Combinatorial matrix theory and spectral analysis

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My research primarily explores the interaction of two areas of mathematics, namely algebra and graph theory. The particular focus is known as combinatorial matrix theory. A significant part of my research involves exploring how graph and digraph structures (e.g. network style diagrams) can give insight into algebraic problems, especially insights involving eigenvalues of matrices. Eigenvalues are key to understanding the long term behaviour of systems modeled with the matrix.  The study of sign pattern matrices has some of its origins in the work of Nobel Laureate P. Samuelson in economics, but has growing interest in other modeling contexts. Recently pattern analysis has been used to detect the possibility of periodicity in biological and ecological systems, and also applied in the context of machine learning. I do not plan to focus directly on an application, but instead the focus of my ongoing research program is on developing the theory that may prove useful in future development of modeling and algorithmic applications. A long term goal is characterizing classes of patterns based on eigenvalue characteristics as above, but also to continue to develop necessary conditions for a pattern to allow the needed properties, focusing especially on digraph conditions. A particularly difficult classification is the determination of when a pattern is potentially stable (allowing only eigenvalues with negative real part)  or spectrally arbitrary (putting no restrictions on the eigenvalues). Other classes of patterns are of interest as they allow for bifurcations in dynamical systems. Developing our understanding of these classes will provide a deeper understanding of how matrix structure affects eigenvalues of a matrix. As an example, further development can provide tools to understand if an equilibrium point of a system is stable or if it is unstable and hence susceptible to small perturbations. Combinatorial matrix structures can also give insight into one of the most fundamental tasks of mathematics, finding the solutions of equations. A particularly curious fact, discovered in the early 19th century, is that relatively simple polynomial equations, of degree five or higher, do not allow for algebraic solutions. Consequently, numerical methods have been developed to find roots of polynomial equations. One method finds the eigenvalues of a Frobenius companion matrix, used for example in the roots command in MATLAB software. My recent research in graph theory has provided some insight into various new classes of companion matrix forms. I plan to continue to explore how these structures can provide new bounds for roots of polynomials and provide structures that may make algorithms more efficient at determining roots of polynomials because of improved condition numbers on the corresponding matrices.