My research interests include stochastic graph theory and network modeling, applications of probabilistic methods to combinatorial problems, extremal combinatorics, graph coloring, and random matrices. A list of publications can be found on my CV.
Among other things, I am currently interested in nonlinear spectra of graphs. Specifically, one can define a constant similar to the first eigenvalue of a graph by embedding the graph into any metric space. The standard first eigenvalue is obtained if the desired metric space is the real numbers. I am particularly interested in the properties of this constant when the metric space involved is another graph. This is a fairly new direction in spectral graph theory, and relatively little is known about how properties of the metric space interact with properties of the graph itself. It can be shown that this constant has a relationship with a version of the k-fold Cheeger constant, which is an intriguing avenue of research that I am exploring.
In a related project, I am working to develop Cheeger inequalities for higher order Cheeger constants. There are many different definitions of a k-fold Cheeger constant, and choosing the “right” definition seems to be among the important questions in front of us. Much work has been done in clustering of graphs, and I have been working to contribute to the literature here with the development of a k-fold Cheeger inequality (together with F. Kenter). I am hoping, over time, to develop tighter bounds for such a quantity, and perhaps connect this to the constants mentioned above.
In addition, I am working with various properties of distance-based random graphs. By this I mean any random graph whose vertices are members of a metric space, and the probability that two vertices are connected is determined as a function of the distance between them. This is related to the Waxman model. Despite widespread use in biological and electrical engineering related problems, the Waxman model has been widely overlooked by the mathematical community. There is little mathematical research available regarding the structure of such graphs, or important graph properties, and much to be gained by this theory for those in the applied communities that prefer such models.
Beyond these primary projects, I have several other projects pertaining to connectivity structure in graphs, clustering coefficient, diameter of randomly perturbed graphs, and other similar ideas. I have been working with multiple algorithms that seek to improve graph invariants by modification with edge-switching, although most of these projects are in their infancy.
Last updated: 25 August 2015.