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Below are some highlighted research topics and interests.
Spanning trees, matroids and their applications
This figure shows a graph and its eight spanning trees, which are central to the study of Laplacian matrices and effective resistances. One way to generalize spanning trees is through matroid theory; in this preprint, we study the class of uniformly dense matroids and here we study tropical affine spaces using matroid theory.
In this paper we define a new class of graphs: a graph is called resistance nonnegative (RN) if there exists a distribution on its spanning trees, such that every vertex has expected degree at most 2 in a random spanning tree. Check out the paper for open questions on RN graphs!
Discrete curvature
In this paper, 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 paper and this paper, use discrete curvature for clusting in this paper and here we explore graphs with given curvatures.
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 between 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.Â
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.
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