Effective resistance in graphs: geometry and combinatorics

The effective resistance is a function that measures the overall connectivity between pairs of vertices in a graph. While it originally served as a practical tool for electrical circuit design, the effective resistance has some striking mathematical features. In this talk, I will give a brief overview of two different perspectives on the effective resistance: a geometric perspective, centered around Fiedler's graph-simplex correspondence, and a combinatorial perspective, centered around Foster's theorem and Kirchhoff's matrix-tree theorem. In both cases, I will discuss a generalization of these classical results in a more modern language. Finally, I will briefly touch on some recent work on discrete curvature that sits at the intersection of these geometric and combinatorial theories.

Background material:

• Biggs, 1997: Algebraic potential theory on graphs, in Bulletin of LMS
• Fiedler, 2011: Matrices and graphs in geometry, Cambridge University Press
• Lyons, 2014: Determinantal probability: Basic properties and conjectures, in Proc. ICM
• Devriendt, 2022: Graph geometry from effective resistances, Oxford PhD thesis
• Devriendt, 2025: Graphs with nonnegative resistance curvature, in Annals of Combinatorics