#### 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.