Spanning Trees, Effective Resistances and Curvature on Graphs

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as the maximal minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees form a regular matroid. In this talk, I will discuss some consequences of this perspective for the study of a related quantity from electrical circuit theory: the effective resistance. I will give a new characterization of effective resistances in terms of a certain polytope and discuss applications to recent work on discrete notions of curvature based on the effective resistance.