Research Interests
Below are some highlighted research topics and interests.
Spanning trees, matchings and matroids
This figure shows a graph and its eight spanning trees. These spanning trees are central to the study of Laplacians and effective resistances and their study is generalized in matroid theory. In this preprint, we study the class of uniformly dense matroids.
Discrete curvature
This is a figure from this paper, where we propose and study a new notion of discrete curvature on graphs based on the effective resistance. We prove some results on graph with positive curvature in this preprint.
Simplex geometry
Miroslav Fiedler gave a bijection between {weighted graphs on n vertices} and {simplices with n vertices and nonobtuse angles}, where the effective resistances in the graph are squared edge lengths of the simplex. The figure on the left shows this correspondence for a tetrahedron and two of its faces, and a graph and two of its Kron reductions.
Simplex geometry plays a central role in my thesis and this paper.
Applications
The figure right is from this paper, where we study measures of polarization in online networks and conversations.
Network dynamics and nonlinearity
The figure left shows the bifurcation diagram of a nonlinear dynamical system associated to a barbell graph. In this paper and this paper we study stationary points and their linear stability using tools from spectral graph theory, and find that stability is controlled by ... the effective resistance.