#### Abstract

Let F be a field, let G be an undirected graph on n vertices, and let SF(G) be the set of all F-valued symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. Let MRF(G) be defined as the set of matrices in SF(G) whose rank achieves the minimum of the ranks of matrices in SF(G). We develop techniques involving Z-hat, a process termed nil forcing, and induced subgraphs, that can determine when diagonal entries corresponding to specific vertices of G must be zero or nonzero for all matrices in MRF(G). We call these vertices nil or nonzero vertices, respectively. If a vertex is not a nil or nonzero vertex, we call it a neutral vertex. In addition, we completely classify the vertices of trees in terms of the classifications: nil, nonzero and neutral. Next we give an example of how nil vertices can help solve the inverse inertia problem. Lastly we give results about the inverse eigenvalue problem and solve a more complex variation of the problem (the λ, µ problem) for the path on 4 vertices. We also obtain a general result for the λ, µ problem concerning the number of λ’s and µ’s that can be equal.

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

http://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Nelson, Curtis G., "Diagonal Entry Restrictions in Minimum Rank Matrices, and the Inverse Inertia and Eigenvalue Problems for Graphs" (2012). *Theses and Dissertations*. 3246.

https://scholarsarchive.byu.edu/etd/3246

#### Date Submitted

2012-06-11

#### Document Type

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd5303

#### Keywords

Combinatorial Matrix Theory, Diagonal Entry Restrictions, Graph, Inverse Eigenvalue Problem, Inverse Inertia Problem, Minimum Rank, Neutral, Nil, Nonzero, Symmetric

#### Language

English