copyright Owner
Taylor & Francis Group, LLC
Copyright Date
2020
Language
en
File Format
application/pdf
Recommended Citation
Hogben, Leslie; Lin, Jephian C.-H.; and Shader, Bryan L., "The Inverse Eigenvalue Problem of a Graph" (2020). Mathematics Publications. 227.
https://lib.dr.iastate.edu/math_pubs/227
Comments
This is a manuscript of a chapter published as Hogben, Leslie, Jephian C.-H. Lin, and Bryan L. Shader. “The Inverse Eigenvalue Problem of a Graph.” In 50 years of Combinatorics, Graph Theory, and Computing (Fan Chung, Ron Graham, Frederick Hoffman, Leslie Hogben, Ronald C. Mullin, and Douglas B. West, eds.). Boca Raton, FL: CRC Press, Taylor & Francis Group (2019): 239-262. Posted with permission.