Grant: Discovery Grants Program - Individual

My research interests are centered around `Positivity' in Linear Algebra and Graph Theory. Along these lines, I have proposed various research projects involving totally positive matrices and on properties of certain matrices associated to graphs. The former is a well studied and fundamental class of matrices that arises in numerous applications, including statistics, mathematical biology, and computer aided geometric design. I intend to investigate a number of key problems associated with this class by making use of the combinatorial framework associated with certain matrix factorizations. The main issue of interest to me concerning matrices associated with graphs comes from a classical problem known as an inverse eigenvalue problem (that is, the eigenvalues are provided and the corresponding matrix is desired). An important part of this inverse eigenvalue problem is the minimum rank problem. The connection between the two follows since rank and nullity are intimately related. The hope here is to appeal to the algebraic properties of matrices and combine them with the combinatorial characteristics of graphs to yield interesting results about this special collection of matrices.