Georgia Institute of Technology College of Sciences

Kirchhoff's celebrated matrix tree theorem expresses the number of spanning trees of a graph as a minor of the Laplacian matrix of the graph. In modern language, this determinantal counting formula reflects the fact that spanning trees in a graph form a regular matroid. In this talk, I will give a short historical overview of the tree-counting problem and a related quantity from electrical circuit theory: the effective resistance. I will describe a characterization of effective resistances in terms of a certain polytope and discuss a recent application to discrete notions of curvature on graphs. The talk is based on the article: https://arxiv.org/abs/2410.07756