Graph geometry from effective resistances

The effective resistance is a concept that originated in the analysis of electrical circuits, where it characterizes how easily an electrical current can flow between two points in the circuit. More abstractly, the effective resistance can be defined between pairs of vertices in a weighted graph. In the graph-theoretic context, the effective resistance has many interpretations: it is a metric between the vertices of a graph and it is related to random spanning trees and random walks.

In this talk, I want to highlight two geometric aspects of effective resistances. First, I will discuss a result by Fiedler that describes a bijection between weighted graphs and hyperacute simplices (simplices with certain angular constraints). Second, I will describe some recent work together with Renaud Lambiotte on discrete curvature based on the effective resistance.