Research Statement

My current research experience is in areas dealing with graph theory, linear algebra and connections between these two fields. Below I will describe three types of general problems that I enjoy working on, however, I am open to working on various types of interesting problems in mathematics.

Inverse Eigenvalue Problems

An inverse eigenvalue problem deals with the construction of a matrix with prescribed eigenvalues. Sometimes in practice, only the signs of the matrices are important. This leads to the idea of sign pattern matrices, a matrix whose entries consist of the symbols {+,-,0}. One particular inverse eigenvalue problem is to find a real matrix (if one exists) with a certain sign pattern that has a prescribed set of eigenvalues. Determining if patterns are spectrally arbitrary or inertially arbitrary is one of my favorite areas of research, and one I am very familiar with.

Inverse eigenvalue problems have many applications. They appear in control theory, system theory, mathematical biology, geophysics and so forth. Properties of sign patterns provide researchers in these fields details about a system based on the signs of the entries in a matrix. In some mathematical models, the stability of an equilibrium point can be decided by looking at a sign pattern matrix.

Partitions and Coverings of Graphs

During my Masters research I worked on a problem dealing with partitions and coverings of graphs. The problem is to partition (resp. cover) the edge set of a graph G into (resp. by) complete graphs, however, similar problems use other types of subgraphs to partition (resp. cover) the edges of G. The goal is to minimize the number of subgraphs being used in a partition or covering.

The techniques that come into play are very interesting to me and include:
  • extremal set theory: intersecting families, Sperner families, the de Bruijn-Erdős Theorem, the Erdős-Ko-Rado Theorem,
  • Steiner systems, designs, latin squares, projective planes,
  • linear programming, duality theory, integer programming,
  • the probabilistic method,
  • recurrence relations, induction and double counting.

Energy of Graphs and the General Randic Index

During the course of my doctorate, I researched a topic that falls under the umbrella of spectral graph theory. In recent years there has been a great deal of interest in this topic as it has applications in chemistry, theoretical physics, quantum mechanics and numerous other fields of study. Spectral graph theory deals with the question of how eigenvalues of adjacency matrices (among other matrix representations) relate to the structure of graphs.

The concept of graph energy was defined by Ivan Gutman in 1978 and originates from theoretical chemistry. To determine the energy of a graph, we essentially add up the eigenvalues (in absolute value) of the graph. The general Randic index of a graph is also a concept originating from chemisty and is a function of the vertex degrees of a graph. In the past ten years, there have been more than 150 papers published on graph energy, and it is a highly researched topic by pure mathematicians and theoretical chemists alike.

Some basic techniques I have used in this research area include the Cauchy-Schwarz inequality, singular value decompositions, interlacing of eigenvalues, and equitable partitions.