#### Title

The Minimum Number of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph is a Tree

#### Date Thesis Awarded

5-2022

#### Access Type

Honors Thesis -- Access Restricted On-Campus Only

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

#### Advisor

Charles Johnson

#### Committee Members

Chi-Kwong Li

Timothy Davis

#### Abstract

Let T be a tree, let S(T) denote the set of real symmetric matrices whose graph is T, and let U(T) be the minimum number of eigenvalues with multiplicity 1 among matrices in S(T). We show that if T ′ is a tree constructed by adding a pendent vertex to a diameter 5 tree T such that d(T) = d(T) ′ , then U(T ′ ) ≤ U(T). We then take this result as motivation to show results that are consistent with and suggest that if T ′ is a tree constructed by adding a pendent vertex to a tree T such that d(T) = d(T) ′ , then U(T ′ ) ≤ U(T). In particular, we show that the result is true for the case where T and T ′ are both linear trees, the case where T is a linear tree and T ′ is a nonlinear tree, and the case where T and T ′ are both minimally nonlinear trees such that U(T) = 2. We then construct a partial proof for the same and similar results where T and T ′ are general trees.

#### Recommended Citation

Zimmerman, Jacob, "The Minimum Number of Multiplicity 1 Eigenvalues Among Real Symmetric Matrices Whose Graph is a Tree" (2022). *Undergraduate Honors Theses.* William & Mary. Paper 1804.

https://scholarworks.wm.edu/honorstheses/1804

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.